Let \(\left( H;\left\langle \cdot ,\cdot \right\rangle \right)\) be a complex Hilbert space. Denote by \(\mathcal{B}\left( H\right)\) the Banach \(C^{\ast }\) -algebra of bounded linear operators on \(H\). For \(A\in \mathcal{B}\left( H\right)\) we define the modulus of \(A\) by \(\left\vert A\right\vert :=\left( A^{\ast }A\right) ^{1/2}.\) We say that the continuous function \(B:\left[ a,b \right] \rightarrow \mathcal{B}\left( H\right)\) is square modulus convex (concave) on \(\left[ a,b\right]\) if \[\begin{equation*} \left\vert B\left( \left( 1-t\right) u+tv\right) \right\vert ^{2}\leq \left( \geq \right) \left( 1-t\right) \left\vert B\left( u\right) \right\vert ^{2}+t\left\vert B\left( v\right) \right\vert ^{2}, \end{equation*}\] in the operator order of \(\mathcal{B}\left( H\right) ,\) for all \(u,\) \(v\in \left[ a,b\right]\) and \(t\in \left[ 0,1\right] .\) In this paper, we show among others that, if \(B:\left[ m,M\right] \subset \mathbb{R\rightarrow } \mathcal{B}\left( H\right)\) is square modulus convex on \(\left[ m,M\right]\) and \(f:\Omega \rightarrow \left[ m,M\right]\) so that \(f,\) \(\left\vert B\circ f\right\vert ^{2},\) \(Re\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) ,\) \(fRe\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) \in L_{w}\left( \Omega ,\mu ,\mathcal{B}\left( H\right) \right) ,\) where \(w\geq 0\) \(\mu\) -a.e. on \(\Omega\) with \(\int_{\Omega }wd\mu =1,\) then \[0 \leq \int_{\Omega }w\left( s\right) \left\vert B\circ f\right\vert ^{2}d\mu \left( s\right) -\left\vert B\left( \int_{\Omega }wfd\mu \right) \right\vert ^{2}\] \[\qquad\qquad\qquad~\qquad\qquad \leq \frac{1}{2}\left( M-m\right) \left\Vert \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) – Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right\Vert .\] The discrete versions are also provided.
Denote by \(\mathcal{B}\left( H\right)\) the Banach \(C^{\ast }\)-algebra of bounded linear operators on Hilbert space \(H\). For \(A\in \mathcal{B}\left( H\right)\) we define the modulus of \(A\) by \(\left\vert A\right\vert :=\left( A^{\ast }A\right) ^{1/2}.\) It is well known that the modulus of operators does not satisfy, in general, the triangle inequality \(\left\vert A+B\right\vert \leq \left\vert A\right\vert +\left\vert B\right\vert ,\) so the classical arguments using this inequality can not be used.
The following Cauchy-Bunyakowsky-Schwarz discrete inequality for the operator modulus holds: \[\begin{equation} \sum_{k=1}^{n}w_{k}\left\vert z_{k}\right\vert ^{2}\sum_{k=1}^{n}w_{k}\left\vert A_{k}\right\vert ^{2}\geq \left\vert \sum_{k=1}^{n}w_{k}z_{k}A_{k}\right\vert ^{2}, \label{CBSd} \end{equation} \tag{1}\] where \(z_{k}\in \mathbb{C}\), \(A_{k}\in \mathcal{B}\left( H\right)\) and \(w_{k}\geq 0\) for \(k\in \left\{ 1,…,n\right\} .\) A more general version of this inequality with a complete proof is given in Lemma 1 below.
We can introduce the following concept of operator convexity:
Definition 1. We say that the continuous function \(B:\left[ a,b\right] \rightarrow \mathcal{B}\left( H\right)\) is square modulus convex (concave) on \(\left[ a,b\right]\) if \[\begin{equation} \left\vert B\left( \left( 1-t\right) u+tv\right) \right\vert ^{2}\leq \left( \geq \right) \left( 1-t\right) \left\vert B\left( u\right) \right\vert ^{2}+t\left\vert B\left( v\right) \right\vert ^{2} , \label{e.1} \end{equation} \tag{2}\] in the operator order of \(\mathcal{B}\left( H\right) ,\) for all \(u,\) \(v\in \left[ a,b\right]\) and \(t\in \left[ 0,1\right] .\)
Let \(A,\) \(B\in \mathcal{B}\left( H\right)\) and \(\alpha \in \left[ 0,1\right] .\) Then by (1), we get \[\begin{align*} \left\vert \left( 1-\alpha \right) A+\alpha B\right\vert ^{2}& =\left\vert \left( 1-\alpha \right) ^{1/2}\left( 1-\alpha \right) ^{1/2}A+\alpha ^{1/2}\alpha ^{1/2}B\right\vert ^{2} \\ & \leq \left[ \left( \left( 1-\alpha \right) ^{1/2}\right) ^{2}+\left( \alpha ^{1/2}\right) ^{2}\right] \left[ \left\vert \left( 1-\alpha \right) ^{1/2}A\right\vert ^{2}+\left\vert \alpha ^{1/2}B\right\vert ^{2}\right] \\ & =\left( 1-\alpha +\alpha \right) \left[ \left( 1-\alpha \right) \left\vert A\right\vert ^{2}+\alpha \left\vert B\right\vert ^{2}\right] \\ & =\left( 1-\alpha \right) \left\vert A\right\vert ^{2}+\alpha \left\vert B\right\vert ^{2}. \end{align*}\]
Consider the function \(C:\left[ 0,1\right] \rightarrow \mathcal{B}\left( H\right) ,\) \(C\left( t\right) =\left\vert \left( 1-t\right) A+tB\right\vert .\) Let \(t_{1},\) \(t_{2}\in \left[ 0,1\right]\) and \(\alpha \in \left[ 0,1 \right] .\) Then \[\begin{align*} \left\vert C\left( \left( 1-\alpha \right) t_{1}+\alpha t_{2}\right) \right\vert ^{2}& =\left\vert \left( 1-\left( 1-\alpha \right) t_{1}-\alpha t_{2}\right) A+\left( \left( 1-\alpha \right) t_{1}+\alpha t_{2}\right) B\right\vert ^{2} \\ & =\left\vert \left( 1-\alpha \right) \left( \left( 1-t_{1}\right) A+t_{1}B\right) +\alpha \left( \left( 1-t_{2}\right) A+t_{2}B\right) \right\vert ^{2} \\ & \leq \left( 1-\alpha \right) \left\vert \left( 1-t_{1}\right) A+t_{1}B\right\vert ^{2}+\alpha \left\vert \left( 1-t_{2}\right) A+t_{2}B\right\vert ^{2} \\ & =\left( 1-\alpha \right) \left\vert C\left( t_{1}\right) \right\vert ^{2}+\alpha \left\vert C\left( t_{2}\right) \right\vert ^{2}, \end{align*}\] which shows that \(C\) is square modulus convex on \(\left[ 0,1\right] .\)
Assume that \(f\) is nonnegative on \(I\) and operator convex, namely \[\begin{equation*} f\left( \left( 1-\alpha \right) A+\alpha B\right) \leq \left( 1-\alpha \right) f\left( A\right) +\alpha f\left( B\right), \end{equation*}\] for all \(\alpha \in \left[ 0,1\right]\) and selfadjoint operators \(A,\) \(B\) with spectra in \(I.\)
For such function and \(A,\) \(B,\) we consider \[\begin{equation*} D\left( t\right) :=\left[ f\left( \left( 1-t\right) A+tB\right) \right] ^{1/2},t\in \left[ 0,1\right] . \end{equation*}\]
Then, using a similar proof as above for the modulus function, we conclude that \(D\) is square modulus convex on \(\left[ 0,1\right] .\)
The function \(f\left( t\right) =t^{r}\) is operator convex on \((0,\infty )\) if either \(1\leq r\leq 2\) or \(-1\leq r\leq 0\) and is operator concave on \(\left( 0,\infty \right)\) if \(0\leq r\leq 1.\) Therefore for \(A,\) \(B>0,\) the function \[\begin{equation*} B_{r}\left( t\right) :=\left( \left( 1-t\right) A+tB\right) ^{r/2},\text{ } t\in \left[ 0,1\right], \end{equation*}\] is square modulus convex on \(\left[ 0,1\right]\) for \(1\leq r\leq 2\) or \(-1\leq r\leq 0\).
Let \(B:\left[ a,b\right] \rightarrow \mathbb{C}\) be defined by \(B\left( t\right) :=x\left( t\right) +y\left( t\right) i,\) \(t\in \left[ a,b\right] .\) Observe that \(\left\vert B\left( t\right) \right\vert ^{2}=x^{2}\left( t\right) +y^{2}\left( t\right) ,\) \(t\in \left[ a,b\right] .\) Now, if \(x^{2},\) \(y^{2}\) are convex on \(\left[ a,b\right] ,\) then obviously that \(x^{2}+y^{2}\) is convex on \(\left[ a,b\right] .\) However, if we take \(x\left( t\right) =t\sin t,\) \(y\left( t\right) =t\cos t,\) \(t\in \left[ 0,2\pi \right] ,\) then neither \(x^{2}\) nor \(y^{2}\) is convex on \(\left[ 0,2\pi \right]\) but \(x^{2}+y^{2}\) is convex on \(\left[ 0,2\pi \right] .\)
Proposition 1. If the continuous function \(B:\left[ a,b\right] \rightarrow \mathcal{B}\left( H\right)\) is square modulus concave on \(\left[ a,b\right]\) then for \(p\in \left( 0,1\right)\), \(\left\vert B\left( \cdot \right) \right\vert ^{p}\) is also square modulus concave on \(\left[ a,b\right] .\)
Proof. By the operator monotonicity and operator concavity of the function \(h\left( t\right) =t^{p}\) for \(p\in \left( 0,1\right) ,\) we have \[\begin{align*} \left\vert B\left( \left( 1-t\right) u+tv\right) \right\vert ^{2p}& \geq \left( \left( 1-t\right) \left\vert B\left( u\right) \right\vert ^{2}+t\left\vert B\left( v\right) \right\vert ^{2}\right) ^{p} \\ & \geq \left( 1-t\right) \left\vert B\left( u\right) \right\vert ^{2p}+t\left\vert B\left( v\right) \right\vert ^{2p}, \end{align*}\] for \(t\in \left[ 0,1\right] ,\) which shows that \(\left\vert B\left( \cdot \right) \right\vert ^{p}\) is also square modulus concave on \(\left[ a,b \right] .\) ◻
For \(A,\) \(B>0,\) the function \[\begin{equation*} B_{q}\left( t\right) :=\left( \left( 1-t\right) A+tB\right) ^{q/2},\text{ } t\in \left[ 0,1\right], \end{equation*}\] is square modulus concave on \(\left[ 0,1\right]\) for \(q\in \left( 0,1\right)\).
Indeed, we have for \(t_{1},\) \(t_{2}\in \left[ 0,1\right]\) and \(\alpha \in \left[ 0,1\right]\) that \[\begin{align*} \left\vert B_{q}\left( \left( 1-\alpha \right) t_{1}+\alpha t_{2}\right) \right\vert ^{2}& =\left( \left( 1-\left( 1-\alpha \right) t_{1}-\alpha t_{2}\right) A+\left( \left( 1-\alpha \right) t_{1}+\alpha t_{2}\right) B\right) ^{q} \\ & =\left[ \left( 1-\alpha \right) \left( \left( 1-t_{1}\right) A+t_{1}B\right) +\alpha \left( \left( 1-t_{2}\right) A+t_{2}B\right) \right] ^{q} \\ & \geq \left( 1-\alpha \right) \left( \left( 1-t_{1}\right) A+t_{1}B\right) ^{q}+\alpha \left( \left( 1-t_{2}\right) A+t_{2}B\right) ^{q} \\ & =\left( 1-\alpha \right) \left\vert B_{q}\left( t_{1}\right) \right\vert ^{2}+\alpha \left\vert B_{q}\left( t_{2}\right) \right\vert ^{2}, \end{align*}\] which shows that \(B_{q}\) is square modulus concave on \(\left[ 0,1 \right]\).
We have the following Hermite-Hadamard type inequalities [1]:
Theorem 1. Assume that the continuous function \(B:\left[ a,b\right] \rightarrow \mathcal{B}\left( H\right)\) is square modulus convex (concave) on \(\left[ a,b\right]\). Then for all \(u,\) \(v\in \left[ a,b\right]\) we have that \[\begin{align} \left\vert B\left( \frac{u+v}{2}\right) \right\vert ^{2}& \leq \left( \geq \right) \int_{0}^{1}\left\vert B\left( \left( 1-t\right) u+tv\right) \right\vert ^{2}dt \label{e.2.1} \\ & \leq \left( \geq \right) \frac{1}{2}\left[ \left\vert B\left( u\right) \right\vert ^{2}+\left\vert B\left( v\right) \right\vert ^{2}\right] . \notag \end{align} \tag{3}\]
For more results of this type, see [1]. Recent inequalities for operator convex functions can be also found in [2]-[7] and [8]-[11].
In this paper, we show among others that, if \(B:\left[ m,M\right] \subset \mathbb{R\rightarrow }\mathcal{B}\left( H\right)\) is square modulus convex on \(\left[ m,M\right]\) and \(f:\Omega \rightarrow \left[ m,M\right]\) so that \(f,\) \(\left\vert B\circ f\right\vert ^{2},\) \(Re\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) ,\) \(f Re\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) \in L_{w}\left( \Omega ,\mu ,\mathcal{B}\left( H\right) \right) ,\) where \(w\geq 0\) \(\mu\)-a.e. on \(\Omega\) with \(\int_{\Omega }wd\mu =1,\) then \[\begin{align*} 0 \leq& \int_{\Omega }w\left( s\right) \left\vert B\circ f\right\vert ^{2}d\mu \left( s\right) -\left\vert B\left( \int_{\Omega }wfd\mu \right) \right\vert ^{2} \\ \leq& \frac{1}{2}\left( M-m\right) \left\Vert \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) – Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right\Vert . \end{align*}\]
The discrete versions are also provided.
Following Roberts and Varberg [12, p.5], we recall that if \(f:I\) \(\rightarrow \mathbb{R}\) is a convex function, then for any \(s_{0}\in \mathring{I}\) (the interior of the interval \(I\)) the limits \[\begin{equation*} f_{-}^{\prime }\left( s_{0}\right) :=\lim_{s\rightarrow s_{0}-}\frac{f\left( s\right) -f\left( s_{0}\right) }{s-s_{0}}\text{ and }f_{+}^{\prime }\left( s_{0}\right) :=\lim_{s\rightarrow s_{0}+}\frac{f\left( s\right) -f\left( s_{0}\right) }{s-s_{0}}\text{ }, \end{equation*}\] exist and \(f_{-}^{\prime }\left( s_{0}\right) \leq f_{+}^{\prime }\left( s_{0}\right) .\) The functions \(f_{-}^{\prime }\) and \(f_{+}^{\prime }\) are monotonic nondecreasing on \(\mathring{I}\) \(\ \)and this property can be extended to the whole interval \(I\) (see [12, p.7]).
From the monotonicity of the lateral derivatives \(f_{-}^{\prime }\) and \(f_{+}^{\prime }\) we also have the gradient inequality \[\begin{equation*} f_{-}^{\prime }\left( s\right) \left( s-\tau \right) \geq f\left( s\right) -f\left( \tau \right) \geq f_{+}^{\prime }\left( \tau \right) \left( s-\tau \right), \end{equation*}\] for any \(s,\) \(\tau \in \mathring{I}.\)
If \(I=\left[ m,M\right] ,\) then at the end points we also have the inequalities \[\begin{equation*} f\left( s\right) -f\left( m\right) \geq f_{+}^{\prime }\left( m\right) \left( s-m\right), \end{equation*}\] for any \(s\in (m,M]\) and \[\begin{equation*} f\left( \tau \right) -f\left( M\right) \geq f_{-}^{\prime }\left( M\right) \left( \tau -M\right), \end{equation*}\] for any \(\tau \in \lbrack m,M).\)
For the operator \(T\in \mathcal{B}\left( H\right)\) we define the selfadjoint operator \[\begin{equation*} Re\left( T\right) =\frac{1}{2}\left( T^{\ast }+T\right) . \end{equation*}\]
Assume that function \(B:\left[ m,M\right] \rightarrow \mathcal{B}\left( H\right)\) is continuous on \(\left[ m,M\right]\). The function \(B:\left[ m,M \right] \rightarrow \mathcal{B}\left( H\right)\) is square modulus convex on \(\left[ m,M\right]\) if and only if for all \(x\in H\setminus \{0\}\) the auxiliary function \(\varphi _{B,x}:\left[ m,M\right] \rightarrow \lbrack 0,\infty ),\) \(\varphi _{B,x}\left( u\right) =\left\Vert B\left( u\right) x\right\Vert ^{2}\) is convex on \(\left[ m,M\right] .\)
Indeed, condition (2) is equivalent to \[\begin{equation*} \left\langle \left\vert B\left( \left( 1-t\right) u+tv\right) \right\vert ^{2}x,x\right\rangle \leq \left( 1-t\right) \left\langle \left\vert B\left( u\right) \right\vert ^{2}x,x\right\rangle +t\left\langle \left\vert B\left( v\right) \right\vert ^{2}x,x\right\rangle , \end{equation*}\] namely \[\begin{align*} \left\langle \left[ B\left( \left( 1-t\right) u+tv\right) \right] ^{\ast }B\left( \left( 1-t\right) u+tv\right) x,x\right\rangle \leq \left( 1-t\right) \left\langle \left[ B\left( u\right) \right] ^{\ast }B\left( u\right) x,x\right\rangle +t\left\langle \left[ B\left( v\right) \right] ^{\ast }B\left( v\right) x,x\right\rangle , \end{align*}\] or \[\begin{equation*} \left\Vert B\left( \left( 1-t\right) u+tv\right) x\right\Vert ^{2}\leq \left( 1-t\right) \left\Vert B\left( u\right) x\right\Vert ^{2}+t\left\Vert B\left( v\right) x\right\Vert ^{2}, \end{equation*}\] for all \(t\in \left[ 0,1\right]\) and \(u,\) \(v\in \left[ m,M\right] .\)
We also have that \[\begin{align*} \varphi _{\pm B,x}^{\prime }\left( u\right) & =\left( \left\langle B\left( u\right) x,B\left( u\right) x\right\rangle \right) _{\pm }^{\prime }=\left\langle B_{\pm }^{\prime }\left( u\right) x,B\left( u\right) x\right\rangle +\left\langle B\left( u\right) x,B_{\pm }^{\prime }\left( u\right) x\right\rangle \\ & =\left\langle \left( B\left( u\right) \right) ^{\ast }B_{\pm }^{\prime }\left( u\right) x,x\right\rangle +\left\langle \left( B_{\pm }^{\prime }\left( u\right) \right) ^{\ast }B\left( u\right) x,x\right\rangle \\ & =\left\langle \left( B\left( u\right) \right) ^{\ast }B_{\pm }^{\prime }\left( u\right) x,x\right\rangle +\left\langle \left( \left( B\left( u\right) \right) ^{\ast }B_{\pm }^{\prime }\left( u\right) \right) ^{\ast }x,x\right\rangle \\ & =\left\langle 2Re\left( \left( B\left( u\right) \right) ^{\ast }B_{\pm }^{\prime }\left( u\right) \right) x,x\right\rangle , \end{align*}\] and \[\begin{align*} \varphi _{+B,x}^{\prime }\left( m\right) & =2\left\langle Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) x,x\right\rangle , \\ \text{ }\varphi _{-B,x}^{\prime }\left( M\right) & =2\left\langle Re \left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) x,x\right\rangle , \end{align*}\] for all \(x\in H\setminus \{0\}.\)
We have for \(t\in \left( m,M\right)\) and small \(h\neq 0\) such that \(t+h\in \left( m,M\right) ,\) \[\begin{align*} \left\langle \frac{\left\vert B\left( t+h\right) \right\vert ^{2}-\left\vert B\left( t\right) \right\vert ^{2}}{h}x,x\right\rangle & =\frac{1}{h}\left[ \left\langle \left\vert B\left( t+h\right) \right\vert ^{2}x,x\right\rangle -\left\langle \left\vert B\left( t\right) \right\vert ^{2}x,x\right\rangle \right] \\ & =\frac{1}{h}\left[ \left\Vert B\left( t+h\right) x\right\Vert ^{2}-\left\Vert B\left( t\right) x\right\Vert ^{2}\right], \end{align*}\] for all \(x\in H\setminus \{0\}.\)
By taking the lateral limits, we get \[\begin{align*} \lim_{h\rightarrow \pm 0}\left\langle \frac{\left\vert B\left( t+h\right) \right\vert ^{2}-\left\vert B\left( t\right) \right\vert ^{2}}{h} x,x\right\rangle & =\lim_{h\rightarrow \pm 0}\frac{1}{h}\left[ \left\Vert B\left( t+h\right) x\right\Vert ^{2}-\left\Vert B\left( t\right) x\right\Vert ^{2}\right] \\ & =\left\langle 2Re\left( \left( B\left( t\right) \right) ^{\ast }B_{\pm }^{\prime }\left( t\right) \right) x,x\right\rangle, \end{align*}\] for all \(x\in H\setminus \{0\}.\)
Therefore for the function \(\varphi \left( t\right) :=\left\vert B\left( t\right) \right\vert ^{2},\) \(t\in \left( m,M\right)\) we have \[\begin{equation*} \varphi _{\pm }^{\prime }\left( t\right) =2Re\left( \left( B\left( t\right) \right) ^{\ast }B_{\pm }^{\prime }\left( t\right) \right), \end{equation*}\] and \[\begin{equation*} \varphi _{+}^{\prime }\left( m\right) =2Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}\left( m\right) \right) ,\text{ }\varphi _{-}^{\prime }\left( M\right) =2Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) . \end{equation*}\] We also have the operator gradient inequality \[\begin{align} \label{e.3.1} 2\left( t-\tau \right) Re\left( \left( B\left( t\right) \right) ^{\ast }B_{-}^{\prime }\left( t\right) \right) & \geq \left\vert B\left( t\right) \right\vert ^{2}-\left\vert B\left( \tau \right) \right\vert ^{2}\notag \\ & \geq 2\left( t-\tau \right) Re\left( \left( B\left( \tau \right) \right) ^{\ast }B_{+}^{\prime }\left( \tau \right) \right) , \end{align} \tag{4}\] for any \(t,\) \(\tau \in \left( m,M\right) .\)
Moreover, at the end points of the interval we have the operator inequalities
\[\begin{equation} \left\vert B\left( t\right) \right\vert ^{2}-\left\vert B\left( m\right) \right\vert ^{2}\geq 2\left( t-m\right) Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right), \label{e.3.2} \end{equation} \tag{5}\] for any \(t\in (m,M]\) and \[\begin{equation} 2\left( M-\tau \right) Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) \geq \left\vert B\left( M\right) \right\vert ^{2}-\left\vert B\left( \tau \right) \right\vert ^{2}, \label{e.3.3} \end{equation} \tag{6}\] for any \(\tau \in \lbrack m,M).\)
Finally, we notice that for \(m\leq t_{1}<t_{2}\leq M,\) we have \[\begin{align} \label{e.3.4} Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) & \leq Re\left( \left( B\left( t_{1}\right) \right) ^{\ast }B_{\pm }^{\prime }\left( t_{1}\right) \right) \notag \\ & \leq Re\left( \left( B\left( t_{2}\right) \right) ^{\ast }B_{\pm }^{\prime }\left( t_{2}\right) \right) \leq Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) , \end{align} \tag{7}\] in the operator order of \(\mathcal{B}\left( H\right) .\)
Let \(\left( \Omega ,\mathcal{A},\mu \right)\) be a measurable space consisting of a set \(\Omega ,\) a \(\sigma\)-algebra \(\mathcal{A}\) of parts of \(\Omega\) and a countably additive and positive measure \(\mu\) on \(\mathcal{A }\) with values in \(\mathbb{R}\cup \left\{ \infty \right\} .\)
For a \(\mu\)-measurable function \(w:\Omega \rightarrow \mathbb{R}\), with \(w\left( s\right) \geq 0\) for \(\mu\)-a.e. (almost every) \(s\in \Omega ,\) consider the Lebesgue space \(L_{w}\left( \Omega ,\mu \right) :=\{f:\Omega \rightarrow \mathbb{R},\;f\) is \(\mu\)-measurable and \(\int_{\Omega }w\left( s\right) \left\vert f\left( s\right) \right\vert d\mu \left( s\right) <\infty \}.\) For simplicity of notation we write everywhere in the sequel \(\int_{\Omega }wd\mu\) instead of \(\int_{\Omega }w\left( s\right) d\mu \left( s\right) .\) We also assume that \(\int_{\Omega }wd\mu =1.\)
We define \(L_{w,p}\left( \Omega ,\mu ,\mathcal{B}\left( H\right) \right) ,\) \(p\geq 1,\) to be the space of all strongly \(\mu\)-measurable functions on \(A:\) \(\Omega \rightarrow \mathcal{B}\left( H\right)\) such that the integral \(\int_{\Omega }w\left( s\right) \left\Vert A\left( s\right) \right\Vert ^{p}d\mu \left( s\right)\) is finite. For \(p=1\) we denote \(L_{w}\left( \Omega ,\mu ,\mathcal{B}\left( H\right) \right) .\)
Theorem 2. Let \(B:\left[ m,M\right] \subset \mathbb{R\rightarrow }\mathcal{ B}\left( H\right)\) be square modulus convex on \(\left[ m,M\right]\) and \(f:\Omega \rightarrow \left[ m,M\right]\) so that \(f,\) \(\left\vert B\circ f\right\vert ^{2},\) \(Re\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) ,\) \(fRe\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) \in L_{w}\left( \Omega ,\mu ,\mathcal{B}\left( H\right) \right) ,\) where \(w\geq 0\) \(\mu\) -a.e. on \(\Omega\) with \(\int_{\Omega }wd\mu =1.\) Then we have the inequality \[\begin{align} \label{e.4.1} 0& \leq \int_{\Omega }w\left( s\right) \left\vert B\circ f\right\vert ^{2}d\mu \left( s\right) -\left\vert B\left( \int_{\Omega }wfd\mu \right) \right\vert ^{2} \notag\\ & \leq 2\left[ \int_{\Omega }wfRe\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) d\mu -\int_{\Omega }wfd\mu \int_{\Omega }wRe\left( \left( B\circ f\right) ^{\ast }\left( B_{-}^{\prime }\circ f\right) \right) d\mu \left( s\right) \right] , \end{align} \tag{8}\] in the operator order of \(\mathcal{B}\left( H\right) .\)
Proof. We have by the operator gradient inequality (4) that \[\begin{align} \label{e.4.2} 2\left( t-\int_{\Omega }wfd\mu \right) Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) & \geq \left\vert B\left( t\right) \right\vert ^{2}-\left\vert B\left( \int_{\Omega }wfd\mu \right) \right\vert ^{2} \notag \\ & \geq 2\left( t-\int_{\Omega }wfd\mu \right) Re\left( \left( B\left( \int_{\Omega }wfd\mu \right) \right) ^{\ast }B^{\prime }\left( \int_{\Omega }wfd\mu \right) \right) , \end{align} \tag{9}\] for almost every \(t\in \left( m,M\right) .\)
If we take in (9) \(t=f\left( s\right) ,\) multiply with \(w\left( s\right) \geq 0\) and integrate, then we get \[\begin{align} & 2\int_{\Omega }w\left( s\right) \left( f\left( s\right) -\int_{\Omega }wfd\mu \right) Re\left( \left( B\left( f\left( s\right) \right) \right) ^{\ast }B^{\prime }\left( f\left( s\right) \right) \right) d\mu \left( s\right) \label{e.4.3} \\ &\quad \geq \int_{\Omega }w\left( s\right) \left[ \left\vert B\left( f\left( s\right) \right) \right\vert ^{2}-\left\vert B\left( \int_{\Omega }wfd\mu \right) \right\vert ^{2}\right] d\mu \left( s\right) \notag \\ &\quad \geq 2\int_{\Omega }w\left( s\right) \left( f\left( s\right) -\int_{\Omega }wfd\mu \right) \notag \\ &\qquad \times Re\left( \left( B\left( \int_{\Omega }wfd\mu \right) \right) ^{\ast }B^{\prime }\left( \int_{\Omega }wfd\mu \right) \right) d\mu \left( s\right) . \notag \end{align} \tag{10}\]
Since \[\begin{align*} & \int_{\Omega }w\left( s\right) \left( f\left( s\right) -\int_{\Omega }wfd\mu \right) Re\left( \left( B\left( f\left( s\right) \right) \right) ^{\ast }B^{\prime }\left( f\left( s\right) \right) \right) d\mu \left( s\right) \\ & \qquad=\int_{\Omega }w\left( s\right) f\left( s\right) Re\left( \left( B\left( f\left( s\right) \right) \right) ^{\ast }B^{\prime }\left( f\left( s\right) \right) \right) d\mu \left( s\right) \\ &\qquad -\int_{\Omega }wfd\mu \int_{\Omega }w\left( s\right) Re\left( \left( B\left( f\left( s\right) \right) \right) ^{\ast }B^{\prime }\left( f\left( s\right) \right) \right) d\mu \left( s\right) , \end{align*}\] \[\begin{align*} & \int_{\Omega }w\left( s\right) \left[ \left\vert B\left( f\left( s\right) \right) \right\vert ^{2}-\left\vert B\left( \int_{\Omega }wfd\mu \right) \right\vert ^{2}\right] d\mu \left( s\right) \\ & \qquad=\int_{\Omega }w\left( s\right) \left\vert B\left( f\left( s\right) \right) \right\vert ^{2}d\mu \left( s\right) -\int_{\Omega }w\left( s\right) d\mu \left( s\right) \left\vert B\left( \int_{\Omega }wfd\mu \right) \right\vert ^{2}, \end{align*}\] and \[\begin{align*} & \int_{\Omega }w\left( s\right) \left( f\left( s\right) -\int_{\Omega }wfd\mu \right) Re\left( \left( B\left( \int_{\Omega }wfd\mu \right) \right) ^{\ast }B^{\prime }\left( \int_{\Omega }wfd\mu \right) \right) d\mu \left( s\right) \\ &\qquad =\int_{\Omega }w\left( s\right) \left( f\left( s\right) -\int_{\Omega }wfd\mu \right) d\mu \left( s\right) Re\left( \left( B\left( \int_{\Omega }wfd\mu \right) \right) ^{\ast }B^{\prime }\left( \int_{\Omega }wfd\mu \right) \right) =0, \end{align*}\] hence by (10) we get (8). ◻
We have the following Cauchy-Bunyakowsky-Schwarz integral inequality for the modulus of operators:
Lemma 1. If \(\alpha \in L_{w}^{2}\left( \Omega ,\mu ,\mathbb{C}\right)\) and \[\begin{equation*} A\in L_{2,w}\left( \Omega ,\mu ,\mathcal{B}\left( H\right) \right) :=\left\{ A:\Omega \rightarrow B\left( H\right) ,\text{ }\int_{\Omega }w\left( s\right) \left\Vert A\left( s\right) \right\Vert ^{2}d\mu \left( s\right) <\infty \right\} , \end{equation*}\] then \[\begin{equation} \left\vert \int_{\Omega }w\alpha Ad\mu \right\vert ^{2}\leq \int_{\Omega }w\left\vert \alpha \right\vert ^{2}d\mu \int_{\Omega }w\left\vert A\right\vert ^{2}d\mu \label{CBS}, \end{equation} \tag{11}\] in the operator order of \(\mathcal{B}\left( H\right) .\)
Proof. We have for \(\alpha \in L_{w}^{2}\left( \Omega ,\mu ,\mathbb{C}\right)\) and \(A\in L_{2,w}\left( \Omega ,\mu ,\mathcal{B}\left( H\right) \right) ,\) that \[\begin{align*} 0& \leq \left\vert \overline{\alpha \left( x\right) }A\left( y\right) – \overline{\alpha \left( y\right) }A\left( x\right) \right\vert ^{2}\\ &=\left\vert \alpha \left( x\right) \right\vert \left\vert A\left( y\right) \right\vert ^{2}-\alpha \left( y\right) \overline{\alpha \left( x\right) }A^{\ast }\left( x\right) A\left( y\right) -\alpha \left( x\right) \overline{\alpha \left( y\right) }A^{\ast }\left( y\right) A\left( x\right) +\left\vert \alpha \left( y\right) \right\vert ^{2}\left\vert A\left( x\right) \right\vert ^{2}, \end{align*}\] which gives that \[\begin{align*} \left\vert \alpha \left( x\right) \right\vert ^{2}\left\vert A\left( y\right) \right\vert ^{2}+\left\vert \alpha \left( y\right) \right\vert ^{2}\left\vert A\left( x\right) \right\vert ^{2} \geq \alpha \left( y\right) \overline{\alpha \left( x\right) }A^{\ast }\left( x\right) A\left( y\right) +\alpha \left( x\right) \overline{\alpha \left( y\right) }A^{\ast }\left( y\right) A\left( x\right) , \end{align*}\] for all \(y,\) \(x\in \Omega .\)
Now, multiply this with \(w\left( y\right) w\left( x\right) \geq 0\) to get \[\begin{align*} & w\left( x\right) \left\vert \alpha \left( x\right) \right\vert ^{2}w\left( y\right) \left\vert A\left( y\right) \right\vert ^{2}+w\left( y\right) \left\vert \alpha \left( y\right) \right\vert ^{2}w\left( x\right) \left\vert A\left( x\right) \right\vert ^{2} \\ &\qquad \geq w\left( x\right) \overline{\alpha \left( x\right) }A^{\ast }\left( x\right) w\left( y\right) \alpha \left( y\right) A\left( y\right) +w\left( y\right) \overline{\alpha \left( y\right) }A^{\ast }\left( y\right) w\left( x\right) \alpha \left( x\right) A\left( x\right) , \end{align*}\] for all \(y,\) \(x\in \Omega .\)
Integrating over \(x\) and \(y\) on \(\Omega ,\) then we get \[\begin{align*} & \int_{a}^{b}w\left( x\right) \left\vert \alpha \left( x\right) \right\vert ^{2}d\mu \left( x\right) \int_{a}^{b}w\left( y\right) \left\vert A\left( y\right) \right\vert ^{2}d\mu \left( y\right) +\int_{a}^{b}w\left( y\right) \left\vert \alpha \left( y\right) \right\vert ^{2}d\mu \left( y\right) \int_{a}^{b}w\left( x\right) \left\vert A\left( x\right) \right\vert ^{2}d\mu \left( x\right) \\ &\quad \geq \int_{a}^{b}w\left( x\right) \overline{\alpha \left( x\right) } A^{\ast }\left( x\right) d\mu \left( x\right) \int_{a}^{b}w\left( y\right) \alpha \left( y\right) A\left( y\right) d\mu \left( y\right) \\ &\qquad +\int_{a}^{b}w\left( y\right) \overline{\alpha \left( y\right) }A^{\ast }\left( y\right) d\mu \left( y\right) \int_{a}^{b}w\left( x\right) \alpha \left( x\right) A\left( x\right) d\mu \left( x\right) \\ &\quad =2\left\vert \int_{a}^{b}w\left( y\right) \alpha \left( y\right) A\left( y\right) d\mu \left( y\right) \right\vert ^{2}, \end{align*}\] and the inequality (11) is obtained. ◻
We recall Löwner-Heinz inequality which says that, if \(0\leq A\leq B,\) then for all \(p\in \left( 0,1\right)\) we have \(0\leq A^{p}\leq B^{p}.\) By using this property, we can state the following result as well:
Corollary 1. With the assumptions of Lemma 1, we have the inequality \[\begin{equation} \left\vert \int_{\Omega }w\alpha Ad\mu \right\vert \leq \left( \int_{\Omega }w\left\vert \alpha \right\vert ^{2}d\mu \right) ^{1/2}\left( \int_{\Omega }w\left\vert A\right\vert ^{2}d\mu \right) ^{1/2}, \label{CBSsq} \end{equation} \tag{12}\] in the operator order of \(\mathcal{B}\left( H\right) .\)
The proof follows by (11) by taking the operator square root.
Remark 1. We remark that, if \(\alpha\) is real valued and \(A\left( s\right) ,\) \(s\in \Omega\) are selfadjoint operators, then we have \[\begin{equation} \left\vert \int_{\Omega }w\alpha Ad\mu \right\vert \leq \left( \int_{\Omega }w\alpha ^{2}d\mu \right) ^{1/2}\left( \int_{\Omega }wA^{2}d\mu \right) ^{1/2}. \label{CBSre} \end{equation} \tag{13}\]
Lemma 2. If \(\alpha :\Omega \rightarrow \left[ m,M\right]\) and \(A\left( s\right) ,\) \(s\in \Omega\) are selfadjoint operators such that \(\alpha \in L_{w}^{2}\left( \Omega ,\mu ,\mathbb{C}\right) ,\) \(A\in L_{2,w}\left( \Omega ,\mu ,\mathcal{B}\left( H\right) \right) ,\) then we have \[\begin{align} \label{e.4.4} \left\vert \int_{\Omega }w\alpha Ad\mu -\int_{\Omega }w\alpha d\mu \int_{\Omega }wAd\mu \right\vert & \leq \left[ \int_{\Omega }w\alpha ^{2}d\mu -\left( \int_{\Omega }w\alpha d\mu \right) ^{2}\right] ^{1/2}\left( \int_{\Omega }wA^{2}d\mu -\left( \int_{\Omega }wAd\mu \right) ^{2}\right) ^{1/2} \notag \\ & \leq \frac{1}{2}\left( M-m\right) \left[ \int_{\Omega }wA^{2}d\mu -\left( \int_{\Omega }wAd\mu \right) ^{2}\right] ^{1/2}. \end{align} \tag{14}\]
Proof. We use the following operator Sonin type identity that can be proved by performing the calculations in the right side \[\begin{equation*} \int_{\Omega }w\alpha Ad\mu -\int_{\Omega }w\alpha d\mu \int_{\Omega }wAd\mu =\int_{\Omega }w\left( \alpha -\int_{\Omega }w\alpha d\mu \right) \left( A-\int_{\Omega }wAd\mu \right) d\mu . \end{equation*}\]
By using (13) we have \[\begin{align} \label{e.4.5} \left\vert \int_{\Omega }w\left( \alpha -\int_{\Omega }w\alpha d\mu \right) \left( A-\int_{\Omega }wAd\mu \right) d\mu \right\vert \leq \left[ \int_{\Omega }w\left( \alpha -\int_{\Omega }w\alpha d\mu \right) ^{2}d\mu \right] ^{1/2}{\left[ \int_{\Omega }w\left( A-\int_{\Omega }wAd\mu \right) ^{2}d\mu \right].}^{1/2} \end{align} \tag{15}\]
Since \[\begin{align} \label{e.4.6} \int_{\Omega }w\left( \alpha -\int_{\Omega }w\alpha d\mu \right) ^{2}d\mu =\int_{\Omega }w\alpha ^{2}d\mu -\left( \int_{\Omega }w\alpha d\mu \right) ^{2} \leq \frac{1}{4}\left( M-m\right) ^{2} , \end{align} \tag{16}\] and \[\begin{equation*} \int_{\Omega }w\left( A-\int_{\Omega }wAd\mu \right) ^{2}d\mu =\int_{\Omega }wA^{2}d\mu -\left( \int_{\Omega }wAd\mu \right) ^{2}, \end{equation*}\] then by (15) and (16) we derive the desired result (14). ◻
Corollary 2. With the assumption of Theorem 2, we have \[\begin{align} \label{e.4.7} 0& \leq \int_{\Omega }w\left( s\right) \left\vert B\circ f\right\vert ^{2}d\mu \left( s\right) -\left\vert B\left( \int_{\Omega }wfd\mu \right) \right\vert ^{2} \notag\\ & \leq \left( M-m\right) \left[ \int_{\Omega }w\left( Re\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) \right) ^{2}d\mu -\left( \int_{\Omega }wRe\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) d\mu \right) ^{2}\right] ^{1/2}. \end{align} \tag{17}\]
Proof. Using (14) for \(\alpha =f,\) we get \[\begin{align*} 0& \leq \int_{\Omega }wfRe\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) d\mu \left. -\int_{\Omega }wfd\mu \int_{\Omega }wRe\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) d\mu \left( s\right) \right] \\ & \leq \frac{1}{2}\left( M-m\right) \left[ \int_{\Omega }w\left( Re \left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) \right) ^{2}d\mu -\left( \int_{\Omega }wRe\left( \left( B\circ f\right) ^{\ast }\left( B^{\prime }\circ f\right) \right) d\mu \right) ^{2}\right] ^{1/2}. \end{align*}\]
By utilising (8) we then obtain the desired result (17). ◻
We have the following reverse of Cauchy-Bunyakowsky-Schwarz norm inequality that is of interest in itself:
Lemma 3. Assume that \(f\in L_{2,w}\left( \Omega ,\mu ,H\right) ,\) then for all \(v\in H,\) \[\begin{align} 0& \leq \int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) \right\Vert ^{2}d\mu \left( s\right) -\left\Vert \int_{\Omega }w\left( s\right) f\left( s\right) d\mu \left( s\right) \right\Vert ^{2} \label{e.4.8} \\ & \leq \int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) -v\right\Vert ^{2}d\mu \left( s\right) . \notag \end{align} \tag{18}\]
Proof. Observe that, for any \(v\in H\) \[\begin{align} \label{e.4.9} 0& \leq \int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) \right\Vert ^{2}d\mu \left( s\right) -\left\Vert \int_{\Omega }w\left( s\right) f\left( s\right) d\mu \left( s\right) \right\Vert ^{2} \notag \\ & =\int_{\Omega }w\left( s\right) \left\langle f\left( s\right) ,f\left( s\right) \right\rangle d\mu \left( s\right) -\left\langle \int_{\Omega }w\left( s\right) f\left( s\right) d\mu \left( s\right) ,\int_{\Omega }w\left( s\right) f\left( s\right) d\mu \left( s\right) \right\rangle \notag \\ & =\int_{\Omega }w\left( s\right) \left\langle f\left( s\right) -\int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) ,f\left( s\right) -v\right\rangle d\mu \left( s\right) =:K , \end{align} \tag{19}\] since, obviously \[\begin{equation*} \int_{\Omega }w\left( s\right) \left\langle f\left( s\right) -\int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) ,v\right\rangle d\mu \left( s\right) =0. \end{equation*}\]
Therefore, by Schwarz inequality in Hilbert spaces and the CBS integral inequality, we have \[\begin{align} \label{e.4.10} K& \leq \int_{\Omega }w\left( s\right) \left\vert \left\langle f\left( s\right) -\int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) ,f\left( s\right) -v\right\rangle \right\vert d\mu \left( s\right) \notag \\ & \leq \int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) -\int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) \right\Vert \left\Vert f\left( s\right) -v\right\Vert d\mu \left( s\right) \notag \\ & \leq \left( \int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) -\int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) \right\Vert ^{2}d\mu \left( s\right) \right) ^{1/2} \times \left( \int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) -v\right\Vert ^{2}d\mu \left( s\right) \right) ^{1/2}. \end{align} \tag{20}\]
Since, by the properties of inner product and integral, \[\begin{align*} &\int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) -\int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) \right\Vert ^{2}d\mu \left( s\right) \\ & =\int_{\Omega }w\left( s\right) \left[ \left\Vert f\left( s\right) \right\Vert ^{2}-2Re\left\langle f\left( s\right) ,\int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) \right\rangle +\left\Vert \int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) \right\Vert ^{2}\right] d\mu \left( s\right) \\ & =\int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) \right\Vert ^{2}d\mu \left( s\right) -2Re\left\langle \int_{\Omega }w\left( s\right) f\left( s\right) d\mu \left( u\right) ,\int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) \right\rangle +\left\Vert \int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) \right\Vert ^{2} \\ & =\int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) \right\Vert ^{2}d\mu \left( s\right) -2\left\Vert \int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) \right\Vert ^{2} +\left\Vert \int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) \right\Vert ^{2} \\ & =\int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) \right\Vert ^{2}d\mu \left( s\right) -\left\Vert \int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) \right\Vert ^{2}, \end{align*}\] then by (19) and (20) we get \[\begin{align*} 0& \leq \int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) \right\Vert ^{2}d\mu \left( s\right) -\left\Vert \int_{\Omega }w\left( s\right) f\left( s\right) d\mu \left( s\right) \right\Vert ^{2} \\ & \leq \left( \int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) \right\Vert ^{2}d\mu \left( s\right) -\left\Vert \int_{\Omega }w\left( u\right) f\left( u\right) d\mu \left( u\right) \right\Vert ^{2}\right) ^{1/2} \times \left( \int_{\Omega }w\left( s\right) \left\Vert f\left( s\right) -v\right\Vert ^{2}d\mu \left( s\right) \right) ^{1/2}, \end{align*}\] which is equivalent to (18). ◻
We can also state the following simpler upper bound for the Jensen’s gap:
Corollary 3. With the assumption of Theorem 2, we have \[\begin{align} \label{e.4.11} 0& \leq \int_{\Omega }w\left( s\right) \left\vert B\circ f\right\vert ^{2}d\mu \left( s\right) -\left\vert B\left( \int_{\Omega }wfd\mu \right) \right\vert ^{2} \notag\\ & \leq \frac{1}{2}\left( M-m\right) \left\Vert \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) – Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right\Vert , \end{align} \tag{21}\] in the operator order of \(\mathcal{B}\left( H\right) .\)
Proof. Observe that, by (7) \[\begin{equation} Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \leq Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) \leq Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) , \label{e.4.12} \end{equation} \tag{22}\] for almost every \(t\in \left[ m,M\right] .\)
This implies that \[\begin{align*} \left\langle Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) x,x\right\rangle & \leq \left\langle Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) x,x\right\rangle \\ & \leq \left\langle Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) x,x\right\rangle, \end{align*}\] for all \(x\in H\) and for almost every \(t\in \left[ m,M\right] .\)
This inequality is equivalent to \[\begin{align*} & \left\vert \left\langle \left( Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) -\frac{1}{2}Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) +Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) \right) x,x\right\rangle \right\vert \\ &\qquad \leq \frac{1}{2}\left\langle \left[ \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) -Re \left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right] x,x\right\rangle , \end{align*}\] for all \(x\in H\) and for almost every \(t\in \left[ m,M\right] .\)
By taking the supremum over \(x\in H,\) \(\left\Vert x\right\Vert =1,\) we derive the norm inequality \[\begin{align*} & \left\Vert Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) -\frac{Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) +Re \left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) }{2}\right\Vert \\ &\qquad \leq \frac{1}{2}\left\Vert \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) -Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right\Vert . \end{align*}\]
Observe that for \(x\in H,\) we have \[\begin{align*} & \left\Vert Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) x-\frac{Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) x+Re \left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) x}{2}\right\Vert ^{2} \\ &\qquad =\left\Vert \left( Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) -\frac{1}{2}Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) +Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) \right) x\right\Vert ^{2} \\ &\qquad \leq \left\Vert \left( Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) -\frac{1}{2}Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) +Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) \right) \right\Vert ^{2}\left\Vert x\right\Vert ^{2} \\ &\qquad \leq \frac{1}{4}\left\Vert \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) -Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right\Vert ^{2}\left\Vert x\right\Vert ^{2}. \end{align*}\]
From (18) we get for \[\begin{equation*} v=\frac{Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) x+Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) x}{2}, \end{equation*}\] that \[\begin{align} \label{e.4.13} 0& \leq \int_{\Omega }w\left( s\right) \left\Vert Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) x\right\Vert ^{2}d\mu \left( s\right) -\left\Vert \int_{\Omega }w\left( s\right) Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) xd\mu \left( s\right) \right\Vert ^{2} \notag \\ & \leq \int_{\Omega }w\left( s\right) \left\Vert Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) x-\frac{1}{2}\left[ Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) x+Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) \right] x\right\Vert ^{2}d\mu \left( s\right) \notag \\ & \leq \frac{1}{4}\left\Vert \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) -Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right\Vert ^{2}\left\Vert x\right\Vert ^{2} , \end{align} \tag{23}\] for \(x\in H.\)
Since \[\begin{align*} &\int_{\Omega }w\left( s\right) \left\Vert Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) x\right\Vert ^{2}d\mu \left( s\right) -\left\Vert \int_{\Omega }w\left( s\right) Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) xd\mu \left( s\right) \right\Vert ^{2} \\ &\qquad =\left\langle \int_{\Omega }w\left( s\right) \left\vert Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) \right\vert ^{2}d\mu \left( s\right) x,x\right\rangle -\left\langle \left\vert \int_{\Omega }w\left( s\right) Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) d\mu \left( s\right) \right\vert ^{2}x,x\right\rangle , \end{align*}\] then \[\begin{align*} 0& \leq \int_{\Omega }w\left( s\right) \left( Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) \right) ^{2}d\mu \left( s\right) -\left( \int_{\Omega }w\left( s\right) Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) d\mu \left( s\right) \right) ^{2} \\ & \leq \frac{1}{4}\left\Vert \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) -Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right\Vert ^{2}, \end{align*}\] which, by taking the square root in the operator inequality, gives that \[\begin{align*} 0& \leq \left[ \int_{\Omega }w\left( s\right) \left( Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) \right) ^{2}d\mu \left( s\right) -\left( \int_{\Omega }w\left( s\right) Re\left( \left( B\left( t\right) \right) ^{\ast }B^{\prime }\left( t\right) \right) d\mu \left( s\right) \right) ^{2}\right] ^{1/2} \\ & \leq \frac{1}{2}\left\Vert \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) -Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right\Vert . \end{align*}\]
By (17) we derive (21). ◻
Now, if we consider the discrete measure, then for \(B:\left[ m,M\right] \subset \mathbb{R\rightarrow }\mathcal{B}\left( H\right) ,\) a square modulus convex function on \(\left[ m,M\right]\) that is also strongly differentiable on \(\left( m,M\right) ,\) \(t_{i}\in \left[ m,M\right] ,\) \(w_{i}\geq 0\) for \(i\in \left\{ 1,…,n\right\} ,\) with \(\sum_{i=1}^{n}w_{i},\) then by (8) we get \[\begin{align} \label{e.4.14} 0& \leq \sum_{i=1}^{n}w_{i}\left\vert B\left( t_{i}\right) \right\vert ^{2}-\left\vert B\left( \sum_{i=1}^{n}w_{i}t_{i}\right) \right\vert ^{2} \notag \\ & \leq 2\left[ \sum_{i=1}^{n}w_{i}t_{i}Re\left[ \left( B\left( t_{i}\right) \right) ^{\ast }\left( B^{\prime }\left( t_{i}\right) \right) \right] -\sum_{i=1}^{n}w_{i}t_{i}\sum_{i=1}^{n}w_{i}Re\left[ \left( B\left( t_{i}\right) \right) ^{\ast }\left( B^{\prime }\left( t_{i}\right) \right) \right] \right] . \end{align} \tag{24}\]
From (17) we get \[\begin{align} \label{e.4.15} 0& \leq \sum_{i=1}^{n}w_{i}\left\vert B\left( t_{i}\right) \right\vert ^{2}-\left\vert B\left( \sum_{i=1}^{n}w_{i}t_{i}\right) \right\vert ^{2} \notag\\ & \leq \left( M-m\right) \left[ \sum_{i=1}^{n}w_{i}\left( Re\left[ \left( B\left( t_{i}\right) \right) ^{\ast }\left( B^{\prime }\left( t_{i}\right) \right) \right] \right) ^{2}d\mu -\left( \sum_{i=1}^{n}w_{i}Re\left[ \left( B\left( t_{i}\right) \right) ^{\ast }\left( B^{\prime }\left( t_{i}\right) \right) \right] d\mu \right) ^{2}\right] ^{1/2}, \end{align} \tag{25}\] while from (21) we derive \[\begin{align} 0& \leq \sum_{i=1}^{n}w_{i}\left\vert B\left( t_{i}\right) \right\vert ^{2}-\left\vert B\left( \sum_{i=1}^{n}w_{i}t_{i}\right) \right\vert ^{2} \label{e.4.16} \\ & \leq \frac{1}{2}\left( M-m\right) \left\Vert \left( Re\left( \left( B\left( M\right) \right) ^{\ast }B_{-}^{\prime }\left( M\right) \right) – Re\left( \left( B\left( m\right) \right) ^{\ast }B_{+}^{\prime }\left( m\right) \right) \right) \right\Vert . \notag \end{align} \tag{26}\]
For distinct operators \(A,\) \(B\in \mathcal{B}\left( H\right)\) we consider the function \(\varphi _{A,B}:\left[ 0,1\right] \rightarrow \mathcal{B}\left( H\right)\) defined by \(\varphi _{A,B}\left( t\right) =\left\vert \left( 1-t\right) A+tB\right\vert ^{2}.\) Let \(t\in \left( 0,1\right)\) and small \(h\neq 0\) such that \(t+h\in \left( 0,1\right) ,\) then we have \[\begin{align*} \varphi _{A,B}\left( t+h\right) & =\left\vert \left( 1-t-h\right) A+\left( t+h\right) B\right\vert ^{2} \\ & =\left( 1-t-h\right) ^{2}\left\vert A\right\vert ^{2}+\left( 1-t-h\right) \left( t+h\right) \left( A^{\ast }B+B^{\ast }A\right) +\left( t+h\right) ^{2}\left\vert B\right\vert ^{2}, \end{align*}\] and \[\begin{align*} \varphi _{A,B}\left( t\right) & =\left\vert \left( 1-t\right) A+tB\right\vert ^{2} \\ & =\left( 1-t\right) ^{2}\left\vert A\right\vert ^{2}+\left( 1-t\right) t\left( A^{\ast }B+B^{\ast }A\right) +t^{2}\left\vert B\right\vert ^{2}. \end{align*}\]
Then we have \[\begin{align*} \varphi _{A,B}\left( t+h\right) -\varphi _{A,B}\left( t\right) & =\left[ \left( 1-t-h\right) ^{2}-\left( 1-t\right) ^{2}\right] \left\vert A\right\vert ^{2} \\ &\quad +\left[ \left( 1-t-h\right) \left( t+h\right) -\left( 1-t\right) t\right] \left( A^{\ast }B+B^{\ast }A\right) +\left[ \left( t+h\right) ^{2}-t^{2} \right] \left\vert B\right\vert ^{2} \\ & =h\left[ -2\left( 1-t\right) +h\right] \left\vert A\right\vert ^{2}+h\left( 1-2t-h\right) \left( A^{\ast }B+B^{\ast }A\right) +h\left( h+2t\right) \left\vert B\right\vert ^{2}, \end{align*}\] which gives that \[\begin{align*} \frac{1}{h}\left[ \varphi _{A,B}\left( t+h\right) -\varphi _{A,B}\left( t\right) \right] =\left[ -2\left( 1-t\right) +h\right] \left\vert A\right\vert ^{2}+\left( 1-2t-h\right) \left( A^{\ast }B+B^{\ast }A\right) +\left( h+2t\right) \left\vert B\right\vert ^{2}, \end{align*}\] for small \(h\neq 0.\)
Taking the strong limit over \(h\rightarrow 0,\) we get \[\begin{align*} \varphi _{A,B}^{\prime }\left( t\right) & =-2\left( 1-t\right) \left\vert A\right\vert ^{2}+\left( 1-2t\right) \left( A^{\ast }B+B^{\ast }A\right) +2t\left\vert B\right\vert ^{2} \\ & =2t\left[ \left\vert A\right\vert ^{2}-\left( A^{\ast }B+B^{\ast }A\right) +\left\vert B\right\vert ^{2}\right] +A^{\ast }B+B^{\ast }A-2\left\vert A\right\vert ^{2} \\ & =2t\left\vert A-B\right\vert ^{2}+A^{\ast }B+B^{\ast }A-\left\vert A\right\vert ^{2}-\left\vert B\right\vert ^{2}+\left\vert B\right\vert ^{2}-\left\vert A\right\vert ^{2} \\ & =2t\left\vert A-B\right\vert ^{2}-\left\vert A-B\right\vert ^{2}+\left\vert B\right\vert ^{2}-\left\vert A\right\vert ^{2} \\ & =\left( 2t-1\right) \left\vert A-B\right\vert ^{2}+\left\vert B\right\vert ^{2}-\left\vert A\right\vert ^{2}, \end{align*}\] for \(t\in \left( 0,1\right) .\)
We also have the lateral derivatives \[\begin{equation*} \varphi _{+A,B}^{\prime }\left( 0\right) =\left\vert B\right\vert ^{2}-\left\vert A\right\vert ^{2}-\left\vert A-B\right\vert ^{2}, \end{equation*}\] and \[\begin{equation*} \varphi _{-A,B}^{\prime }\left( 1\right) =\left\vert A-B\right\vert ^{2}+\left\vert B\right\vert ^{2}-\left\vert A\right\vert ^{2}. \end{equation*}\]
Consider the \(\mu\)-measurable function \(f:\Omega \rightarrow \left[ 0,1 \right]\) such that \(f\in L_{2,w}\left( \Omega ,\mu \right) .\) Then by using the inequality (21) we derive \[\begin{align} \label{e.5.1} 0& \leq \int_{\Omega }w\left( s\right) \left\vert \left( 1-f\left( s\right) \right) A+f\left( s\right) B\right\vert ^{2}d\mu \left( s\right) -\left\vert \left( 1-\int_{\Omega }w\left( s\right) f\left( s\right) d\mu \left( s\right) \right) A+\left( \int_{\Omega }w\left( s\right) f\left( s\right) d\mu \left( s\right) \right) B\right\vert ^{2} \notag \\ & \leq \left\Vert A-B\right\Vert ^{2} , \end{align} \tag{27}\] for all distinct \(A,\) \(B\in \mathcal{B}\left( H\right) .\)
Dragomir, S. S. (2021). Hermite-Hadamard type inequalities for the operator modulus in Hilbert spaces, Preprint. Research Group in Mathematical Inequalities and Applications, 24, Art. 51, 16 pp.
Alomari, M. W. (2019). Operator Popoviciu’s inequality for superquadratic and convex functions of selfadjoint operators in Hilbert spaces. Advances in Pure and Applied Mathematics, 10(4), 313-324.
Anjidani, E. (2017). On operator inequalities of Jensen type for convex functions. Linear and Multilinear Algebra, 65(7), 1493-1502.
Li, B. (2013). Refinements of Hermite-Hadamard’s type inequalities for operator convex functions. International Journal of Contemporary Mathematical Sciences, 8(9-12), 463-467.
Dragomir, S. S. (2017). Inequalities for quadratic operator perspective of convex functions and bounded linear operators on Hilbert spaces. SUT Journal of Mathematics, 53(1), 39-58.
Dragomir, S. S. (2020). Some inequalities for weighted and integral means of operator convex functions. SUT Journal of Mathematics, 56(2), 109-127.
Dragomir, S. S. (2020). On Jensen’s additive inequality for positive convex functions of selfadjoint operators in Hilbert spaces. Jordan Journal of Mathematics and Statistics, 13(4), 601-623.
Erdaş, Y., Ünlüyol, E., & Salaş, S. (2019). Some New Inequalities of Operator m-Convex Functions and Applications for Synchronous–Asynchronous Functions. Complex Analysis and Operator Theory, 13(8), 3871-3881.
Ghazanfari, A. G. (2016). Hermite–Hadamard type inequalities for functions whose derivatives are operator convex. Complex Analysis and Operator Theory, 10(8), 1695-1703.
Rooin, J., Alikhani, A., & Moslehian, M. S. (2018). Operator m-convex functions. Georgian Mathematical Journal, 25(1), 93-107.
Taghavi, A., Darvish, V., Nazari, H. M., & Dragomir, S. S. (2019). Some quantum \(f\)-divergence inequalities for convex functions of self-adjoint operators. Boletim da Sociedade Paranaense de Matematica, 37(1), 125-139.
Roberts, A. W. (1993). Convex Functions. In Handbook of convex geometry (pp. 1081-1104). North-Holland.
