This article presents a generalization of the Hardy–Littlewood–Pólya majorization theorem by employing a weighted Montgomery identity derived from Taylor’s formula. We establish new identities and inequalities for n-convex functions, provide Čebyšev-type bounds for the remainders, and derive associated Ostrowski and Grüss-type inequalities. Our results significantly extend the classical theory of majorization and provide a comprehensive framework for analyzing n-convex functions in the context of weighted integral inequalities.
The theory of majorization provides a fundamental framework for comparing the diversity of vectors and has profound implications across mathematics, economics, and the sciences. The classical Hardy–Littlewood–Pólya majorization theorem establishes that the inequality \(\sum\limits_{i=1}^m \digamma(x_i) \leq \sum\limits_{i=1}^m \digamma({y}_i)\) holds for all continuous convex functions \(\digamma\) if and only if \((x_1,\ldots,x_m)\) is majorized by \(({y}_1,\ldots,{y}_m)\). This seminal result, along with its subsequent integral and weighted variants [1– 4], characterizes the way convex functions preserve the order induced by majorization.
A natural and compelling question is whether this powerful relationship can be extended beyond the realm of convex functions. The class of \(n\)-convex functions, defined by the nonnegativity of the \(n\)th-order divided differences (or the \(n\)th derivative, for smooth functions), offers a significant generalization. While all convex functions are \(2\)-convex, the converse is not true, making \(n\)-convexity a strictly broader and richer class. Establishing majorization-type inequalities for \(n\)-convex functions requires more sophisticated tools that can capture the behavior of higher-order derivatives.
In this paper, we achieve this generalization by employing an extension of the weighted Montgomery identity based on Taylor’s formula [5– 7]. This identity provides a precise representation of a function in terms of its derivatives at the boundary points and a remainder involving a specially constructed kernel, \(K_{\rho,n}(x,{u})\). This kernel serves as the bridge connecting the function’s local behavior to its global properties.
The primary aim of this work is to derive a new, generalized identity for the difference \(\int \digamma({y}) dB_v – \int \digamma(x) dB_v\) (Theorem 1). From this identity, we establish a majorization inequality for \(n\)-convex functions under a novel and natural condition on the kernel \(K_{\rho,n}\) (Theorem 2). Furthermore, recognizing the importance of quantitative estimates, we provide sharp bounds for the remainder terms using Čebyšev (Theorem 3) and Grüss-type (Theorem 4) inequalities. Finally, we present Ostrowski-type inequalities (Theorem 5) that offer error estimates for our approximations under various \(L_p\) norms. Our results not only contain the classical majorization theorems as special cases but also open new avenues for analysis in the theory of inequalities.
We begin by recalling some fundamental definitions and results from [4]. Throughout this article, \(I\) denotes an interval in \(\mathbb{R}\). Throughout the article, \(BV[\alpha_0, \alpha_1]\) denotes the class of real-valued functions of bounded variation defined on \([\alpha_0,\alpha_1]\), and \(AC[\beta_0,\beta_1]\) stands for the class of real-valued absolutely continuous functions defined on the interval \([\beta_0,\beta_1]\).
Definition 1. A function \(\digamma:I\rightarrow \mathbb{R}\) is convex (concave) if \[\digamma(\sigma x_1 +(1-\sigma)x_2)\leq (\ge) \sigma \digamma(x_1)+ (1-\sigma)\digamma(x_2),\] holds for all \(\sigma \in [0,1]\) and all \(x_1, x_2 \in I\).
For fixed \(m \geq 2\), let \(\mathbf{ x}=(x_1, \ldots, x_m)\) and \(\mathbf{ y}=({y}_1, \ldots, {y}_m)\) be real \(m\)-tuples with \(x_{[1]} \geq x_{[2]} \geq \ldots \geq x_{[m]}\), \({y}_{[1]} \geq {y}_{[2]} \geq \ldots \geq {y}_{[m]}\) being their ordered components.
Definition 2. For \(\mathbf{x} , \mathbf{y} \in \mathbb{R}^m\), we say \(\mathbf{x}\) is majorized by \(\mathbf{y}\) (denoted \(\mathbf{x} \prec \mathbf{y}\)) if \[\begin{cases} \sum\limits_{i=1}^{k}x_{[i]} \leq \sum\limits_{i=1}^{k}{y}_{[i]}, & k\in \{1,\ldots,m-1\}\\ \sum\limits_{i=1}^{m}x_{[i]}= \sum\limits_{i=1}^{m}{y}_{[i]}. & \end{cases}\]
The majorization theorem states the following.
Proposition 1(See [4, p.320]).Let \(\mathbf{ x}, \mathbf{ y} \in[\alpha_0, \alpha_1]^m\). The inequality \[\label{e6} \sum\limits_{i=1}^{m}\digamma(x_i)\leq \sum\limits_{i=1}^{m}\digamma({y}_i), \tag{1}\] holds for every continuous convex function \(\digamma: [\alpha_0,\alpha_1] \rightarrow \mathbb{R}\) if and only if \(\mathbf{ x} \prec \mathbf{ y}\).
We state some integral majorization results from [8].
Proposition 2(See [4, p.325]).Let \(x, {y}: [\alpha_0 , \alpha_1] \rightarrow I\) be two nonincreasing continuous functions and \(B_v\in BV[\alpha_0, \alpha_1]\). If \[\int_{\alpha_0}^{k} x({\tau}) dB_v({\tau}) \leq \int_{\alpha_0}^{k} {y}({\tau})dB_v({\tau}) \quad \text{{for all} } \quad k \in (\alpha_0, \alpha_1),\label{ee1} \tag{2}\] and \[\int_{\alpha_0}^{\alpha_1} x({\tau})dB_v({\tau})= \int_{\alpha_0}^{\alpha_1} {y}({\tau})dB_v({\tau}),\label{ee1.5} \tag{3}\] then for every continuous convex function \(\digamma:I\rightarrow \mathbb{R}\), we have \[\label{e7} \int_{\alpha_0}^{\alpha_1} \digamma(x({\tau}))dB_v({\tau})\leq \int_{\alpha_0}^{\alpha_1} \digamma({y}({\tau}))dB_v({\tau}). \tag{4}\]
Remark 1(See [3, p.584]).If \(x, {y} :[\alpha_0, \alpha_1]\rightarrow I\) are increasing continuous functions in Proposition 2, and if (3) is replaced by \[\int_{k}^{\alpha_1} x({\tau}) dB_v ({\tau}) \leq \int_{k}^{\alpha_1} {y}({\tau})dB_v({\tau}) \quad \text{{for all} } \quad k \in (\alpha_0, \alpha_1),\] then (4) also holds.
We now state the key tool for our generalizations, the extension of Montgomery’s identity via Taylor’s formula from [5]. Suppose \(\rho:[\beta_0, \beta_1]\rightarrow [0,\infty)\) is a pdf satisfying \(\int_{\beta_0}^{\beta_1} \rho({t}) d{t} = 1\) and define \(\wp({t})= \int_{\beta_0}^{{t}} \rho(x) d{x}\) for \({t} \in [\beta_0, \beta_1]\), with \(\wp({t})= 0\) for \({t} < \beta_0\) and \(\wp({t})=1\) for \({t} > \beta_1\).
Proposition 3 (See [5, Theorem 1]).Fix \(n\in \mathbb{N}\setminus\{1\}\). Let \(\digamma^{(n-1)}\in AC[\beta_0, \beta_1]\) and suppose \(\rho:[\beta_0, \beta_1]\rightarrow [0,\infty)\) is a pdf. Then we have \[\begin{aligned} \label{wtdmain} \digamma(x) =& \int_{\beta_0}^{\beta_1}\rho({t})\digamma({t})d{t} + \sum\limits_{i=0}^{n-2}\frac{\digamma^{(i+1)}(\beta_0)}{(i+1)!}\int_{\beta_0}^{x}\rho({u})((x-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})d{u}\notag\\ &+ \sum\limits_{i=0}^{n-2}\frac{\digamma^{(i+1)}(\beta_1)}{(i+1)!}\int_{x}^{\beta_1} \rho({u})((x-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})d{u}\notag\\ &+ \frac{1}{(n-1)!}\int_{\beta_0}^{\beta_1}K_{\rho,n}(x,{u})\digamma^{(n)}({u}) d{u}, \end{aligned} \tag{5}\] where \[\begin{aligned} \label{e23} K_{\rho,n}(x ,{u})= \begin{cases} \displaystyle\int_{x}^{{u}}\rho(\mu)(\mu-{u})^{n-1}d\mu + \wp(x)(x-{u})^{n-1}, & \beta_0\leq {u}\leq x,\\ \displaystyle\int_{x}^{{u}}\rho(\mu)(\mu-{u})^{n-1}d\mu +(\wp(x)-1)(x-{u})^{n-1}, & x < {u} \leq \beta_1. \end{cases} \end{aligned} \tag{6}\]
We refer to (5) as the generalized weighted Montgomery identity.
Remark 2. When \(\rho({t})=\frac{1}{\beta_1-\beta_0}\) for \({t} \in [\beta_0, \beta_1]\), (5) reduces to \[\begin{aligned} \label{e24} \digamma(x)=&\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1}\digamma({t}) d{t} + \sum\limits_{i=0}^{n-2}\frac{\digamma^{(i+1)}(\beta_0)}{i!(i+2)}\frac{(x-\beta_0)^{i+2}}{\beta_1-\beta_0}\notag\\ &- \sum\limits_{i=0}^{n-2}\frac{\digamma^{(i+1)}(\beta_1)} {i!(i+2)}\frac{(x-\beta_1)^{i+2}}{\beta_1-\beta_0}+ \frac{1}{(n-1)!}\int_{\beta_0}^{\beta_1}K_{n}(x,{u})\digamma^{(n)}({u}) d{u}, \end{aligned} \tag{7}\] where \[\begin{aligned} \label{e25} K_{n}(x ,{u})= \begin{cases} -\frac{(x-{u})^n}{n(\beta_1-\beta_0)} + \frac{x-\beta_0}{\beta_1-\beta_0}(x-{u})^{n-1}, & \beta_0 \leq {u} \leq x,\\ -\frac{(x-{u})^n}{n(\beta_1-\beta_0)} + \frac{x-\beta_1} {\beta_1-\beta_0}(x-{u})^{n-1}, & x < {u} \leq \beta_1. \end{cases} \end{aligned} \tag{8}\]
For \(n=1\), (7) becomes the well-known Montgomery identity (for details see [9] and [10]).
Theorem 1. Let all the assumptions of Proposition 3 be valid. Further let \(x, {y} : [\alpha_{0} , \alpha_{1}] \rightarrow [\beta_0, \beta_1]\) be two functions, and assume \(B_v \in\) BV \([\alpha_{0},\alpha_{1}]\). Then we have \[\begin{aligned} \label{3.1} \int_{\alpha_0}^{\alpha_1}&\digamma({y}({\tau}))dB_v({\tau})-\int_{\alpha_0}^{\alpha_1}\digamma(x({\tau}))dB_v({\tau})\notag\\ =&\sum\limits_{i=0}^{n-2}\frac{1}{(i+1)!}\left[\int_{\alpha_0}^{\alpha_1}\digamma^{(i+1)}(\beta_0)\left(\int_{\beta_0}^{y(\tau)}\rho({u})(({y}({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right.\right.\notag\\ &\left.-\int_{\beta_0}^{x(\tau)}\rho({u})((x({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right)d{u} dB_v({\tau})\notag\\ &\left.+\int_{\alpha_0}^{\alpha_1}\digamma^{(i+1)}(\beta_1)\left(\int_{y(\tau)}^{\beta_1}\rho({u})(({y}({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})\right.\right.\notag\\ &\left.\left.-\int_{x(\tau)}^{\beta_1}\rho({u})((x({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})\right)d{u} dB_v({\tau})\right]\notag\\ &+\frac{1}{(n-1)!}\int_{\beta_0}^{\beta_1}\int_{\alpha_0}^{\alpha_1}\left[K_{\rho,n}({y}({\tau}),{u})dB_v({\tau})-K_{\rho,n}(x({\tau}),{u})dB_v({\tau})\right]\digamma^{(n)}({u}) d{u}, \end{aligned} \tag{9}\] where \(K_{\rho,n}(\cdot,{u})\) is as defined in (6).
Proof. We use (5) to obtain \[\begin{aligned} \int_{\alpha_0}^{\alpha_1}&\digamma({y}({\tau}))dB_v({\tau})-\int_{\alpha_0}^{\alpha_1}\digamma(x({\tau}))dB_v({\tau})\\ =&\sum\limits_{i=0}^{n-2}\frac{\digamma^{(i+1)}(\beta_0)}{(i+1)!}\int_{\alpha_0}^{\alpha_1}\int_{\beta_0}^{y(\tau)}\rho({u})({y}({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})d{u} dB_v({\tau})\\ &+\sum\limits_{i=0}^{n-2}\frac{\digamma^{(i+1)}(\beta_1)}{(i+1)!}\int_{\alpha_0}^{\alpha_1}\int_{y(\tau)}^{\beta_1} \rho({u})({y}(({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})d{u} dB_v({\tau})\\ &-\sum\limits_{i=0}^{n-2}\frac{\digamma^{(i+1)}(\beta_0)}{(i+1)!}\int_{\alpha_0}^{\alpha_1}\int_{\beta_0}^{x(\tau)}\rho({u})(x({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})d{u} dB_v({\tau})\\ &-\sum\limits_{i=0}^{n-2}\frac{\digamma^{(i+1)}(\beta_1)}{(i+1)!}\int_{\alpha_0}^{\alpha_1}\int_{x(\tau)}^{\beta_1} \rho({u})(x(({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})d{u} dB_v({\tau})\\ &+\frac{1}{(n-1)!}\int_{\alpha_0}^{\alpha_1}\int_{\beta_0}^{\beta_1}K_{\rho,n}({y}({\tau}),{u})\digamma^{(n)}({u}) d{u} dB_v({\tau})\\ &-\frac{1}{(n-1)!}\int_{\alpha_0}^{\alpha_1}\int_{\beta_0}^{\beta_1}K_{\rho,n}(x({\tau}),{u})\digamma^{(n)}({u}) d{u} dB_v({\tau}). \end{aligned}\]
Then using Fubini’s theorem and after simplification, we get (9). ◻
Remark 3. By putting \(dB_v({\tau})=\rho({\tau})d{\tau}\) and \(\rho({u})=\frac{1}{\beta_1-\beta_0}\) for all \({u}\in [\beta_0,\beta_1]\) in Theorem 1, we get the following result, which is [8, Theorem 2].
Corollary 1. Under the assumptions of Theorem 1, we have the identity \[\begin{aligned} \int_{\alpha_0}^{\alpha_1}&\rho\left( {\tau}\right) \digamma({y}({\tau}))\,d{\tau}-\int_{\alpha_0}^{\alpha_1}\rho\left( {\tau}\right) \digamma(x({\tau}))\,d{\tau}\\ =&\frac{1}{\beta_1-\beta_0}\left[ \sum\limits_{k=0}^{n-2}\frac{1}{k!\left( k+2\right) } \int_{\alpha_0}^{\alpha_1}\rho\left( {\tau}\right) \left[ \digamma^{\left( k+1\right) }\left( \beta_0\right) \left[ \left( y({\tau})-\beta_0\right) ^{k+2}-\left( x({\tau})-\beta_0\right) ^{k+2}\right] \right. \right. \\ & \left. \left. -\digamma^{\left( k+1\right) }\left( \beta_1\right) \left[ \left( y\left( {\tau}\right) -\beta_1\right) ^{k+2}-\left( x\left( {\tau}\right) -\beta_1\right) ^{k+2}\right] \right] d{\tau}\right] \\ & +\frac{1}{\left( n-1\right) !}\int_{\beta_0}^{\beta_1}\left( \int_{\alpha_0}^{\alpha_1 }\rho\left( {\tau}\right) \left( K_{n}\left( y({\tau}),{u}\right) -K_{n}\left( x({\tau}),{u}\right) \right) d{\tau}\right) \digamma^{\left( n\right) }\left( s\right) d{u}, \end{aligned}\] where \(K_{n}{(\cdot,{u})}\) is as defined in (8).
Corollary 2. Fix \(m, n\in\mathbb{N}\setminus \{1\}\). Assume \(\digamma^{\left( n-1\right)}\in AC[\beta_0,\beta_1]\). Let \(x_{i},{y}_{i}\in\left[ \beta_0,\beta_1\right]\) and \(\rho_{i}\in\mathbb{R}\) for \(i\in\{1,2,\ldots,m\}\). Then \[\begin{aligned} \sum\limits_{i=1}^{m}\rho_i \digamma({y}_i)-\sum\limits_{i=1}^{m}\rho_i \digamma(x_i) = & \sum\limits_{k=0}^{n-2}\frac{1}{(k+1)!}% \left[\sum\limits_{i=1}^{m}\rho_i \digamma^{\left( k+1\right) }\left( \beta_0\right)\left[\int_{\beta_0}^{{y}_i}\rho({u}) \left[ \left( {y}_i-\beta_0\right) ^{k+1}-\left( {u}-\beta_0\right) ^{k+1}\right]d{u} \right.\right. \\ & \left.- \int_{\beta_0}^{x_i}\rho({u}) \left[ \left( x_i-\beta_0\right) ^{k+1}-\left( {u}-\beta_0\right) ^{k+1}\right]d{u} \right] \\ & +\sum\limits_{i=1}^{m}\rho_i \digamma^{\left( k+1\right) }\left( \beta_1\right)\left[\int_{{y}_i}^{\beta_1}\rho({u}) \left[ \left( {y}_i-\beta_1\right) ^{k+1}-\left( {u}-\beta_1\right) ^{k+1}\right]d{u} \right. \\ & \left.\left.- \int_{x_i}^{\beta_1}\rho({u}) \left[ \left( x_i-\beta_1\right) ^{k+1}-\left( {u}-\beta_1\right) ^{k+1}\right]d{u} \right] \right] \\ & +\frac{1}{\left( n-1\right) !}\int_{\beta_0}^{\beta_1}\left( \sum\limits_{i=1}^{m }\rho_i \left( K_{\rho,n}\left( {y}_i,{u}\right) -K_{\rho,n}\left( x_i,{u}\right) \right) \right) \digamma^{\left( n\right) }\left( s\right) d{u}, \end{aligned}\] where \(K_{\rho,n}(\cdot,{u})\) is as defined in (6).
Proof. The result follows from Theorem 1 by taking step functions. More concretely, for \(\alpha_0=1\) and \(\alpha_1=m+1\), we let \(x({\tau})=\sum^m_{i=1}x_{i}\,\chi_{[i,i+1)}({\tau})\), \(y({\tau})=\sum^m_{i=1}{y}_i\,\chi_{[i,i+1)}({\tau})\), \(w({\tau})=\sum^m_{i=1}w_i\,\chi_{[i,i+1)}({\tau})\), and \(dB_v({\tau}) = \sum\limits_{i=1}^m \rho_i\chi_{[i,i+1)}({\tau}) d{\tau}\). ◻
Remark 4. If in Corollary 2 we simply put \(\rho({u})=\frac{1}{\beta_1-\beta_0}\) for all \({u}\in [\beta_0,\beta_1]\), then we get [8, Theorem 1].
With the help of identity (9), we state the following generalization of the majorization inequality.
Theorem 2. Let all the assumptions of Theorem 1 hold with the condition \[\label{e30} \int_{\alpha_0}^{\alpha_1}K_{\rho,n}(x({\tau}),{u})dB_v({\tau})\leq \int_{\alpha_0}^{\alpha_1}K_{\rho,n}({y}({\tau}),{u})dB_v({\tau})\quad \text{for all}\quad {u} \in [\beta_0,\beta_1], \tag{10}\] where \(K_{\rho,n}(\cdot,{u})\) is as defined in (6). Then for every \(n\)-convex function \(\digamma: I\rightarrow \mathbb{R}\), we have \[\begin{aligned} \label{e31} \int_{\alpha_0}^{\alpha_1}&\digamma({y}({\tau}))dB_v({\tau})-\int_{\alpha_0}^{\alpha_1}\digamma(x({\tau}))dB_v({\tau}) \notag\\ \geq&\sum\limits_{i=0}^{n-2}\frac{1}{(i+1)!}\int_{\alpha_0}^{\alpha_1}\left[\digamma^{(i+1)}(\beta_0)\left[\int_{\beta_0}^{y(\tau)}\rho({u})(({y}({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right.\right. \notag\\ &\left.\left.-\int_{\beta_0}^{x(\tau)}\rho({u})((x({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right]d{u}\right] dB_v({\tau}) \notag\\ &\left.+\digamma^{(i+1)}(\beta_1)\left[\int_{y(\tau)}^{\beta_1}\rho({u})(({y}({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})\right.\right. \notag\\ &\left.\left.-\int_{x(\tau)}^{\beta_1}\rho({u})((x({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})\right]d{u}\right]dB_v({\tau}). \end{aligned} \tag{11}\]
Proof. Since \(\digamma\) is an \(n\)-convex function, and without loss of generality, assuming \(\digamma\) is \(n\)-times differentiable, we have \(\digamma^{(n)}\geq 0\). By putting (10) in (9), we get \[\begin{aligned} \int_{\alpha_0}^{\alpha_1}&\digamma({y}({\tau}))dB_v({\tau})-\int_{\alpha_0}^{\alpha_1}\digamma(x({\tau}))dB_v({\tau})\\ =&\sum\limits_{i=0}^{n-2}\frac{1}{(i+1)!}\int_{\alpha_0}^{\alpha_1}\left[\digamma^{(i+1)}(\beta_0)\left[\int_{\beta_0}^{y(\tau)}\rho({u})(({y}({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right.\right. \\ &\left.\left.-\int_{\beta_0}^{x(\tau)}\rho({u})((x({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right]d{u}\right] dB_v({\tau}) \\ &\left.+\digamma^{(i+1)}(\beta_1)\left[\int_{y(\tau)}^{\beta_1}\rho({u})(({y}({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})\right.\right. \\ &\left.\left.-\int_{x(\tau)}^{\beta_1}\rho({u})((x({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})\right]d{u}\right]dB_v({\tau}) \\ &+\frac{1}{(n-1)!}\int_{\beta_0}^{\beta_1}\left[\int_{\alpha_0}^{\alpha_1}K_{\rho,n}({y}({\tau}),{u})dB_v({\tau})-K_{\rho,n}(x({\tau}),{u})dB_v({\tau})\right]\digamma^{(n)}({u}) d{u}. \end{aligned}\]
Since the last term above is nonnegative by (10), we arrive at (11). ◻
Remark 5. If the reverse inequality holds in (10), then the reverse inequality holds in (11) as well.
Remark 6. If in Theorem 2, we simply put \(dB_v({\tau})=\rho({\tau})d{\tau}\) and \(\rho({u})=\frac{1}{\beta_1-\beta_0}\) for all \({u}\in [\beta_0,\beta_1]\), then we get the following result, which is in fact [8, Theorem 4].
Corollary 3. Under the assumptions of Theorem 2, we have \[\begin{aligned} \int_{\alpha_0}^{\alpha_1}&\rho\left( {\tau}\right) \digamma({y}({\tau}))\,d{\tau}-\int_{\alpha_0}^{\alpha_1}\rho\left( {\tau}\right) \digamma(x({\tau}))\,d{\tau}\\ \geq &\frac{1}{\beta_1-\beta_0}\left[ \sum\limits_{k=0}^{n-2}\frac{1}{k!\left( k+2\right) }% \int_{\alpha_0}^{\alpha_1}\rho\left( {\tau}\right) \left[ \digamma^{\left( k+1\right) }\left( \beta_0\right) \left[ \left( y({\tau})-\beta_0\right) ^{k+2}-\left( x({\tau})-\beta_0\right) ^{k+2}\right] \right. \right.\\ & \left. \left. -\digamma^{\left( k+1\right) }\left( \beta_1\right) \left[ \left( y\left( {\tau}\right)-\beta_1\right) ^{k+2}-\left( x\left( {\tau}\right) -\beta_1\right) ^{k+2}\right] \right] d{\tau}\right]. \end{aligned}\]
Corollary 4. Fix \(m,n\in\mathbb{N}\setminus\{1\}\). Let \(\digamma^{\left( n-1\right)}\in AC[\beta_0<\beta_1]\). Further let \(x_{i},{y}_{i}\in\left[ \beta_0,\beta_1\right]\) and \(\rho_{i}\in\mathbb{R}\) for \(i\in\{1,2,\ldots,m\}\). Then \[\begin{aligned} \sum\limits_{i=1}^{m}&\rho_i \digamma({y}_i)-\sum\limits_{i=1}^{m}\rho_i \digamma(x_i)\\ \geq & \sum\limits_{k=0}^{n-2}\frac{1}{(k+1)!}% \left[\sum\limits_{i=1}^{m}\rho_i \digamma^{\left( k+1\right) }\left( \beta_0\right)\left[\int_{\beta_0}^{{y}_i}\rho({u}) \left[ \left( {y}_i-\beta_0\right) ^{k+1}-\left( {u}-\beta_0\right) ^{k+1}\right]d{u} \right.\right. \\ & \left.- \int_{\beta_0}^{x_i}\rho({u}) \left[ \left( x_i-\beta_0\right) ^{k+1}-\left( {u}-\beta_0\right) ^{k+1}\right]d{u} \right] \\ & +\sum\limits_{i=1}^{m}\rho_i \digamma^{\left( k+1\right) }\left( \beta_1\right)\left[\int_{{y}_i}^{\beta_1}\rho({u}) \left[ \left( {y}_i-\beta_1\right) ^{k+1}-\left( {u}-\beta_1\right) ^{k+1}\right]d{u} \right. \\ & \left.\left.- \int_{x_i}^{\beta_1}\rho({u}) \left[ \left( x_i-\beta_1\right) ^{k+1}-\left( {u}-\beta_1\right) ^{k+1}\right]d{u} \right] \right]. \end{aligned}\]
Remark 7. If in Corollary 4, we simply put \(\rho({u})=\frac{1}{\beta_1-\beta_0}\) for all \({u}\in [\beta_0,\beta_1]\), then we get [8, Theorem 3].
We now state an important consequence as follows.
Corollary 5. Suppose all assumptions of Theorem \(\ref{t1}\) hold true. Additionally assume that \(x\) and \({y}\) are nonincreasing and satisfy (2) and (3). If \(f\) is \(2n\)-convex, then we have \[\begin{aligned} \label{e34} \int_{\alpha_0}^{\alpha_1}&\digamma({y}({\tau}))dB_v({\tau})-\int_{\alpha_0}^{\alpha_1}\digamma(x({\tau}))dB_v({\tau}) \notag\\ \geq&\sum\limits_{i=0}^{2n-2}\frac{1}{(i+1)!}\int_{\alpha_0}^{\alpha_1}\left[\digamma^{(i+1)}(\beta_0)\left[\int_{\beta_0}^{y(\tau)}\rho({u})(({y}({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right.\right. \notag\\ &\left.\left.-\int_{\beta_0}^{x(\tau)}\rho({u})((x({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right]d{u}\right] dB_v({\tau}) \notag\\ &\left.+\digamma^{(i+1)}(\beta_1)\left[\int_{y(\tau)}^{\beta_1}\rho({u})(({y}({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})\right.\right. \notag\\ &\left.\left.-\int_{x(\tau)}^{\beta_1}\rho({u})((x({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})\right]d{u}\right]dB_v({\tau}). \end{aligned} \tag{12}\]
Moreover, if \(\digamma^{(k)}(\beta_0)\geq 0\) and \((-1)^{k}\digamma^{(k)}(\beta_1)\geq 0\) for \(k \in \{1, \ldots, 2 n – 1\}\), then \[\label{e35} \int_{\alpha_0}^{\alpha_1}\digamma({y}({\tau}))dB_v({\tau})\geq \int_{\alpha_0}^{\alpha_1}\digamma(x({\tau}))dB_v({\tau}). \tag{13}\]
Proof. Starting with the definition of the kernel \(\mathrm{K}_{\rho,n}(\cdot,u)\) from (6), we obtain \[\begin{aligned} \frac{d^{2}}{dx^{2}} \mathrm{K}_{\rho,n}(x,u) = \begin{cases} (n-1)\rho(x)(x-u)^{\,n-2} + (n-1)(n-2)\wp(x)(x-u)^{\,n-3}, & \beta_0 \le u \le x \le \beta_1, \\[2mm] (n-1)\rho(x)(x-u)^{\,n-2} + (n-1)(n-2)\big(\wp(x)-1\big)(x-u)^{\,n-3}, & \beta_0 \le x \le u \le \beta_1 . \end{cases} \end{aligned}\]
Since \(\mathrm{K}_{\rho,n}(\cdot,u)\) is continuous for every \(n \ge 2\), and convex whenever \(n\) is even, it satisfies (10) by the generalized majorization theorem (Proposition 2). Consequently, applying Theorem 2 to (10) yields (11) with \(2n\) in place of \(n\). Moreover, we impose the conditions \[\digamma^{(k)}(\beta_0) \geq 0 \quad \text{and} \quad (-1)^{k}\digamma^{(k)}(\beta_1) \geq 0 \quad \text{for all} \quad \, k \in \{1,\dots,2n-1\}.\]
Using Proposition 2 with the continuous convex function \[\digamma(x) = (x – \beta_0)^{i+2}, \quad x \in [\beta_0,\beta_1],\] we obtain \[\label{e38} \int_{\alpha_0}^{\alpha_1}({y}({\tau})-\beta_0)^{i+2}dB_v({\tau})\geq \int_{\alpha_0}^{\alpha_1}(x({\tau})-\beta_0)^{i+2}dB_v({\tau}).\]
Since the continuous function \(\digamma\) is convex on \(x\in [\beta_0,\beta_1]\) if \(i\) is even and concave if \(i\) is odd, by Proposition 2, we have \[\begin{aligned}\label{e38.1} &\int_{\alpha_0}^{\alpha_1}({y}({\tau})-\beta_1)^{i+2}dB_v({\tau})\geq \int_{\alpha_0}^{\alpha_1}(x({\tau})-\beta_1)^{i+2}dB_v({\tau}) \quad \text{if $i$ is even},\\ &\int_{\alpha_0}^{\alpha_1}({y}({\tau})-\beta_1)^{i+2}dB_v({\tau})\leq \int_{\alpha_0}^{\alpha_1}(x({\tau})-\beta_1)^{i+2}dB_v({\tau}) \quad \text{if $i$ is odd}. \end{aligned}\]
Now considering the assumptions \(\digamma^{(k)}(\beta_1) \geq 0\) and \((-1)^{k}\digamma^{(k)}(\beta_0)\) \(\geq 0\) for all \(k \in \{1,\ldots 2n-1\}\), we have that \[\begin{aligned}%\label{e39} \int_{\alpha_0}^{\alpha_1}&\left[\digamma^{(i+1)}(\beta_0) \left[\int_{\beta_0}^{y({\tau})}\left(\rho({u})(({y}({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right)d{u} \right.\right. \\ &\left.\left.-\int_{\beta_0}^{x({\tau})}\rho({u}) \left((x({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1}\right)d{u}\right] +\digamma^{i+1}(\beta_1)\left[\int_{y({\tau})}^{\beta_1}\rho({u})(({y}({\tau})-\beta_1)^{i+1}\right.\right. \\ &\left.\left.-({u}-\beta_1)^{i+1})d{u}-\int_{x({\tau})}^{\beta_1} \rho({u})((x({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})d{u}\right]\right]dB_v({\tau}) \\ =& \digamma^{(i+1)}(\beta_0) \int_{\alpha_0}^{\alpha_1} \left[\int_{\beta_0}^{y(\tau)}\left(\rho({u})(({y}({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right)d{u}\right. \\ &\left.-\int_{\beta_0}^{x({\tau})} \rho({u})\left((x({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1}\right)d{u}\right]dB_v({\tau}) \\ &+\digamma^{i+1}(\beta_1)\int_{\alpha_0}^{\alpha_1}\left[\int_{y({\tau})}^{\beta_1}\rho({u}) (({y}({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})d{u}\right. \\ &\left.-\int_{x({\tau})}^{\beta_1}\rho({u})((x({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})d{u}\right]dB_v({\tau}) , \end{aligned}\] is positive for all \(i \in \{0,1,\ldots,2n-2\}\). Therefore, the right-hand side of (12) is positive and (13) holds. ◻
Remark 8. Since, in the case \(\beta_0 \leq {u} \leq x \leq \beta_1\), the second derivative \[\frac{d^{2}}{d{x}^{2}}K_{\rho,n}(x,{u}),\] is always positive, the function \(K_{\rho,n}(\cdot,{u})\) cannot be concave, and therefore the reverse inequalities do not hold. Furthermore, analogous corollaries and remarks can be formulated, corresponding to Corollary 1, Corollary 2, and Remark 4 for Theorem 1, as well as Corollary 3, Corollary 4, and Remark 7 for Theorem 2. These results also allow us to capture parallel findings to those presented in [8].
If \(g, h: [\beta_0,\beta_1]\rightarrow \mathbb{R}\) are two Lebesgue integrable functions, then we consider the Čebyšev functional [11] \[\label{e10} T(g,h)=\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1} g(x) h(x) d{x}- \left(\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1} g(x)d{x}\right)\left(\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1} h(x)d{x}\right). \tag{14}\]
Proposition 4(See [12]).If \(g: [\beta_0,\beta_1]\rightarrow \mathbb{R}\) is Lebesgue integrable and \(h\in BV[\beta_0,\beta_1]\) with \((\cdot-\beta_0)(\beta_1- \cdot)[h']^{2}\in L[\beta_0,\beta_1]\), then \[\label{e11} |T(g,h)|\leq \frac{1}{\sqrt{2}}\left(\frac{1}{\beta_1-\beta_0}|T(g,g)|\int_{\beta_0}^{\beta_1}(x-\beta_0)(\beta_1-x)[h'(x)]^{2}d{x} \right)^{\frac{1}{2}}. \tag{15}\]
The constant \(\frac{1}{\sqrt{2}}\) in (15) is the best possible.
Proposition 5. If \(g\in BV[\beta_0,\beta_1]\) such that \(g'\in L_{\infty}[\beta_0,\beta_1]\) and \(h: [\beta_1,\beta_0]\rightarrow \mathbb{R}\) is monotonic nondecreasing, then \[\label{e12} |T(g,h)|\leq \frac{1}{2}||g' ||_{\infty}\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1}(x-\beta_0)(\beta_1-x)dh(x). \tag{16}\]
The constant \(\frac{1}{2}\) in (16) is the best possible.
Now by using \(\mathbf{w}=(\rho_1,\ldots,\rho_m)\), \(\mathbf{ x}=(x_1,\ldots,x_m)\) and \(\mathbf{ y}=({y}_1,\ldots,{y}_m)\) with \(x_i,{y}_i \in [\beta_0,\beta_1]\), \(\rho_i \in \mathbb{R}\) for \(i\in \{1,\ldots,m\}\), and \(K_{\rho,n}\) defined as in (6), denote \[\triangle_1({u})= \sum\limits_{i=1}^{m}\rho_i K_{\rho,n}({y}_i,{u})- \sum\limits_{i=1}^{m}\rho_i K_{\rho,n}(x_i,{u}) \quad \text{for all} \quad {u} \in [\beta_0,\beta_1],\] and \[\triangle({u})= \sum\limits_{i=1}^{m}K_{\rho,n}({y}_i,{u})- \sum\limits_{i=1}^{m}K_{\rho,n}(x_i,{u}) \quad \text{for all} \quad {u} \in [\beta_0,\beta_1].\]
Similarly for continuous functions \(x,{y}:[\alpha_0,\alpha_1]\rightarrow[\beta_0,\beta_1]\) and \(\rho:[\alpha_0,\alpha_1]\rightarrow \mathbb{R}\), we let \[\label{e69} \gamma_1({u})= \int_{\alpha_0}^{\alpha_1}w({t})K_{\rho,n}({y}({t}),{u})d{t}- \int_{\alpha_0}^{\alpha_1}w({t})K_{\rho,n}(x({t}),{u})d{t} \quad \text{for all} \quad s \in [\beta_0,\beta_1], \tag{17}\] and \[\label{e69.1} \gamma({u})= \int_{\alpha_0}^{\alpha_1}K_{\rho,n}({y}({\tau}),{u})dB_v({\tau})- \int_{\alpha_0}^{\alpha_1}K_{\rho,n}(x({\tau}),{u})dB_v({\tau})\quad \text{for all} \quad s \in [\beta_0,\beta_1]. \tag{18}\]
Using (17), we define a Čebyšev functional by \[T(\gamma,\gamma)= \frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1}\gamma^{2}({u})d{u} – \left(\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1}\gamma({u})d{u}\right)^{2}.\]
We now present bounds for the remainders in the new generalizations of the weighted majorization inequality.
Theorem 3. Let \(\digamma\in C^{n}[\beta_0,\beta_1]\) for \(n \in \mathbb{N}\) with \((\cdot-\beta_0)(\beta_1-\cdot)[\digamma^{(n+1)}]^{2}\) \(\in L [\beta_0,\beta_1]\) and \(B_v \in BV[\alpha_0,\alpha_1]\) and \(x,{y} :[\alpha_0,\alpha_1]\) \(\rightarrow\) \([\beta_0,\beta_1]\). Suppose the functions \(K_{\rho,n}\), \(T\) and \(\gamma\) are defined in (6), (14) and (18) respectively. Then we have \[\begin{aligned} \label{e71} \int_{\alpha_0}^{\alpha_1}&\digamma({y}({\tau}))dB_v({\tau})-\int_{\alpha_0}^{\alpha_1}\digamma(x({\tau}))dB_v({\tau}) \notag\\ =&\sum\limits_{i=0}^{n-2}\frac{1}{(i+1)!}\left[\int_{\alpha_0}^{\alpha_1}\digamma^{(i+1)}(\beta_0)\left(\int_{\beta_0}^{y(\tau)}\rho({u})(({y}({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right.\right. \notag\\ &\left.-\int_{\beta_0}^{x(\tau)}\rho({u})((x({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})\right)d{u} dB_v({\tau}) \notag\\ &\left.+\int_{\alpha_0}^{\alpha_1}\digamma^{(i+1)}(\beta_1)\left(\int_{y(\tau)}^{\beta_1}\rho({u})(({y}({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})\right.\right. \notag \\ &\left.\left.-\int_{x(\tau)}^{\beta_1}\rho({u})((x({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})\right)d{u} dB_v({\tau})\right] \notag\\ &+\frac{\digamma^{(n-1)}(\beta_1)-\digamma^{(n-1)}(\beta_0)}{(n-1)!(\beta_1-\beta_0)}\int_{\beta_0}^{\beta_1}\gamma({u})d{u}+ R^{1}_n(f; \beta_0,\beta_1), \end{aligned} \tag{19}\] where the remainder \(R^{1}_n(f; \beta_0,\beta_1)\) satisfies the estimate \[\begin{aligned}\label{e72} \left|R^{1}_n(f; \beta_0,\beta_1)\right|\leq \frac{1}{(n-1)!}\left(\left|\frac{\beta_1-\beta_0}{2}T(\gamma,\gamma)\int_{\beta_0}^{\beta_1}({u} – \beta_0)(\beta_1-{u})[\digamma^{(n+1)}({u})]^{2}\right|\right)^{1/2}. \end{aligned} \tag{20}\]
Proof. If we apply Proposition 4 for \(g\rightarrow\gamma\) and \(h\rightarrow \digamma^{n}\), then we obtain \[\begin{aligned}\label{e73} &\left|\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1} g(x) h(x)d{x}-\left(\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1} g(x)d{x}\right)\left(\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1} h(x)d{x}\right)\right| \\ &\qquad\leq\frac{1}{\sqrt{2}}\left(\left|T(\gamma,\gamma)\right|\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1} ({u}-\beta_0)(\beta_1-{u})[\digamma^{(n+1)}({u})]^{2}d{u}\right)^{\frac{1}{2}}, \end{aligned}\] and therefore we have \[\begin{aligned}\label{e74} \frac{1}{(n-1)!}\int_{\beta_0}^{\beta_1}\gamma({u}) \digamma^{(n)}({u}) d{u}= \frac{[\digamma^{(n-1)}(\beta_1)-\digamma^{(n-1)}(\beta_0)]}{(\beta_1-\beta_0)(n-1)!}\int_{\beta_0}^{\beta_1}\gamma({u})d{u}+ R^{1}_n(f; \beta_0,\beta_1), \end{aligned}\] where \(R^{1}_n(f;\beta_0, \beta_1)\) satisfies (20). Now from (9), we obtain (19). ◻
Remark 9. If in Theorem 3, we simply put \(dB_v({\tau})=\rho({\tau})d{\tau}\) and \(\rho({u})=\frac{1}{\beta_1-\beta_0}\) for all \({u}\in [\beta_0,\beta_1]\), then we get the following result of [8, Theorem 6].
Corollary 6. Under the assumptions of Theorem 3, we have \[\begin{aligned} \int_{\alpha_0}^{\alpha_1}&\rho\left( {\tau}\right) \digamma({y}({\tau}))\,d{\tau}-\int_{\alpha_0}^{\alpha_1}\rho\left( {\tau}\right) \digamma(x({\tau}))\,d{\tau}\\ = &\frac{1}{\beta_1-\beta_0}\left[ \sum\limits_{k=0}^{n-2}\frac{1}{k!\left( k+2\right) }% \int_{\alpha_0}^{\alpha_1}\rho\left( {\tau}\right) \left[ \digamma^{\left( k+1\right) }\left( \beta_0\right) \left[ \left( y({\tau})-\beta_0\right) ^{k+2}-\left( x({\tau})-\beta_0\right) ^{k+2}\right] \right. \right. \\ & \left. \left. -\digamma^{\left( k+1\right) }\left( \beta_1\right) \left[ \left( y\left( {\tau}\right) -\beta_1\right) ^{k+2}-\left( x\left( {\tau}\right) -\beta_1\right) ^{k+2}\right] \right] d{\tau}\right] \\ &+\frac{\digamma^{(n-1)}(\beta_1)-\digamma^{(n-1)}(\beta_0)}{(n-1)!(\beta_1-\beta_0)}\int_{\beta_0}^{\beta_1}\gamma_1({u})d{u}+ R^{2}_n(f; \beta_0,\beta_1), \end{aligned}\] where the remainder \(R^{1}_n(f; \beta_0,\beta_1)\) satisfies the estimate \[\begin{aligned} \left|R^{2}_n(f; \beta_0,\beta_1)\right|\leq \frac{1}{(n-1)!}\left(\left|\frac{\beta_1-\beta_0}{2}T(\gamma_1,\gamma_1)\int_{\beta_0}^{\beta_1}({u} – \beta_0)(\beta_1-{u})[\digamma^{(n+1)}({u})]^{2}\right|\right)^{1/2}. \end{aligned}\]
Corollary 7. Let \(\digamma\in C^{n}[\beta_0,\beta_1]\) for \(n \in \mathbb{N}\) with \((\cdot-\beta_0)(\beta_1-\cdot)[\digamma^{(n+1)}]^{2}\in L [\beta_0,\beta_1]\). Additionally suppose that \(x_{i},{y}_{i}\in\left[ \beta_0,\beta_1\right]\) and \(\rho_{i}\in\mathbb{R}\) for \(i\in\{1,2,\ldots,m\}\). Suppose the functions \(K_{n}\), \(T\) and \(\gamma\) are defined in (8),(14) and (18) respectively. Then we have
\[\begin{aligned} \sum\limits_{i=1}^{m}\rho_i \digamma({y}_i)-\sum\limits_{i=1}^{m}\rho_i \digamma(x_i) = & \sum\limits_{k=0}^{n-2}\frac{1}{(k+1)!}% \left[\sum\limits_{i=1}^{m}\rho_i \digamma^{\left( k+1\right) }\left( \beta_0\right)\left[\int_{\beta_0}^{{y}_i}\rho({u}) \left[ \left( {y}_i-\beta_0\right) ^{k+1}-\left( {u}-\beta_0\right) ^{k+1}\right]d{u} \right.\right. \\ & \left.- \int_{\beta_0}^{x_i}\rho({u}) \left[ \left( x_i-\beta_0\right) ^{k+1}-\left( {u}-\beta_0\right) ^{k+1}\right]d{u} \right] \\ & +\sum\limits_{i=1}^{m}\rho_i \digamma^{\left( k+1\right) }\left( \beta_1\right)\left[\int_{{y}_i}^{\beta_1}\rho({u}) \left[ \left( {y}_i-\beta_1\right) ^{k+1}-\left( {u}-\beta_1\right) ^{k+1}\right]d{u} \right. \\ & \left.\left.- \int_{x_i}^{\beta_1}\rho({u}) \left[ \left( x_i-\beta_1\right) ^{k+1}-\left( {u}-\beta_1\right) ^{k+1}\right]d{u} \right] \right] \\ &+\frac{\digamma^{(n-1)}(\beta_1)-\digamma^{(n-1)}(\beta_0)}{(n-1)!(\beta_1-\beta_0)}\int_{\beta_0}^{\beta_1}\triangle_1({u})d{u}+ R^{3}_n(f; \beta_0,\beta_1), \end{aligned}\] where the remainder \(R^{3}_n(f; \beta_0,\beta_1)\) satisfies the estimate \[\begin{aligned} \left|R^{3}_n(f; \beta_0,\beta_1)\right|\leq \frac{1}{(n-1)!}\left(\left|\frac{\beta_1-\beta_0}{2}T(\triangle_1,\triangle_1)\int_{\beta_0}^{\beta_1}({u} – \beta_0)(\beta_1-{u})[\digamma^{(n+1)}({u})]^{2}\right|\right)^{1/2}. \end{aligned}\]
Remark 10. If in Corollary 7, we simply put \(\rho({u})=\frac{1}{\beta_1-\beta_0}\) for \({u}\in [\beta_0,\beta_1]\), then we get [8, Theorem 5].
By using Proposition 5, we acquire a Grüss-type inequality as follows.
Theorem 4. Suppose \(\digamma\in C^{n}[\beta_0, \beta_1]\) for n \(\in\) \(\mathbb{N}\) with \(\digamma^{(n+1)}\) \(\geq0\) on \([\beta_0, \beta_1]\) and suppose the functions T and \(\gamma\) are given in (14) and (18) respectively. Then we have the representation (19) and the remainder \(R^{4}_n\) \((f,\beta_0, \beta_1)\) satisfies the condition \[\begin{aligned}\label{e75} &\left|R^{4}_n(f;\beta_0,\beta_1)\right|\leq \frac{1}{(n-1)!}||\gamma^{'}||_\infty \left[\frac{\beta_1-\beta_0}{2}\left(\digamma^{(n-1)}(\beta_1)+\digamma^{(n-1)}(\beta_0)\right) -\left(\digamma^{(n-2)}(\beta_1)-\digamma^{(n-2)}(\beta_0)\right)\right]. \end{aligned} \tag{21}\]
Proof. If we apply Proposition 5 for \(g \rightarrow \gamma\) and \(h \rightarrow \digamma^{(n)}\), then we obtain \[\begin{aligned}%\label{e76} &\left|\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1} g(x) h(x)d{x}-\left(\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1} g(x)d{x}\right) \left(\frac{1}{\beta_1-\beta_0}\int_{\beta_0}^{\beta_1} h(x)d{x}\right)\right| \\ &\qquad\leq \frac{1}{2(\beta_1-\beta_0)}||\gamma^{'}||_\infty \int_{\beta_0}^{\beta_1}({u}-\beta_0)(\beta_1-{u})\digamma^{(n+1)}({u})d{u}. \end{aligned}\]
Note \[\begin{aligned} \label{e77} \int_{\beta_0}^{\beta_1}({u}-\beta_0)(\beta_1-{u})\digamma^{(n+1)}({u})d{u} &= \int_{\beta_0}^{\beta_1}(2{u}-(\beta_1-\beta_0))\digamma^{(n)}({u})\notag\\ &=(\beta_1-\beta_0)\left[\digamma^{(n-1)}(\beta_1)+\digamma^{(n-1)}(\beta_0)\right]-2\left[\digamma^{(n-2)}(\beta_1)-\digamma^{(n-2)}(\beta_0)\right]. \end{aligned} \tag{22}\]
Therefore, by using (9) and (22), we deduce (21). ◻
Remark 11. If in Theorem 4, we simply put \(dB_v({\tau})=\rho({\tau})d{\tau}\) and \(\rho({u})=\frac{1}{\beta_1-\beta_0}\) for all \({u}\in [\beta_0,\beta_1]\), then we get the following result of [8, Theorem 7].
Corollary 8. Let \(x,{y}:[\alpha_{0} , \alpha_{1}] \rightarrow [\beta_0, \beta_1]\) be two functions and assume \(\rho:[\alpha_0,\alpha_1]\rightarrow \mathbb{R}\) is a continuous function. If \(\digamma^{\left( n-1\right)}\in BV[\beta_0,\beta_1]\), then for all \({u}\in\lbrack \beta_0,\beta_1]\), we have \[\begin{aligned}%\label{c61} &\left|R^{5}_n(f;\beta_0,\beta_1)\right|\leq \frac{1}{(n-1)!}||\gamma_1^{'}||_\infty \left[\frac{\beta_1-\beta_0}{2}\left(\digamma^{(n-1)}(\beta_1)+\digamma^{(n-1)}(\beta_0)\right) -\left(\digamma^{(n-2)}(\beta_1)-\digamma^{(n-2)}(\beta_0)\right)\right]. \end{aligned}\]
Corollary 9. Fix \(n \in\mathbb{N}\setminus \{1\}\). Let \(\digamma^{\left( n-1\right)} \in AC[\beta_0,\beta_1]\). Further let \(m\in\mathbb{N}\), \(x_{i},{y}_{i}\in\left[ \beta_0,\beta_1\right]\) and \(\rho_{i}\in\mathbb{R}\) for \(i\in\{1,2,\ldots,m\}\). Then the remainder \(R^{6}_n(f; \beta_0,\beta_1)\) satisfies the estimate \[\begin{aligned}%\label{c62} & \left|R^{6}_n(f;\beta_0,\beta_1)\right|\leq \frac{1}{(n-1)!}||\triangle_1^{'}||_\infty \left[\frac{\beta_1-\beta_0}{2}\left(\digamma^{(n-1)}(\beta_1)+\digamma^{(n-1)}(\beta_0)\right) -\left(\digamma^{(n-2)}(\beta_1)-\digamma^{(n-2)}(\beta_0)\right)\right]. \end{aligned}\]
Remark 12. If in Corollary 9, we simply put \(\rho({u})=\frac{1}{\beta_1-\beta_0}\) for \({u}\in [\beta_0,\beta_1]\), then we get [8, Theorem 8].
On the interval \([\beta_0,\beta_1]\), the space of \(q\)-power integrable functions is symbolised by \(L_q[\beta_0,\beta_1]\), \(1 \leq q <\infty\), where the norm is given by \[|| f ||_q = \left(\int_{\beta_0}^{\beta_1}|\digamma({\tau})|^{q}d {\tau}\right)^{\frac{1}{q}},\] and on \([\beta_0, \beta_1]\), the space of essentially bounded functions is symbolised by \(L_{\infty}[\beta_0,\beta_1]\) with the norm \[||f ||_{\infty}=\mathop{\mathrm{ess\,\, sup} }\limits_{{\tau}\in[\beta_0,\beta_1]}|\digamma({\tau})|.\]
Now we present some Ostrowski-type inequalities related to the generalized majorization inequalities.
Theorem 5. Suppose all the assumptions of Theorem 1 hold. Also, let \((q,r)\) be a pair of conjugate exponents, i.e., \(1\leq\) q ,r \(\leq \infty\), \(\frac{1}{q}+ \frac{1}{r}= 1\). Let \(\digamma^{(n)}\in L_q[\beta_0, \beta_1]\) for some \(n \in \mathbb{N}\setminus\{1\}\). Then we have \[\begin{aligned} \label{e78} &\left|\int_{\alpha_0}^{\alpha_1}\digamma({y}({\tau}))dB_v({\tau})-\int_{\alpha_0}^{\alpha_1}\digamma(x({\tau}))dB_v({\tau})\right.\notag\\ &-\left.\sum\limits_{i=0}^{n-2}\frac{1}{(i+1)!}\int_{\alpha_0}^{\alpha_1}\left[\digamma^{(i+1)}(\beta_0)\left[\int_{\beta_0}^{y(\tau)}\rho({u})(({y}({\tau})-\beta_0)^{i+1} -({u}-\beta_0)^{i+1})d{u}\right.\right.\right. \notag\\ &\left.\left.\left.-\int_{\beta_0}^{x(\tau)}\rho({u})((x({\tau})-\beta_0)^{i+1}-({u}-\beta_0)^{i+1})d{u}\right]\right.\right. \notag\\ &\left.\left.-\digamma^{(i+1)}(\beta_1)\left[\int_{y(\tau)}^{\beta_1}\rho({u})(({y}({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})d{u} \right.\right.\right. \notag\\ &\left.\left.\left.- \int_{x(\tau)}^{\beta_1}\rho({u})((x({\tau})-\beta_1)^{i+1}-({u}-\beta_1)^{i+1})\right]d{u}\right]dB_v({\tau})\right| \notag\\ &\qquad\leq \frac{1}{(n-1)!}||\digamma^{(n)}\|_q\left\|\int_{\alpha_0}^{\alpha_1}[K_{\rho,n}({y}({\tau}),{u})- K_{\rho,n}(x({\tau},{u}))]dB_v({\tau})\right\|_r. \end{aligned} \tag{23}\]
The constant on the right-hand side of (23) is sharp for \(1< q \leq \infty\) and the best possible for \(q = 1\).
Remark 13. For the idea of the proof, we refer the reader to [13].
Remark 14. If in Theorem 5, we simply put \(dB_v({\tau})=\rho({\tau})d{\tau}\) and \(\rho({u})=\frac{1}{\beta_1-\beta_0}\) for all \({u}\in [\beta_0,\beta_1]\), then we get the following result which is, in fact, [8, Theorem 10].
Corollary 10. Let \(x,{y}:[\alpha_{0} , \alpha_{1}] \rightarrow [\beta_0, \beta_1]\) be two functions and assume \(\rho:[\alpha_0,\alpha_1]\rightarrow \mathbb{R}\) is a continuous function. If \(\digamma:I\rightarrow\mathbb{R}\) is such that \(\digamma^{\left( n-1\right) }\) is absolutely continuous for some \(n\in\mathbb{N}\), \(I\subset\mathbb{R}\) an open interval, \(\beta_0,\beta_1\in I\), \(\beta_0<\beta_1\), then for all \({u}\in\lbrack \beta_0,\beta_1]\), we have the identity \[\begin{aligned} & \left| \int_{\alpha_0}^{\alpha_1}\rho\left( {\tau}\right) \digamma({y}({\tau}))\,d{\tau}-\int_{\alpha_0}^{\alpha_1}\rho\left( {\tau}\right) \digamma(x({\tau}))\,d{\tau}\right.\\ & -\frac{1}{\beta_1-\beta_0}\left[ \sum\limits_{k=0}^{n-2}\frac{1}{k!\left( k+2\right) }% \int_{\alpha_0}^{\alpha_1}\rho\left( {\tau}\right) \left[ \digamma^{\left( k+1\right) }\left( \beta_0\right) \left[ \left( y({\tau})-\beta_0\right) ^{k+2}-\left( x({\tau})-\beta_0\right) ^{k+2}\right] \right. \right. \\ & \left. \left. \left. -\digamma^{\left( k+1\right) }\left( \beta_1\right) \left[ \left( y\left( {\tau}\right) -\beta_1\right) ^{k+2}-\left( x\left( {\tau}\right) -\beta_1\right) ^{k+2}\right] \right] d{\tau}\right]\right| \\ &\qquad\leq \frac{1}{(n-1)!}||\digamma^{(n)}\|_q\left\|\int_{\alpha_0}^{\alpha_1}[K_{n}({y}({\tau}),{u})- K_{n}(x({\tau},{u}))]dB_v({\tau})\right\|_r, \end{aligned}\] where \(K_{n}{(\cdot,{u})}\) is as defined in (8).
Corollary 11. Let \(m,n\in\mathbb{N}\setminus\{1\}\). Let \(\digamma^{\left( n-1\right)} AC[\beta_0,\beta_1]\). Additionally, suppose that \(x_{i},{y}_{i}\in\left[ \beta_0,\beta_1\right]\) and \(\rho_{i}\in\mathbb{R}\) for \(i\in\{1,2,\ldots,m\}\). Then \[\begin{aligned} & \left| \sum\limits_{i=1}^{m}\rho_i \digamma({y}_i)-\sum\limits_{i=1}^{m}\rho_i \digamma(x_i)\right. – \sum\limits_{k=0}^{n-2}\frac{1}{(k+1)!}% \left[\sum\limits_{i=1}^{m}\rho_i \digamma^{\left( k+1\right) }\left( \beta_0\right)\left[\int_{\beta_0}^{{y}_i}\rho({u}) \left[ \left( {y}_i-\beta_0\right) ^{k+1}-\left( {u}-\beta_0\right) ^{k+1}\right]d{u} \right.\right. \\ & \left.- \int_{\beta_0}^{x_i}\rho({u}) \left[ \left( x_i-\beta_0\right) ^{k+1}-\left( {u}-\beta_0\right) ^{k+1}\right]d{u} \right] -\sum\limits_{i=1}^{m}\rho_i \digamma^{\left( k+1\right) }\left( \beta_1\right)\left[\int_{{y}_i}^{\beta_1}\rho({u}) \left[ \left( {y}_i-\beta_1\right) ^{k+1}-\left( {u}-\beta_1\right) ^{k+1}\right]d{u} \right. \\ & \left.\left.\left.- \int_{x_i}^{\beta_1}\rho({u}) \left[ \left( x_i-\beta_1\right) ^{k+1}-\left( {u}-\beta_1\right) ^{k+1}\right]d{u} \right] \right]\right| \\ &\qquad\qquad\leq \frac{1}{(n-1)!}||\digamma^{(n)}\|_q\left\|\int_{\alpha_0}^{\alpha_1}[K_{\rho,n}({y}({\tau}),{u})- K_{\rho,n}(x({\tau},{u}))]dB_v({\tau})\right\|_r, \end{aligned}\] where \(K_{\rho,n}(\cdot,{u})\) is as defined in (6).
Remark 15. If in Corollary 11, we simply put \(\rho({u})=\frac{1}{\beta_1-\beta_0}\) for \({u}\in [\beta_0,\beta_1]\), then we get [8, Theorem 9].
In this article, we have successfully developed a comprehensive framework for generalizing the classical theory of majorization. By leveraging a generalized weighted Montgomery identity, we have established new identities and inequalities that are applicable to the broad class of \(n\)-convex functions. Our main results, encapsulated in Theorems 1 through 5, provide:
1. A precise identity for the difference of integrals (Theorem 1).
2. A sufficient condition for a majorization-type inequality for \(n\)-convex functions (Theorem 2).
3. Quantitative error bounds via Čebyšev and Grüss-type functionals (Theorems 3 and 4).
4. Ostrowski-type inequalities with sharp constants (Theorem 5).
These results significantly extend the existing literature, particularly the work in [8], by introducing a weight function \(\rho\) and handling functions of bounded variation \(dB_v({\tau})\), thereby enhancing the applicability and generality of the theory.
The methodology and results presented here suggest several promising directions for future research. By pursuing these avenues, the framework established in this paper can be extended and applied to an even wider spectrum of mathematical and applied problems.
A highly fruitful direction would be to replace the Taylor-type identity with representations based on Green’s functions associated with two-point boundary value problems [14, 15]. Different boundary conditions would generate different Green’s functions, potentially leading to a new family of majorization identities and inequalities with distinct kernel properties.
The linearity of our identities in the higher derivative \(\digamma^{(n)}\) makes them ideal for exploring exponential convexity and constructing new Cauchy-type mean value theorems. By considering specific families of functions, one can derive further inequalities and explore the monotonicity of means generated by our functionals.
The connection between majorization and entropy is well-known. A compelling application would be to explore the implications of our \(n\)-convex majorization inequalities for generalized entropies (e.g., Rényi or Tsallis entropies) and other measures of disorder in information theory.
The natural progression of this work is to extend it to the settings of fractional integrals and derivatives (e.g., using Riemann–Liouville or Caputo operators) and quantum calculus (\(q\)-calculus). Developing majorization theorems within these modern frameworks would be a novel and significant contribution.
Fuchs, L. (1947). A new proof of an inequality of Hardy-Littlewood-Pólya. Matematisk Tidsskrift. B, 53-54.
Hardy, G. H., Littlewood, J. E., & Pólya, G. (1952). Inequalities. Cambridge university press.
Marshall, A. W., Olkin, I., & Arnold, B. C. (1979). Inequalities: Theory of Majorization and Its Applications.
Peajcariaac, J. E., & Tong, Y. L. (1992). Convex Functions, Partial Orderings, and Statistical Applications. Academic press.
Aljinovic, A. A., Pecaric, J., & Vukelic, A. (2005). On some Ostrowski type inequalities via Montgomery identity and Taylor’s formula II. Tamkang Journal of Mathematics, 36(4), 279-301.
Khan, A. R., Nabi, H., & Pečarić, J. (2024). Popoviciu type inequalities for n-convex functions via extension of weighted Montgomery identity. Studia Universitatis Babeş-Bolyai, Mathematica, 69(1), 15–48.
Khan, A. R., Nabi, H., & Pečarić, J. E. (2022). On Ostrowski Type Inequalities via the Extended Version of Montgomery’s Identity. Mathematics, 10(7), 1113.
Aljinović, A. A., Khan, A. R., & Pečarić, J. E. (2015). Weighted majorization theorems via generalization of Taylor’s formula. Journal of Inequalities and Applications, 2015(1), 196.
Mitrinovic, D. S., Pecaric, J. E., & Fink, A. M. (1993). Classical and new inequalities in analysis, volume 61 of. Mathematics and its Applications (East European Series).
Irshad, N., Khan, A. R., Mehmood, F., & Pečarić, J. (2022). New Perspectives on the Theory of Inequalities for Integral and Sum. Springer International Publishing AG.
Barnett, N., & Dragomir, S. (2001). An Ostrowski type inequality for double integrals and applications for cubature formulae. Soochow Journal of Mathematics, 27(1), 1-10.
Khan, A. R., Pečarić, J., Praljak, M., & Varošanec, S. (2017). General Linear Inequalities and Positivity. Higher order convexity. Zagreb: Element.
Khan, A. R., & Pečarić, J. (2021). Positivity of sums and integrals for n-convex functions via the Fink identity and new Green functions. Studia Universitatis Babeş-Bolyai, Mathematica, 66(4), 613–627.
Khan, A. R., PECARIC, J., Praljak, M., & Varosanec, S. (2019). Positivity of sums and integrals for \(n\)-convex functions via the Fink identity. Turkish Journal of Mathematics, 43(2), 579-594.
