In this paper, we define a new generalized class of preinvex functions which includes harmonically \((s,m)\)-convex functions as a special case and establish a new identity. Using this identity, we introduce some new integral inequalities for harmonically \((s,m)\)-preinvex functions.
Definition 1.1 . A set \(K\) in \(\mathbb{R}^{n}\) is said to be a convex set, if and only if, $$ (1 – t) u + tv \in K,\;\; for\; all\;u,v\;\in K\;,\;t\;\in[0,1]. $$
Definition 1.2. A function \(f\) on the convex set \(K\) is said to be a convex function if and only if $$ f((1 – t)u + tv) \leq (1 – t)f(u) + t f(v),\;\; for\; all \;u,v\;\in K\;,\;t\;\in[0,1]. $$
For the differentiable convex function, we have the following interesting result.Theorem 1.3 .[3] Let \(K\) be a nonempty convex set in \(\mathbb{R}^{n}\), and let \(f\) be a differentiable convex function on the set \(K\). Then \(u \in K\) is the minimum of \(f\) if and only if \(u \in K\) satisfies the inequality $$ \langle f'(u), v – u \rangle \geq 0, \; for \;all\;v \in K. $$
Definition 1.4 . [4] A set \(K_{\eta} \subseteq \mathbb{R}\) is said to be invex set with respect to the bifunction \(\eta(.,.)\) if and only if $$ x + t \eta(y,x) \in K_{\eta},\; for\; all\; x, y \in K_{\eta},\; t \in [0,1]. $$
The invex set \(K_{\eta}\) is also called \(\eta\)-connected set. Note that, if \(\eta(b,a) = b – a\), then invex set becomes the convex set. Clearly, every convex set is an invex set but converse is not true in general.Definition 1.5 . [5] Let \(K_{\eta}\) be an invex set in \(\mathbb{R}\). Then, a function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is said to be preinvex function with respect to the bifunction \(\eta(.,.)\) if and only if $$ f(x + t \eta(y,x)) \leq (1 – t) f(x) + t f(y) \;\;for\; all\; x, y \in K_{\eta},\; t \in [0,1]. $$
Theorem 1.6 . [6] Let \(K_{\eta}\) be an invex set in \(\mathbb{R}\) and let \(f\) be a differentiable preinvex function on set \(K_{\eta}\). Then \(u \in K_{\eta}\) is the minimum of \(f\) if and only if \(u \in K_{\eta}\) satisfies the inequality $$ \langle f'(u), \eta(v , u )\rangle \geq 0, \; for \;all\;v \in K_{\eta}. $$
Definition 1.7 . [7] A set \(K_{h} \subset \mathbb{R}/ \{0\} \rightarrow \mathbb{R}\) is said to be a harmonically convex set if and only if $$ \frac{uv}{v + t (u – v)} \in K_{h}, \;\; for \; all\; u, v \in K_{h},\; t \in [0,1].$$
Definition 1.8 . [7] A function \(f : K_{h} \subset \mathbb{R}/ \{0\} \rightarrow \mathbb{R}\) is said to be harmonically convex function if and only if $$ f \left(\frac{xy}{tx + (1 – t)y} \right) \leq (1 – t)f(x) + t f(y),\;\;for \;all \;x,y\in K_{h},\;t\in[0,1].$$
Definition 1.9 . [8] The function \(f: I \subset (0, \infty) \rightarrow \mathbb{R}\) is said to be harmonically \((s,m)\)-convex in second sense, where \(s \in (0,1]\) and \(m \in (0,1]\) if $$f \big(\frac{mxy}{mty + (1 – t)x}\big) = f \big( (\frac{t}{x} + \frac{1 – t}{my})^{-1} \big) \leq t^{s} f(x) + m (1 – t)^{s} f(y)$$ \(\forall x, y \in I\) and \(t \in [0,1]\).
Remark 1.1 . Note that for \(s = 1\), \((s,m)\)-convexity reduces to harmonically \(m\)-convexity and for \(m = 1\), harmonically \((s,m)\)-convexity reduces to harmonically \(s\)-convexity in second sense and for \(s,m = 1\), harmonically \((s,m)\)-convexity reduces to ordinary harmonically convexity .
Definition 1.10. A set \(I = [a, a + \eta(b,a)] \subseteq \mathbb{R}/ \{0\}\) is said to be a harmonic invex set with respect to the bifunction \(\eta(,)\) if and only if $$ \frac{x(x + \eta(y,x))}{x + (1 – t)\eta(y,x)} \in I,\;\; for\;all\;x,y\in I,\;t\in[0,1]$$
Definition 1.11. Let \(h:[0,1] \subseteq J \rightarrow \mathbb{R}\) be a non-negative function. A function \(f: I \rightarrow [a, a + \eta (b, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R}\) is relative harmonic preinvex function with respect to an arbitrary nonnegative function \(h\) and an arbitrary bifunction \(\eta(,)\) if $$ f\left(\frac{x(x + \eta(y,x))}{x + (1 – t)\eta(y,x)}\right) \leq h(1 – t)f(x) + h(t)f(y), \;\; for\;all\;x,y\in I,\;t\in[0,1] $$
Definition 2.1. A function \(f: [a, a + \eta (b, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R}\) is said to be harmonically \((s,m)\)-preinvex functions with respect to the bifunction \(\eta(,)\), if $$ f\left( \frac{x(x + \eta(my, x))}{x + t \eta(my, x)}\right) = f \left( \frac{t}{x} + \frac{1 – t}{x + \eta (my,x)} \right)^{-1} \leq t^{s} f(x) + m(1 – t)^{s} f(y) $$ for all \(x , y \in [a, a+ \eta(b , a)]\), with \(x < my\), \(t \in [0, 1]\), \(s \in (0, 1]\), \(m \in (0, 1]\).
Note: if \(\eta(y,x) = y – x\), then harmonic \((s,m)\)-preinvexity reduce to harmonic \((s,m)\)-convexity. We need the following identity, which plays an important role in the derivations of our main results.Lemma 2.2. Let \(f: [a, a + \eta (mb, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R}\) be a differentiable function on the interior of \(I^{\circ}\) of \(I\). If \(f’ \in [a, a + \eta (mb, a)]\) and \(\lambda \in [0,1]\), then \begin{eqnarray*} &&\Upsilon_{f}(a,a + \eta(mb,a);\lambda)\\ &=& \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \left[\int_{0}^{\frac{1}{2}}\frac{\lambda – 2t}{(a + t\eta(mb,a))^{2}} f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)dt\right.\\ &&\left.+ \int^{1}_{\frac{1}{2}}\frac{2 – 2t – \lambda}{(a + t\eta(mb,a))^{2}} f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)dt\right] \end{eqnarray*} where $$ \Upsilon_{f}(a,a + \eta(mb,a);\lambda) $$ $$= (1 – \lambda) f\left( \frac{2a(a + \eta(mb,a))}{2a + \eta(mb,a)} \right) + \lambda\left[\frac{f(a) + f(a + \eta(mb,a))}{2}\right]$$ $$ -\frac{2a(a + \eta(mb,a))}{\eta(mb,a)} \int_{a}^{a + m\eta(b,a)} \frac{f(x)}{x^{2}}dx $$
Proof. Integrating by parts, we have \begin{eqnarray*} &&I_{1} = \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \int_{0}^{\frac{1}{2}}\frac{\lambda – 2t}{(a + t\eta(mb,a))^{2}} f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)dt\\ &&=\frac{1 – \lambda}{2} f\left( \frac{2a(a + \eta(mb,a))}{2a + \eta(mb,a)} \right) + \frac{\lambda}{2} f(a + \eta(mb,a)) \\ &&- \int_{0}^{\frac{1}{2}}f\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)dt, \end{eqnarray*} and $$ I_{2} = \frac{a(a,a + \eta(mb,a))\eta(b,a)}{2}\int^{1}_{\frac{1}{2}}\frac{2t – 2 + \lambda}{(a + t\eta(mb,a))^{2}} f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)dt$$ $$= \frac{1 – \lambda}{2} f\left( \frac{2a(a + \eta(mb,a))}{2a + \eta(mb,a)} \right) + \frac{\lambda}{2} f(a )- \int^{1}_{\frac{1}{2}}f\left( \frac{a(a + \eta(mb,a))}{x + t\eta(mb,a)}\right)dt$$ Thus $$I_{1} + I_{2}$$ $$= (1 – \lambda) f\left( \frac{2a(a + \eta(mb,a))}{2a + \eta(mb,a)} \right) + \lambda\left[\frac{f(a) + f(a + \eta(mb,a))}{2}\right]$$ $$ -\frac{2a(a + \eta(mb,a))}{\eta(mb,a)} \int_{a}^{a + m\eta(b,a)} \frac{f(x)}{x^{2}}dx$$ which is the required result.
Theorem 2.3. Let \(f: [a, a + \eta (mb, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R}\) be a differentiable function on the interior \(I^{\circ}\) of \(I\). If \(f’ \in [a, a + \eta (mb, a)]\) and \(|f’|^{q}\) is harmonic \((s,m)\)-preinvex function on \(I\) for \(q \geq 1\) and \(\lambda \in[0,1]\), then $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2}\sigma_{1}(a,b;\lambda)^{1 – \frac{1}{q}}\{\sigma_{2}(a,b;\lambda, s)|f'(a)|^{q}$$ $$ + m \sigma_{3}(a,b;\lambda, s)|f'(b)|^{q}\}^{\frac{1}{q}}$$ $$ + \sigma_{4}(a,b;\lambda)^{1 – \frac{1}{q}}\{\sigma_{5}(a,b;\lambda, s)|f'(a)|^{q} + m \sigma_{6}(a,b;\lambda, s)|f'(b)|^{q}\}^{\frac{1}{q}},$$ where one can evaluate these integrals using any mathematical software (i.e maple). $$\sigma_{1}(a,b;\lambda) = \int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|}{(a + t\eta(mb,a))^{2}} dt , $$ $$ \sigma_{2}(a,b;\lambda, s) = \int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|(1 – t)^{s}}{(a + t\eta(mb,a))^{2}} , $$ $$ \sigma_{3}(a,b;\lambda, s) = \int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|t^{s}}{(a + t\eta(mb,a))^{2}} , $$ $$\sigma_{4}(a,b;\lambda) = \int^{1}_{\frac{1}{2}}\frac{|2 – 2t – \lambda|}{(a + t\eta(mb,a))^{2}} dt $$ $$ \sigma_{5}(a,b;\lambda, s) = \int_{0}^{\frac{1}{2}}\frac{|2 – 2t – \lambda|(1 – t)^{s}}{(a + t\eta(mb,a))^{2}} , $$ $$ \sigma_{6}(a,b;\lambda, s) = \int_{0}^{\frac{1}{2}}\frac{|2 – 2t – \lambda|t^{s}}{(a + t\eta(mb,a))^{2}} . $$
Proof. Using Lemma 2.2 and the power mean inequality, we have $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$\leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|}{(a + t\eta(mb,a))^{2}} \left|f’\left( \frac{a(a + \eta(mb,a))}{a +t\eta(mb,a)}\right)\right|dt $$ $$ + \int^{1}_{\frac{1}{2}}\frac{|2 – 2t – \lambda|}{(a + t\eta(mb,a))^{2}} \left|f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|dt $$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \left( \int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|}{(a + t\eta(mb,a))^{2}} dt\right)^{1 – \frac{1}{q}}$$ $$ \times \left(\int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|}{(a + t\eta(mb,a))^{2}} \left|f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|^{q}dt\right)^{\frac{1}{q}}$$ $$+ \left( \int^{1}_{\frac{1}{2}}\frac{|2 – 2t – \lambda|}{(a + t\eta(mb,a))^{2}}\right)^{ 1 – \frac{1}{q}}$$ $$ \times \left(\int^{1}_{\frac{1}{2}}\frac{|2 – 2t – \lambda|}{(a + t\eta(mb,a))^{2}} \left|f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|^{q}dt \right)^{\frac{1}{q}}$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(b,a)}{2}\left( \int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|}{(a + t\eta(mb,a))^{2}} dt\right)^{1 – \frac{1}{q}}$$ $$ \times \left(\int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|}{(a + t\eta(mb,a))^{2}} \{t^{s}|f'(a)|^{q} + m(1 – t)^{s} |f'(b)|^{q} \}dt\right)^{\frac{1}{q}}$$ $$+ \left( \int^{1}_{\frac{1}{2}}\frac{| 2 – 2t – \lambda|}{(a + t\eta(mb,a))^{2}}\right)^{ 1 – \frac{1}{q}}$$ $$ \times \left(\int^{1}_{\frac{1}{2}}\frac{|2 – 2t – \lambda|}{(a + t\eta(mb,a))^{2}} \{t^{s}|f'(a)|^{q} + m(1 – t)^{s} |f'(b)|^{q} \}dt\right)^{\frac{1}{q}}$$ $$ = \frac{a(a + \eta(mb,a))\eta(mb,a)}{2}\sigma_{1}(a,b;\lambda)^{1 – \frac{1}{q}}\{\sigma_{2}(a,b;\lambda, s)|f'(a)|^{q} $$ $$+ m \sigma_{3}(a,b;\lambda, s)|f'(b)|^{q}\}^{\frac{1}{q}}$$ $$ + \sigma_{4}(a,b;\lambda)^{1 – \frac{1}{q}}\{\sigma_{5}(a,b;\lambda, s)|f'(a)|^{q} + m \sigma_{6}(a,b;\lambda, s)|f'(b)|^{q}\}^{\frac{1}{q}},$$ which is the required result.
If \(q=1\), then Theorem 1.6 reduces to the following result, which appears to be a better new one than already exists.Corollary 2.4 . Let \(f: [a, a + \eta (m b, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R}\) be a differentiable function on the interior of \(I^{\circ}\) of \(I\). If \(f’ \in [a, a + \eta (mb, a)]\) and \(|f’|\) is harmonic \((s,m)\)-preinvex function on \(I\) and \(\lambda \in[0,1]\), then $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2}\{\sigma_{2}(a,b;\lambda, s) + m \sigma_{3}(a,b;\lambda, s)\}|f'(a)|$$ $$ + \{\sigma_{5}(a,b;\lambda, s) + m\sigma_{6}(a,b;\lambda, s)\}|f'(b)|,$$ where \(\sigma_{2}(a,b;\lambda, s)\), \(\sigma_{3}(a,b;\lambda, s)\), \(\sigma_{5}(a,b;\lambda, s)\), \(\sigma_{6}(a,b;\lambda, s)\) are given as in Theorem 1.6.
Theorem 2.5. Let \(f: [a, a + \eta (mb, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R} \) be a differentiable function on the interior of \(I^{\circ}\) of \(I\). If \(f’ \in [a, a + \eta (mb, a)]\) and \(|f’|^{q}\) is harmonic \((s,m)\)-preinvex function on \(I\) for \(p,q > 1\), \(\frac{1}{p} + \frac{1}{q} = 1\) and \(\lambda \in[0,1]\), then $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$\leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} (\sigma_{7}(a,b;\lambda, p))^{\frac{1}{p}}$$$$\times \left(\{(1 – \frac{1}{2^{s + 1}} )|f'(a)|^{q} + \frac{m}{2^{s + 1}} |f'(b )|^{q} \}\right)^{\frac{1}{q}} $$ $$ + (\sigma_{8}(a,b;\lambda, p))^{\frac{1}{p}} \left(\{m(1 – \frac{1}{2^{s + 1}} )|f'(b)|^{q} + \frac{1}{2^{s + 1}} |f'(a )|^{q} \}\right)^{\frac{1}{q}} , $$ where $$ \sigma_{7}(a,b;\lambda, p) = \int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|^{p}}{(a + t\eta(mb,a))^{2p}} dt, $$ $$\sigma_{8}(a,b;\lambda, p) = \int^{1}_{\frac{1}{2}}\frac{|2 – 2t – \lambda|^{p}}{(a + t\eta(mb,a))^{2p}}dt . $$
Proof. Using Lemma 2.2 and Holder’s integral inequality, we have $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|}{(a + t\eta(mb,a))^{2}} \left|f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|dt $$ $$ + \int^{1}_{\frac{1}{2}}\frac{|2 – 2t – \lambda|}{(a + t\eta(mb,a))^{2}} \left|f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|dt $$ $$ \leq \frac{a(a,a + \eta(mb,a))\eta(mb,a)}{2} \left(\int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|^{p}}{(a + t\eta(mb,a))^{2p}} dt\right)^{\frac{1}{p}}$$$$\times \left(\int_{0}^{\frac{1}{2}}\left|f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|^{q}dt\right)^{\frac{1}{q}} $$ $$ + \left( \int^{1}_{\frac{1}{2}}\frac{|2 – 2t – \lambda|^{p}}{(a + t\eta(mb,a))^{2p}}dt\right)^{\frac{1}{p}} \left(\int^{1}_{\frac{1}{2}} \left|f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)\right|^{q}dt\right)^{\frac{1}{q}} $$ $$ \leq \frac{a(a + \eta(mb,a))\eta(b,a)}{2} \left(\int_{0}^{\frac{1}{2}}\frac{|\lambda -2t|^{p}}{(a + t\eta(mb,a))^{2p}} dt\right)^{\frac{1}{p}}$$ $$\times \left(\int_{0}^{\frac{1}{2}}\{t^{s} |f'(a)|^{q} + m(1 – t)^{s} |f'(b)|^{q}\}dt\right)^{\frac{1}{q}} $$ $$ + \left( \int^{1}_{\frac{1}{2}}\frac{|2 – 2t – \lambda|^{p}}{(a + t\eta(mb,a))^{2p}}dt\right)^{\frac{1}{p}} \left(\int^{1}_{\frac{1}{2}} \{t^{s} |f'(a)|^{q} + m(1 – t)^{s} |f'(b)|^{q}\}dt\right)^{\frac{1}{q}} $$ $$= \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \left(\int_{0}^{\frac{1}{2}}\frac{|\lambda – 2t|^{p}}{(a + t\eta(mb,a))^{2p}} dt\right)^{\frac{1}{p}}$$ $$\times \left(\{(1 – \frac{1}{2^{s + 1}} )|f'(a)|^{q} + m \frac{1}{2^{s + 1}} |f'(b )|^{q} \}\right)^{\frac{1}{q}} $$ $$ + \left( \int^{1}_{\frac{1}{2}}\frac{|2 – 2t – \lambda|^{p}}{(a + t\eta(mb,a))^{2p}}dt\right)^{\frac{1}{p}}$$$$ \times \left(\{m(1 – \frac{1}{2^{s + 1}} )|f'(b)|^{q} + \frac{1}{2^{s + 1}} |f'(a )|^{q} \}\right)^{\frac{1}{q}} $$ The proof completes.
Theorem 2.6 . Let \(f: [a, a + \eta (mb, a)] \subseteq \mathbb{R} / \{0\} \rightarrow \mathbb{R} \) be a differentiable function on the interior \(I^{\circ}\) of \(I\). If \(f’ \in [a, a + \eta (mb, a)]\) and \(|f’|^{q}\) is harmonic \((s,m)\)-preinvex function on \(I\) for \(p,q > 1\), \(\frac{1}{p} + \frac{1}{q} = 1\) and \(\lambda \in[0,1]\), then $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \times \left( \frac{\lambda^{p + 1} + ( 1 – \lambda)^{ p + 1}}{2(p + 1)} \right)^{\frac{1}{p}} $$ $$ \times (\sigma_{9}(a,b;\lambda, q)|f'(a)|^{q} + m \sigma_{10}(a,b;\lambda, q)|f'(b)|^{q})^{\frac{1}{q}}$$ $$ + (\sigma_{11}(a,b;\lambda, q)|f'(a)|^{q} + m \sigma_{12}(a,b;\lambda, q)|f'(b)|^{q})^{\frac{1}{q}} ,$$ where $$ \sigma_{9}(a,b;\lambda, p) = \int^{\frac{1}{2}}_{0}\frac{t^{s}}{(a + t\eta(mb,a))^{2q}}dt$$ $$ \sigma_{10}(a,b;\lambda, p) = \int^{\frac{1}{2}}_{0}\frac{(1 – t)^{s}}{(a + t\eta(mb,a))^{2q}}dt $$ $$ \sigma_{11}(a,b;\lambda, p) = \int_{\frac{1}{2}}^{1}\frac{t^{s}}{(a + t\eta(mb,a))^{2q}}dt$$ $$ \sigma_{12}(a,b;\lambda, p) = \int_{\frac{1}{2}}^{1}\frac{(1 – t)^{s}}{(a + t\eta(mb,a))^{2q}}dt $$
Proof. Using Lemma 2.2 and the Holder’s integral inequality, we have $$\left| \Upsilon_{f}(a,a + \eta(mb,a);\lambda) \right|$$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \int_{0}^{\frac{1}{2}}|\lambda – 2t| \left|\frac{f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)}{(a + t\eta(mb,a))^{2}}\right|dt $$ $$ + \int^{1}_{\frac{1}{2}}| 2 – 2t – \lambda| \left|\frac{f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)}{(a + t\eta(mb,a))^{2}}\right|dt $$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2}\left( \int_{0}^{\frac{1}{2}}| \lambda- 2t|^{p}dt\right)^{\frac{1}{p}} $$ $$\times \left(\int_{0}^{\frac{1}{2}}\left|\frac{f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)}{(a + t\eta(mb,a))^{2}}\right|^{q}dt \right)^{\frac{1}{q}} $$ $$ + \left(\int^{1}_{\frac{1}{2}}|2 -2t – \lambda|^{p}dt\right)^{\frac{1}{p}} \left( \int^{1}_{\frac{1}{2}}\left|\frac{f’\left( \frac{a(a + \eta(mb,a))}{a + t\eta(mb,a)}\right)}{(a + t)\eta(mb,a))^{2}}\right|^{q}dt\right)^{\frac{1}{q}} $$ $$ \leq \frac{a(a + \eta(mb,a))\eta(mb,a)}{2}\left( \int_{0}^{\frac{1}{2}}|\lambda – 2t|^{p}dt\right)^{\frac{1}{p}} $$ $$ \times \left( |f'(a)|^{q}\int^{\frac{1}{2}}_{0}\frac{t^{s}}{(a + t\eta(mb,a))^{2q}}dt + m|f'(b)|^{q} \int^{\frac{1}{2}}_{0}\frac{(1 – t)^{s}}{(a + t\eta(mb,a))^{2q}}dt \right)^{\frac{1}{q}} $$ $$ + \left( \int^{1}_{\frac{1}{2}}|2 – 2t – \lambda|^{p}dt\right)^{\frac{1}{p}} $$ $$\times \left( |f'(a)|^{q}\int^{1}_{\frac{1}{2}}\frac{t^{s}}{(a + t\eta(mb,a))^{2q}}dt + m |f'(b)|^{q} \int^{1}_{\frac{1}{2}}\frac{(1 – t)^{s}}{(a + t\eta(mb,a))^{2q}}dt \right)^{\frac{1}{q}} $$ $$= \frac{a(a + \eta(mb,a))\eta(mb,a)}{2} \times \left( \frac{\lambda^{p + 1} + ( 1 – \lambda)^{ p + 1}}{2(p + 1)} \right)^{\frac{1}{p}} $$ $$\times (\sigma_{9}(a,b;\lambda, q)|f'(a)|^{q} + m \sigma_{10}(a,b;\lambda, q)|f'(b)|^{q})^{\frac{1}{q}}$$ $$ + (\sigma_{11}(a,b;\lambda, q)|f'(a)|^{q} + m \sigma_{12}(a,b;\lambda, q)|f'(b)|^{q})^{\frac{1}{q}} .$$ This completes the proof.