This work is dedicated to some generalized upper bounds obtained for the Jensen gap by using different generalized convex functions including strongly convex functions, s-convex functions, η-convex functions, strongly η-convex functions, m-convex functions, and (α, m)-convex functions. The results are then extended to the integral form of Jensen’s inequality. The main results enable us to establish such bounds for Hölder and Hermite-Hadamard inequalities as well. Finally, estimates for the Csiszár divergence are presented as direct applications of the main outcomes.
The field of mathematical inequalities and their applications has an important place in the literature of modern applied analysis. Mathematical inequalities have a strong background regarding their deep connection with convex functions. These two areas collectively have a lot of applications in various fields of art, science, engineering, and technology [1– 7]. Among these mathematical inequalities, Jensen’s inequality is the most prominent. This inequality was first presented by Danish mathematician Johan Jensen in 1906, it generalizes the classical convexity, and it is described as [8, p.43]:
Theorem 1. If \(\Psi:[\delta_{1},\delta_{2}]\rightarrow \mathbb{R}\) is a convex function and \(y_{k}\in[\delta_{1},\delta_{2}], \rho_{k}\geq 0\) for \(k=1,2,…,n\) with \(0<\rho_{n}=\sum\limits_{k=1}^{n}\rho_{k},\) and \(\overline{y}=\frac{1}{\rho_{n}}\sum\limits_{k=1}^{n}\rho_{k}y_{k}\). Then \[\label{JI} \Psi\left(\overline{y}\right)\leq \frac{1}{\rho_{n}}\sum\limits_{k=1}^{n}\rho_{k}\Psi(y_{k}), \tag{1}\] known as Jensen’s inequality in the literature.
The generalization of the above celebrated inequality in the Riemann integral sense is provided as [9]:
Theorem 2. Let \([\delta_{1},\delta_{2}]\subset\mathbb{R}\) and \(\beta_{1},\beta_{2}:[\alpha_{1},\alpha_{2}]\rightarrow \mathbb{R}\) be two measurable functions such that \(\beta_{1}(x)\in[\delta_{1},\delta_{2}],~\forall~ x\in[\alpha_{1},\alpha_{2}].\) Further, suppose that the functions \(\beta_{2},~\beta_{1}\beta_{2}\) and \((\Psi\circ \beta_{1}).\beta_{2}\) are integrable on \([\alpha_{1},\alpha_{2}]\) for \(\Psi:[\delta_{1},\delta_{2}]\rightarrow \mathbb{R}\) as convex function. Also, suppose that \(\beta_{2}(x)\geq0\) on \([\alpha_{1},\alpha_{2}],\) \(\int_{\alpha_{1}}^{\alpha_{2}}\beta_{2}(x)dx=\beta_{n}>0,\) and \(\bar{\beta}=\frac{1}{\beta_{n}}\int_{\alpha_{1}}^{\alpha_{2}}\beta_{1}(x)\beta_{2}(x)dx,\) then \[\label{Jensenine1} \Psi\left(\bar{\beta}\right) \leq\frac{1}{\beta_{n}}\int_{\alpha_{1}}^{\alpha_{2}}\left(\Psi\circ \beta_{1}\right)(x)\beta_{2}(x)dx. \tag{2}\]
We recall the well-known Hölder inequality, which is given by [8, p.124]:
Theorem 3. For exponents \(t_{1},t_{2} > 1\) with \(\frac{1}{t_{1}} + \frac{1}{t_{2}} = 1\) and positive tuples \(\mathbf{p} = (p_1,p_2, \dots, p_n)\) and \(\mathbf{q} = (q_1, q_2, \dots, q_n)\), Hölder’s inequality states: \[\begin{aligned} \sum\limits_{k=1}^n p_{k} q_{k} &\leq \left( \sum\limits_{k=1}^n p_{k}^{t_{1}} \right)^{\!1/t_{1}} \left( \sum\limits_{k=1}^n q_{k}^{t_{2}} \right)^{\!1/t_{2}}. \label{eq:holder} \end{aligned} \tag{3}\]
The following celebrated inequality, due to Hermite and Hadamard [10], holds:
Theorem 4. For a convex function \(\Psi:[\alpha_1,\alpha_2]\subset\mathbb{R}\rightarrow\mathbb{R}\), the mathematical form of Hermite-Hadamard inequality is \[\Psi\left(\frac{\alpha_1+\alpha_2}{2}\right)\leq\frac{1}{\alpha_2-\alpha_1}\int_{\alpha_1}^{\alpha_2}\Psi(x)dx\leq\frac{\Psi(\alpha_1)+\Psi(\alpha_2)}{2}. \tag{4}\]
Here, we recall some well-known generalizations of convex functions that will be used throughout this work.
Definition 1(Convex Function).A function \(f : D \subset \mathbb{R} \to \mathbb{R}\) is said to be convex, if for all \(x, y \in D\) and \(t \in [0,1]\), the following inequality holds: \[f(tx + (1-t)y) \le t f(x) + (1-t) f(y).\]
Definition 2(Strongly Convex Function).A function \(f : D \subset \mathbb{R} \to \mathbb{R}\) is called strongly convex with modulus \(C > 0\) if \[f(tx + (1-t)y) \le t f(x) + (1-t) f(y) – \frac{C}{2} t(1-t)\|x – y\|^2,\] for all \(x, y \in D\) and \(t \in [0,1]\).
Definition 3(s-Convex Function in the Second Sense).Let \(s \in (0,1]\). A function \(f : D \to [0,\infty)\) is said to be \(s\)-convex in the second sense if \[f(tx + (1-t)y) \le t^s f(x) + (1-t)^s f(y),\] for all \(x, y \ge 0\) and \(t \in [0,1]\).
Definition 4 (m-Convex Function).A function \(f : [0,b] \to \mathbb{R}\) is said to be \(m\)-convex, where \(m \in [0,1]\), if \[f(tx + m(1-t)y) \le t f(x) + m(1-t) f(y),\] for all \(x, y \in [0,b]\) and \(t \in [0,1]\).
Definition 5 ((α, m)-Convex Function).Let \(\alpha, m \in [0,1]\). A function \(f : [0,b] \to \mathbb{R}\) is said to be \((\alpha,m)\)-convex, if for all \(x, y \in [0,b]\) and \(t \in [0,1]\), the following inequality holds: \[f(tx + m(1-t)y) \le t^{\alpha} f(x) + m(1-t^{\alpha}) f(y).\]
Definition 6(η-Convex Function).f Let \(\eta : \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) be a bifunction. A mapping \(f : D \subset \mathbb{R} \to \mathbb{R}\) is called \(\eta\)-convex if \[f(tx + (1-t)y) \le t f(x) + (1-t) f(y) + t(1-t)\,\eta(f(x), f(y)),\] for all \(x, y \in D\) and \(t \in [0,1]\).
Definition 7(Strongly η-Convex Function).A function \(f :D \subset \mathbb{R} \to \mathbb{R}\) is said to be strongly \(\eta\)-convex with modulus \(C > 0\) if \[f(tx + (1-t)y) \le t f(x) + (1-t) f(y) + t(1-t)\,\eta(f(x), f(y)) – C\,t(1-t)\,(x-y)^2,\] for all \(x, y \in D\) and \(t \in [0,1]\).
This section is dedicated to some new generalized upper bounds for the gap of left and right hand sides of the celebrated Jensen inequality through various generalized convex functions. The following theorem is about an upper bound for the Jensen gap (JG) for strongly convex functions. This function actually strengthens the results.
Theorem 5. Let \(\Psi:[\delta_1,\delta_2] \rightarrow \mathbb{R}\) be the first order differential function such that \(|\Psi'|^q\) is strongly-convex with modulus C and for \(q>1\).Let \(y_k\in[\delta_1,\delta_2], \rho_k \in \mathbb{R}\) for each \(k\in\{1,2,…,n\}\) with \(\sum\limits_{k=1}^n \rho_k=\rho_n\neq0\), and \(\bar{y}=\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_ky_k\in[\delta_1,\delta_2],\) then \[\label{h1} \begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(\frac{|\Psi'(\bar{y})|^q+|\Psi'(y_k)|^q-C(\bar{y}-y_k)^2}{2}\right)^\frac{1}{q}. \end{aligned} \tag{5}\]
Proof. Without loss of generality, if \(\bar{y}\neq y_k\) for \(k=\{1,2,…,n\}\) then through integration by parts we have \[\sum\limits_{k=1}^n \frac{\rho_k(\bar{y}-y_k)}{\rho_n}\int_{0}^{1}\Psi'(t\bar{y}+(1-t)y_k)dt=\sum\limits_{k=1}^n \frac{\rho_k(\bar{y}-y_k)}{\rho_n}\left[\frac{\Psi'(t\bar{y}+(1-t)y_k)}{\bar{y}-y_k}\Big|^{1}_{0}\right].\]
Implies that \[\sum\limits_{k=1}^n \frac{\rho_k(\bar{y}-y_k)}{\rho_n}\int_{0}^{1}\Psi'(t\bar{y}+(1-t)y_k)dt=\Psi(\bar{y})-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k}).\]
Taking the absolute value of the above inequality, and applying triangular inequality, we deduce \[\label{h2} \begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\int_{0}^{1}|\Psi'(t\bar{y}+(1-t)y_k)|dt. \end{aligned} \tag{6}\]
Applying Hölder inequality on the right-hand side of (6), we obtain \[\label{h3} \begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(\int_{0}^{1}|\Psi'(t\bar{y}+(1-t)y_k)|^q dt\right)^\frac{1}{q} . \end{aligned} \tag{7}\]
Given that, \(|\Psi'|^q\) is strongly-convex with mod C, therefore (7) inequality becomes \[\begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(\int_{0}^{1}(t\left|\Psi'(\bar{y})|^q+(1-t)|\Psi'(y_k)\right|^q -Ct(1-t)(\bar{y}-y_k)^2) dt\right)^\frac{1}{q}. \end{aligned}\]
Calculating the integral in the above inequality, we obtain (5). ◻
The following theorem presents a generalized upper bound for the JG through s-convex functions.
Theorem 6. Let \(\Psi:[\delta_1,\delta_2] \rightarrow \mathbb{R}\) be the first order differential function such that \(|\Psi'|^q\) is s-convex for \(q>1\). Let \(y_k\in[\delta_1,\delta_2], \rho_k \in \mathbb{R}\) for each \(k\in\{1,2,…,n\}\) with \(\sum\limits_{k=1}^n \rho_k=\rho_n\neq0\), and \(\bar{y}=\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_ky_k\in[\delta_1,\delta_2],\) then \[\label{h4} \begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(\frac{|\Psi'(\bar{y})|^q+|\Psi'(y_k)|^q}{s+1}\right)^\frac{1}{q}. \end{aligned} \tag{8}\]
Proof. Given that, \(|\Psi'|^q\) is s-convex therefore (7) inequality becomes \[\left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\leq\sum\limits_{k=1}^n\left|\frac{\Psi_k(\bar{y}-y_k)}{\rho_n}\right|\left(\int_{0}^{1}(t^s|\Psi'(\bar{y})|^q+(1-t)^s|\Psi'(y_k)|^q dt\right)^\frac{1}{q}.\]
Calculating the integral in the above inequality, we get the desired inequality (8) ◻
The following theorem presents a bound in generalized form for the JG through m-convex functions.
Theorem 7. Let \(\Psi:[\delta_1,\delta_2] \rightarrow \mathbb{R}\) be the first order differential function such that \(|\Psi'|^q\) is m-convex for \(q>1\). Let \(y_k\in[\delta_1,\delta_2], \rho_k \in \mathbb{R}\) for each \(k\in\{1,2,…,n\}\) with \(\sum\limits_{k=1}^n \rho_k=\rho_n\neq0\), and \(\bar{y}=\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_ky_k\in[\delta_1,\delta_2],\) then \[\label{h5} \begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(\frac{m|\Psi'(\bar{y})|^q+|\Psi'(y_k)|^q}{2}\right)^\frac{1}{q}. \end{aligned} \tag{9}\]
Proof. Given that, \(|\Psi'|^q\) is m-convex therefore (7) inequality becomes \[\left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\leq\sum\limits_{k=1}^n\left|\frac{\Psi_k(\bar{y}-y_k)}{\rho_n}\right|\left(\int_{0}^{1}(mt|\Psi'(\bar{y})|^q+(1-t)|\Psi'(y_k)|^q dt\right)^\frac{1}{q}.\]
Calculating the integral in the above inequality, we obtain (9) as follows. \[\begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right| &\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(m|\Psi'(\bar{y})|^q\int_{0}^{1}tdt+|\Psi'(y_k)|^q \int_{0}^{1}(1-t)dt\right)^\frac{1}{q}\\ &=\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(m|\Psi'(\bar{y})|^q(\frac{1}{2})+|\Psi'(y_k)|^q (\frac{1}{2})\right)^\frac{1}{q}\\ &=\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(\frac{m|\Psi'(\bar{y})|^q+|\Psi'(y_k)|^q}{2}\right)^\frac{1}{q}. \end{aligned}\]
Which concludes the required result. ◻
The following theorem proposes another generalized bound for JG, mentioned therein.
Theorem 8. Let \(\Psi:[\delta_1,\delta_2] \rightarrow \mathbb{R}\) be the first order differential function such that \(|\Psi'|^q\) is \((\alpha,m)\)-convex for \(q>1\).Let \(y_k\in[\delta_1,\delta_2], \rho_k \in \mathbb{R}\) for each \(k\in\{1,2,…,n\}\) with \(\sum\limits_{k=1}^n \rho_k=\rho_n\neq0\), and \(\bar{y}=\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_ky_k\in[\delta_1,\delta_2],\) then \[\label{h6} \begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(\frac{|\Psi'(\bar{y})|^q+m|\Psi'(y_k)|^q}{\alpha+1}\right)^\frac{1}{q}. \end{aligned} \tag{10}\]
Proof. Given that, \(|\Psi'|^q\) is \((\alpha,m)\)-convex therefore (7) inequality becomes \[\left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(\int_{0}^{1}(t^\alpha|\Psi'(\bar{y})|^q+m(1-t^\alpha)|\Psi'(y_k)|^q dt\right)^\frac{1}{q}.\]
Calculating the integral in the above inequality, we obtain (10) as follows: \[\begin{aligned} \left|\psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right| &\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(|\Psi'(\bar{y})|^q\int_{0}^{1}t^\alpha dt+m|\Psi'(y_k)|^q \int_{0}^{1}(1-t^\alpha)dt\right)^\frac{1}{q}\\ &=\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(|\Psi'(\bar{y})|^q(\frac{1}{\alpha+1})+m|\Psi'(y_k)|^q (\frac{1}{\alpha+1})\right)^\frac{1}{q}\\ &=\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(\frac{|\Psi'(\bar{y})|^q+m|\Psi'(y_k)|^q}{\alpha+1}\right)^\frac{1}{q}. \end{aligned}\]
Which concludes the final result. ◻
The following theorem also gives a new generalized bound for JG, for \(\eta-convex\) functions.
Theorem 9. Let \(\Psi:[\delta_1,\delta_2] \rightarrow \mathbb{R}\) be the first order differential function such that \(|\Psi'|^q\) is \(\eta-convex\) for \(q>1\). Let \(y_k\in[\delta_1,\delta_2], \rho_k \in \mathbb{R}\) for each \(k\in\{1,2,…,n\}\) with \(\sum\limits_{k=1}^n \rho_k=\rho_n\neq0\), and \(\bar{y}=\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_ky_k\in[\delta_1,\delta_2],\) then \[\label{h7} \begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(|\psi '(y_k)|^q+\frac{\eta(|\Psi'(\bar{y})|^q,|\Psi'(y_k)|^q)}{2}\right)^\frac{1}{q}. \end{aligned} \tag{11}\]
Proof. Given that, \(|\Psi'|^q\) is \(\eta-convex\) therefore (7) inequality becomes \[\begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right| \leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(\int_{0}^{1}(|\Psi'(y_k)|^q+t\eta(|\Psi'(\bar{y})|^q,|\Psi '(y_k)|^q))dt\right)^\frac{1}{q}. \end{aligned}\]
Calculating the integral in the above inequality, we obtain (11) as given by the following calculations. \[\begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right| &\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(\int_{0}^{1}|\Psi'(y_k)|^q dt+\int_{0}^{1}t\eta(|\Psi'(\bar{y})|^q,|\Psi'(y_k)|^q)dt\right)^\frac{1}{q}\\ &=\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(|\Psi'(y_k)|^q \int_{0}^{1}1dt+\eta(|\Psi'(\bar{y})|^q,|\Psi'(y_k)|^q)\int_{0}^{1}tdt\right)^\frac{1}{q}\\ &=\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right|\left(|\Psi'(y_k)|^q+\frac{\eta(|\Psi'(\bar{y})|^q,|\Psi'(y_k)|^q)}{2}\right)^\frac{1}{q}. \end{aligned}\]
Thus concludes the desired result. ◻
Another theorem which presents a generalized bound for JG through strongly \(\eta\)-convex functions.
Theorem 10. Let \(\Psi:[\delta_1,\delta_2] \rightarrow \mathbb{R}\) be the first order differential function such that \(|\Psi'|^q\) is strongly \(\eta\)-convex for \(q>1\). Let \(y_k\in[\delta_1,\delta_2], \rho_k \in \mathbb{R}\) for each \(k\in\{1,2,…,n\}\) with \(\sum\limits_{k=1}^n \rho_k=\rho_n\neq0\), and \(\bar{y}=\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_ky_k\in[\delta_1,\delta_2],\) then \[\begin{aligned} \label{imaf} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right| \leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right| \left(|\Psi'(y_k)|^q+\frac{\eta(|\Psi'(\bar{y})|^q,|\Psi'(y_k)|^q)}{2}-\frac{n(\bar{y}-y_k)^2}{6}\right)^\frac{1}{q}. \end{aligned} \tag{12}\]
Proof. Given that, \(|\Psi'|^q\) is strongly \(\eta\)-convex therefore (7) inequality becomes \[\begin{aligned} \left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right| &\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right| \left(\int_{0}^{1}(|\Psi'(y_k)|^q+t\eta(|\Psi'(\bar{y})|^q,|\Psi'(y_k)|^q)-nt(1-t)(\bar{y}-y_k)^2)dt\right)^\frac{1}{q}. \end{aligned}\]
Calculating the integral in the above inequality, we obtain (12) \[\begin{aligned} &\left|\Psi{(\bar{y})}-\frac{1}{\rho_n}\sum\limits_{k=1}^n \rho_k\Psi({y_k})\right|\\ &\leq\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right| \times\left(|\Psi'(y_k)|^q\int_{0}^{1}1dt+\eta(|\Psi'(\bar{y})|^q,|\Psi'(y_k)|^q)\int_{0}^{1}tdt-n(\bar{y}-y_k)^2\int_{0}^{1}t(1-t)dt\right)^\frac{1}{q}\\ &=\sum\limits_{k=1}^n\left|\frac{\rho_k(\bar{y}-y_k)}{\rho_n}\right| \left(|\Psi'(y_k)|^q+\frac{\eta(|\Psi'(\bar{y})|^q,|\Psi'(y_k)|^q)}{2}-\frac{n(\bar{y}-y_k)^2}{6}\right)^\frac{1}{q}. \end{aligned}\]
Thus concludes the desired result. ◻
Now, we present integral versions of the above theorems without proof. The proofs are straightforward as in the discrete cases.
Theorem 11. Let \(\Psi:[\delta_1,\delta_2] \rightarrow
\mathbb{R}\) be the differential function such that \(|\Psi'|^q\) is strongly-convex with
modulus C for \(q>1\) and let \(\beta_1,\beta_2\) be two integrable
functions defined on the interval \([\alpha_1,\alpha_2]\) such that \(\beta_1(x)\in[\delta_1,\delta_2]\) for all
\(x\in[\alpha_1,\alpha_2]\) and
\(\int_{\alpha_1}^{\alpha_2}\beta_2(x)dx=\beta_n\neq0.\)
Assume that \(\bar{\beta}=\frac{1}{\bar{\beta_n}}\int_{\alpha_1}^{\alpha_2}\beta_1(x)\beta_2(x)dx\in[\delta_1,\delta_2]\),
then \[\label{h9}
\begin{aligned}
&\left|\Psi{(\bar{\beta})}-\frac{1}{\beta_n}\int_{\alpha_1}^{\alpha_2}(\Psi\circ\beta_1)(x)\beta_2(x)dx\right|\\
&\quad\leq\frac{1}{|\beta_n|}\int_{\alpha_1}^{\alpha_2}|\beta_2(x)(\bar{\beta}-\beta_1(x))|\left(\frac{|\Psi'(\bar{\beta})|^q+|\Psi'(\beta_1(x))|^q-C(\bar{\beta}-\beta_1)^2}{2}\right)^\frac{1}{q}dx.
\end{aligned} \tag{13}\]
Theorem 12. Let \(\Psi:[\delta_1,\delta_2] \rightarrow \mathbb{R}\) be the differential function such that \(|\Psi'|^q\) is s-convex for \(q>1\) and let \(\beta_1,\beta_2\) be two integrable functions defined on the interval \([\alpha_1,\alpha_2]\) such that \(\beta_1(x)\in[\delta_1,\delta_2]\) for all \(x\in[\alpha_1,\alpha_2]\) and \(\int_{\alpha_1}^{\alpha_2}\beta_2(x)dx=\beta_n\neq0.\) Assume that \(\bar{\beta}=\frac{1}{\bar{\beta_n}}\int_{\alpha_1}^{\alpha_2}\beta_1(x)\beta_2(x)dx\in[\delta_1,\delta_2]\), then \[\label{h10} \begin{aligned} \left|\Psi{(\bar{\beta})}-\frac{1}{\beta_n}\int_{\alpha_1}^{\alpha_2}(\Psi\circ\beta_1)(x)\beta_2(x)dx\right| \leq\frac{1}{|\beta_n|}\int_{\alpha_1}^{\alpha_2}|\beta_2(x)(\bar{\beta}-\beta_1(x))|\left(\frac{|\Psi'(\bar{\beta})|^q+|\Psi'(\beta_1(x))|^q}{s+1}\right)^\frac{1}{q}dx. \end{aligned} \tag{14}\]
Theorem 13. Let \(\Psi:[\delta_1,\delta_2] \rightarrow \mathbb{R}\) be the differential function such that \(|\Psi'|^q\) is m-convex for \(q>1\) and let \(\beta_1,\beta_2\) be two integrable functions defined on the interval \([\alpha_1,\alpha_2]\) such that \(\beta_1(x)\in[\delta_1,\delta_2]\) for all \(x\in[\alpha_1,\alpha_2]\) and \(\int_{\alpha_1}^{\alpha_2}\beta_2(x)dx=\beta_n\neq0.\) Assume that \(\bar{\beta}=\frac{1}{\bar{\beta_n}}\int_{\alpha_1}^{\alpha_2}\beta_1(x)\beta_2(x)dx\in[\delta_1,\delta_2]\), then \[\label{h11} \begin{aligned} \left|\Psi{(\bar{\beta})}-\frac{1}{\beta_n}\int_{\alpha_1}^{\alpha_2}(\Psi\circ\beta_1)(x)\beta_2(x)dx\right|\leq\frac{1}{|\beta_n|}\int_{\alpha_1}^{\alpha_2}|\beta_2(x)(\bar{\beta}-\beta_1(x))|\left(\frac{m|\Psi'(\bar{\beta})|^q+|\Psi'(\beta_1(x))|^q}{2}\right)^\frac{1}{q}dx. \end{aligned} \tag{15}\]
Theorem 14. Let \(\Psi:[\delta_1,\delta_2] \rightarrow \mathbb{R}\) be the differential function such that \(|\Psi'|^q\) is \((\alpha,m)\)-convex for \(q>1\) and let \(\beta_1,\beta_2\) be two integrable functions defined on the interval \([\alpha_1,\alpha_2]\) such that \(\beta_1(x)\in[\delta_1,\delta_2]\) for all \(x\in[\alpha_1,\alpha_2]\) and \(\int_{\alpha_1}^{\alpha_2}\beta_2(x)dx=\beta_n\neq0.\) Assume that \(\bar{\beta}=\frac{1}{\bar{\beta_n}}\int_{\alpha_1}^{\alpha_2}\beta_1(x)\beta_2(x)dx\in[\delta_1,\delta_2]\), then \[\label{h12} \begin{aligned} \left|\Psi{(\bar{\beta})}-\frac{1}{\beta_n}\int_{\alpha_1}^{\alpha_2}(f\circ\beta_1)(x)\beta_2(x)dx\right| \leq\frac{1}{|\beta_n|}\int_{\alpha_1}^{\alpha_2}|\beta_2(x)(\bar{\beta}-\beta_1(x))|\left(\frac{|\Psi'(\bar{\beta})|^q+m|\Psi'(\beta_1(x))|^q}{\alpha+1}\right)^\frac{1}{q}dx. \end{aligned} \tag{16}\]
Theorem 15. Let \(\Psi:[\delta_1,\delta_2] \rightarrow \mathbb{R}\) be the differential function such that \(|\Psi'|^q\) is \(\eta\)-convex for \(q>1\) and let \(\beta_1,\beta_2\) be two integrable functions defined on the interval \([\alpha_1,\alpha_2]\) such that \(\beta_1(x)\in[\delta_1,\delta_2]\) for all \(x\in[\alpha_1,\alpha_2]\) and \(\int_{\alpha_1}^{\alpha_2}\beta_2(x)dx=\beta_n\neq0.\) Assume that \(\bar{\beta}=\frac{1}{\bar{\beta_n}}\int_{\alpha_1}^{\alpha_2}\beta_1(x)\beta_2(x)dx\in[\delta_1,\delta_2]\), then \[\label{h13} \begin{aligned} &\left|\Psi{(\bar{\beta})}-\frac{1}{\beta_n}\int_{\alpha_1}^{\alpha_2}(\Psi\circ\beta_1)(x)\beta_2(x)dx\right|\\ &\quad\leq\frac{1}{|\beta_n|}\int_{\alpha_1}^{\alpha_2}|\beta_2(x)(\bar{\beta}-\beta_1(x))|\left(|\Psi'({\beta_1})|^q+\frac{\eta(|\Psi'(\bar{\beta})|^q,|\Psi'(\beta_1(x))|^q)}{2}\right)^\frac{1}{q}dx. \end{aligned} \tag{17}\]
Theorem 16. Let \(\Psi:[\delta_1,\delta_2] \rightarrow \mathbb{R}\) be the differential function such that \(|\Psi'|^q\) is strongly \(\eta\)-convex for \(q>1\) and let \(\beta_1,\beta_2\) be two integrable functions defined on the interval \([\alpha_1,\alpha_2]\) such that \(\beta_1(x)\in[\delta_1,\delta_2]\) for all \(x\in[\alpha_1,\alpha_2]\) and \(\int_{\alpha_1}^{\alpha_2}\beta_2(x)dx=\beta_n\neq0.\) Assume that \(\bar{\beta}=\frac{1}{\bar{\beta_n}}\int_{\alpha_1}^{\alpha_2}\beta_1(x)\beta_2(x)dx\in[\delta_1,\delta_2]\), then \[\begin{aligned} \label{h14} &\left|\Psi{(\bar{\beta})}-\frac{1}{\beta_n}\int_{\alpha_1}^{\alpha_2}(\Psi\circ\beta_1)(x)\beta_2(x)dx\right|\nonumber\\ &\leq\frac{1}{|\beta_n|}\int_{\alpha_1}^{\alpha_2}|\beta_2(x)(\bar{\beta}-\beta_1(x))|\times\left(|\Psi'({\beta_1})|^q+\frac{\eta(|\Psi'(\bar{\beta})|^q,|\Psi'(\beta_1(x))|^q)}{2}-\frac{n(\bar{\beta}-\beta_1)^2}{6}\right)^\frac{1}{q}dx. \end{aligned} \tag{18}\]
Proposition 1. Let \((p_1,…,p_n),\) and \((q_1,…,q_n),\) be two positive n-tuples and \([\delta_1,\delta_2]\) be a positive interval. Then
1. for \(t_1>1, t_2\in(1,2)\) such that \(\frac{1}{t_1}+\frac{1}{t_2}=1.\) Following inequality holds \[\begin{aligned} \label{h15} & \left(\sum\limits_{k=1}^n p_k^{t_1} \right)^\frac{1}{t_1}\left(\sum\limits_{k=1}^n q_k^{t_2} \right)^\frac{1}{t_2}-\left(\sum\limits_{k=1}^n p_kq_k\right)\nonumber\\ &\leq \Bigg[\frac{t_2}{2^\frac{1}{q}}\sum\limits_{k=1}^n \left|p_k^{t_1}\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n p_k^{t_1}}-q_k p_k^{\frac{-t_1}{t_2}}\right)\right|\hspace{7cm}\nonumber\\ &\times\left\{\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n p_k^{t_1}}\right)^{q(t_2-1)} +\frac{q_k^{q(t_2-1)}}{p_k^{q}}-\frac{1}{2}(q^{2}t_2^{2}-2q^{2}t_2-qt_2+q^{2}+q)\delta_2^{q(t_2-1)-2}\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n p_k^{t_1}}-q_k p_k^{\frac{-t_1}{t_2}}\right)^2\right\}^{\frac{1}{q}}\Bigg]^{\frac{1}{t_2}}\nonumber\\ &\times \left(\sum\limits_{k=1}^n p_k^{t_1}\right)^{\frac{1}{t_1}}. \end{aligned} \tag{19}\]
2. If the statement of part 1 is satisfied, then the following inequality holds for \(t_2>2\) \[\begin{aligned} \label{h16} & \left(\sum\limits_{k=1}^n p_k^{t_1} \right)^\frac{1}{t_1}\left(\sum\limits_{k=1}^n q_k^{t_2} \right)^\frac{1}{t_2}-\left(\sum\limits_{k=1}^n p_kq_k\right)\nonumber\\ &\leq \Bigg[\frac{t_2}{2^\frac{1}{q}}\sum\limits_{k=1}^n\left|p_k^{t_1}\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n p_k^{t_1}}-q_k p_k^{\frac{-t_1}{t_2}}\right)\right|\nonumber\\ &\times\left\{\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n p_k^{t_1}}\right)^{q(t_2-1)}+\frac{q_k^{q(t_2-1)}}{p_k^{q}}-\frac{1}{2}(q^{2}t_2^{2}-2q^{2}t_2-qt_2+q^{2}+q)\delta_1^{q(t_2-1)-2}\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n p_k^{t_1}}-q_k p_k^{\frac{-t_1}{t_2}}\right)^2\right\}^{\frac{1}{q}}\Bigg]^{\frac{1}{t_2}}\nonumber\\ &\times \left(\sum\limits_{k=1}^n p_k^{t_1}\right)^{\frac{1}{t_1}}. \end{aligned} \tag{20}\]
Proof. 1. Let \(\Psi(x)=x^{t_2}, \ x\in[\delta_1,\delta_2]\) then \(\Psi''(x)=t_2(t_2-1)x^{t_2-2}>0, \ (|\Psi'|^q)^{''}=t_2^{q}q(t_2-1)(q(t_2-1)-1)x^{q(t_2-1)-2},\) which show that the function \(\Psi\) is convex, and for the given value of \(t_2\), the function \((|\Psi'|^q)^{''}(x)\) is decreasing while \((|\Psi'|^q)^{''}(x)\geq 2\left(\frac{t_2^{q}q(t_2-1)(q(t_2-1)-1)\delta_2^{q(t_2-1)-2}}{2}\right) \ \ \forall x\in[\delta_1,\delta_2].\) Therefore the function \(|\Psi'|^q\) is strongly convex with \(C=\frac{t_2^{q}q(t_2-1)(q(t_2-1)-1)\delta_2^{q(t_2-1)-2}}{2}\), so using (5) for \(\Psi(x)=x^{t_2},\rho_k=p_k^{t_1},\) and \(y_k=q_kp_k^{\frac{-t_1}{t_2}}\), we derive \[\begin{aligned} \label{h17} & \Bigg\{\left(\sum\limits_{k=1}^n q_k^{t_2} \right)\left(\sum\limits_{k=1}^n p_k^{t_1} \right)^{t_2-1}-\left(\sum\limits_{k=1}^np_kq_k\right)^{{t_2}}\Bigg\}^{\frac{1}{t_2}}\nonumber\\ &\leq \Bigg[\frac{t_2}{2^\frac{1}{q}}\sum\limits_{k=1}^n\left|p_k^{t_1}\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n p_k^{t_1}}-q_k p_k^{\frac{-t_1}{t_2}}\right)\right|\nonumber\\ &\times\left\{\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n p_k^{t_1}}\right)^{q(t_2-1)}+\frac{q_k^{q(t_2-1)}}{p_k^{q}}-\frac{1}{2}(q^{2}t_2^{2}-2q^{2}t_2-qt_2+q^{2}+q)\delta_2^{q(t_2-1)-2}\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n p_k^{t_1}}-q_k p_k^{\frac{-t_1}{t_2}}\right)^2\right\}^{\frac{1}{q}}\Bigg]^{\frac{1}{t_2}}\nonumber\\ &\times \left(\sum\limits_{k=1}^n p_k^{t_1}\right)^{\frac{1}{t_1}}. \end{aligned} \tag{21}\]
By applying the inequality \(x^n-y^n\leq(x-y)^n\) holds for \(n\in[0,1]\) and \(0\leq y\leq x,\) therefore by putting \(x=\left(\sum\limits_{k=1}^nq_k^{t_2}\right)\left(\sum\limits_{k=1}^n p_k^{t_1} \right)^{t_2-1}\), \(y=\left(\sum\limits_{k=1}^np_kq_k\right)^{{t_2}}\), \(n=\frac{1}{t_2}\) we obtain \[\begin{aligned} \label{h18} \left(\sum\limits_{k=1}^n p_k^{t_1} \right)^\frac{1}{t_1}\left(\sum\limits_{k=1}^n q_k^{t_2} \right)^\frac{1}{t_2}-\sum\limits_{k=1}^np_kq_k\leq \left(\left(\sum\limits_{k=1}^n q_k^{t_2} \right)\left(\sum\limits_{k=1}^n p_k^{t_1} \right)^{t_2-1}-\left(\sum\limits_{k=1}^np_kq_k\right)^{{t_2}}\right)^{\frac{1}{t_2}}. \end{aligned} \tag{22}\]
From (21) and (22), we get (19).
2. For \(C=\frac{t_2^{q}q(t_2-1)(q(t_2-1)-1)\delta_1^{q(t_2-1)-2}}{2}\), as by same proposed value of \(t_2\), the function \((|\Psi'|^q)^{''}(x)\) become increasing function. Now by applying the same method of part 1 the inequality (20) can be obtained. ◻
Proposition 2. Let \((p_1,…,p_n),\) and \((q_1,…,q_n),\) be two positive n-tuples
and \([\delta_1,\delta_2]\) be a
positive interval. Then for \(t_1>1\), \(t_2
\notin(1,1+\frac{1}{q})\) for \(q>1\) such that \(\frac{1}{t_1}+\frac{1}{t_2}=1\), the
following inequality holds
\[\begin{aligned}
\label{h19}
& \left(\sum\limits_{k=1}^n p_k^{t_1}
\right)^\frac{1}{t_1}\left(\sum\limits_{k=1}^n q_k^{t_2}
\right)^\frac{1}{t_2}-\left(\sum\limits_{k=1}^n p_kq_k\right)\nonumber\\
&\leq \left[\frac{t_2}{(s+1)^\frac{1}{q}}\sum\limits_{k=1}^n
\left|p_k^{t_1}\left(\frac{\sum\limits_{k=1}^n p_k
q_k}{\sum\limits_{k=1}^n p_k^{t_1}}-q_k
p_k^{\frac{-t_1}{t_2}}\right)\right|
\left\{\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n
p_k^{t_1}}\right)^{q(t_2-1)}
+\frac{q_k^{q(t_2-1)}}{p_k^{q}}\right\}^{\frac{1}{q}}\right]^{\frac{1}{t_2}}\times\left(\sum\limits_{k=1}^n
p_k^{t_1}\right)^{\frac{1}{t_1}}.
\end{aligned} \tag{23}\]
Proof. Let \(\Psi(x)=x^{t_2}, \ x\in[\delta_1,\delta_2],\) then \(\Psi''(x)=t_2^{q}q(t_2-1)(q(t_2-1)-1)x^{q(t_2-1)-2},\) which show that \(\Psi\) is a convex function. Also it can be easily verified that \(|\Psi'|^q\) is s-convex, and so using (8) for \(\Psi(x)=x^{t_2}, \rho_k=p_k^{t_1},\) and \(y_k=q_k p_k^{\frac{-t_1}{t_2}}\), we derive \[\begin{aligned} \label{h20} & \left\{\left(\sum\limits_{k=1}^n q_k^{t_2} \right)\left(\sum\limits_{k=1}^n p_k^{t_1} \right)^{t_2-1}-\left(\sum\limits_{k=1}^np_kq_k\right)^{{t_2}}\right\}^{\frac{1}{t_2}}\nonumber\\ &\leq \left[\frac{t_2}{(s+1)^\frac{1}{q}}\sum\limits_{k=1}^n\left|p_k^{t_1}\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n p_k^{t_1}}-q_k p_k^{\frac{-t_1}{t_2}}\right)\right|\left\{\left(\frac{\sum\limits_{k=1}^n p_k q_k}{\sum\limits_{k=1}^n p_k^{t_1}}\right)^{q(t_2-1)}+\frac{q_k^{q(t_2-1)}}{p_k^{q}}\right\}^{\frac{1}{q}}\right]^{\frac{1}{t_2}}\times\left(\sum\limits_{k=1}^n p_k^{t_1}\right)^{\frac{1}{t_1}}. \end{aligned} \tag{24}\]
By applying the inequality \(x^n-y^n\leq(x-y)^n\) holds for \(n\in[0,1]\) and \(0\leq y\leq x,\) therefore by putting \(x=\left(\sum\limits_{k=1}^nq_k^{t_2}\right)\left(\sum\limits_{k=1}^n p_k^{t_1} \right)^{t_2-1}\), \(y=\left(\sum\limits_{k=1}^np_kq_k\right)^{{t_2}}\), \(n=\frac{1}{t_2}\) we obtain \[\begin{aligned} \label{h21} &\left(\sum\limits_{k=1}^n p_k^{t_1} \right)^\frac{1}{t_1}\left(\sum\limits_{k=1}^n q_k^{t_2} \right)^\frac{1}{t_2}-\sum\limits_{k=1}^np_kq_k\leq \left(\left(\sum\limits_{k=1}^n q_k^{t_2} \right)\left(\sum\limits_{k=1}^n p_k^{t_1} \right)^{t_2-1}-\left(\sum\limits_{k=1}^np_kq_k\right)^{{t_2}}\right)^{\frac{1}{t_2}}. \end{aligned} \tag{25}\]
From (24) and (25), we get (23). ◻
Remark 1. Analogously, we can present results for the Hölder difference as applications of Theorem 7–Theorem 10.
Remark 2. Similarly, integral versions of the above propositions can be established as direct applications of Theorem 11–Theorem 16.
The following corollaries propose bounds for the Hermite-Hadamard gap via aforementioned generalized convex functions.
Corollary 1. Let \([\alpha_1,\alpha_2]\) be a positive interval, consider that \(\Psi:[\alpha_1,\alpha_2] \rightarrow \mathbb{R}\) such that \(|\Psi'|^q\) is a strongly convex function with modulus C, then \[\begin{aligned} \label{h31} &\left|\frac{1}{\alpha_2-\alpha_1}\int_{\alpha_1}^{\alpha_2}\Psi(x)dx-f\left(\frac{\alpha_1+\alpha_2}{2}\right) \right|\nonumber\\ &\leq\frac{1}{\alpha_2-\alpha_1}\int_{\alpha_1}^{\alpha_2}\left|\left(\frac{\alpha_1+\alpha_2}{2}\right)-x\right|\left(\frac{|\Psi'(\frac{\alpha_1+\alpha_2}{2})|^q+|\Psi'(x)|^q-C((\frac{\alpha_1+\alpha_2}{2})-x)^2}{2}\right)^\frac{1}{q}dx. \end{aligned} \tag{26}\]
Proof. As function \(|\Psi'|^q\) is strongly convex on \([\alpha_1,\alpha_2]\). Therefore, using (13) i.e \[\begin{aligned} &\ \left|\Psi{(\bar{\beta})}-\frac{1}{\beta_n}\int_{\alpha_1}^{\alpha_2}(\Psi\circ\beta_1)(x)\beta_2(x)dx \right|\\ &\leq\frac{1}{|\beta_n|}\int_{\alpha_1}^{\alpha_2}|\beta_2(x)\left(\bar{\beta}-\beta_1(x)\right)|\left(\frac{|\Psi'(\bar{\beta})|^q+|\Psi'(\beta_1(x))|^q-C(\bar{\beta}-\beta_1)^2}{2}\right)^\frac{1}{q}dx. \end{aligned}\]
Substituting \(\beta_1(x)=x\), \(\beta_2(x)=1\), so \(\beta_n=\alpha_2-\alpha_1\) and \(\bar{\beta}=\frac{\alpha_1+\alpha_2}{2}\), thus the above inequality becomes \[\begin{aligned} &\ \Big|\Psi{\left(\frac{\alpha_1+\alpha_2}{2}\right)}-\frac{1}{(\alpha_2-\alpha_1)}\int_{\alpha_1}^{\alpha_2}\Psi(x)dx\Big|\\ &\leq\frac{1}{\alpha_2-\alpha_1}\int_{\alpha_1}^{\alpha_2}\left|\left(\frac{\alpha_1+\alpha_2}{2}\right)-x\right|\left(\frac{|\Psi'(\frac{\alpha_1+\alpha_2}{2})|^q+|\Psi'(x)|^q-C(\frac{\alpha_1+\alpha_2}{2})-x)^2}{2}\right)^\frac{1}{q}dx. \end{aligned}\]
After some calculations, we get the required result. ◻
Corollary 2. Let \([\alpha_1,\alpha_2]\) be a positive interval, consider that \(\Psi:[\alpha_1,\alpha_2] \rightarrow \mathbb{R}\) such that \(|\Psi'|^q\) is an s-convex function, then \[\begin{aligned} \label{h32} \left|\frac{1}{\alpha_2-\alpha_1}\int_{\alpha_1}^{\alpha_2}\Psi(x)dx-\Psi\left(\frac{\alpha_1+\alpha_2}{2}\right) \right|\leq\frac{1}{\alpha_2-\alpha_1}\int_{\alpha_1}^{\alpha_2}\Big|\left(\frac{\alpha_1+\alpha_2}{2}\right)-x\Big|\left(\frac{|\Psi'(\frac{\alpha_1+\alpha_2}{2})|^q+|\Psi'(x)|^q}{s+1}\right)^\frac{1}{q}dx. \end{aligned} \tag{27}\]
Proof. As function \(|\Psi'|^q\) is s-convex on \([\alpha_1,\alpha_2]\). Hence, using (14) i.e \[\begin{aligned} \left|\Psi{(\bar{\beta})}-\frac{1}{\beta_n}\int_{\alpha_1}^{\alpha_2}(\Psi\circ\beta_1)(x)\beta_2(x)dx \right| \leq\frac{1}{|\beta_n|}\int_{\alpha_1}^{\alpha_2}|\beta_2(x)(\bar{\beta}-\beta_1(x))|\left(\frac{|\Psi'(\bar{\beta})|^q+|\Psi'(\beta_1(x))|^q}{s+1}\right)^\frac{1}{q}dx. \end{aligned}\]
Substituting \(\beta_1(x)=x\) and \(\beta_2(x)=1\), so \(\beta_n=\alpha_2-\alpha_1\) and \(\bar{\beta}=\frac{\alpha_1+\alpha_2}{2},\) therefore the above inequality becomes \[\begin{aligned} \left|\Psi{\left(\frac{\alpha_1+\alpha_2}{2}\right)}-\frac{1}{(\alpha_2-\alpha_1)}\int_{\alpha_1}^{\alpha_2}\Psi(x)dx\right|\leq\frac{1}{\alpha_2-\alpha_1}\int_{\alpha_1}^{\alpha_2}\left|\left(\frac{\alpha_1+\alpha_2}{2}\right)-x\right| \left(\frac{|\Psi'(\frac{\alpha_1+\alpha_2}{2})|^q+|\Psi'(x)|^q}{s+1}\right)^\frac{1}{q}dx. \end{aligned}\]
After some calculations, we obtain the required result. ◻
Remark 3. We can establish similar results around the Hermite-Hadamard inequality as direct applications of Theorem 11–Theorem 16.
In this section, we present applications of the main integral results
to Csis\(\acute{z}\)r divergence, which
is the key tool in information theory as this gives a platform to other
divergences, entropies, or distances. Before we proceed, first we recall
Csiszár divergence [9]: Csiszár divergence: Let
\([\omega_1,\omega_2]\subseteq
\mathbb{R}\) and \(\Psi:[\omega_1,\omega_2] \rightarrow
\mathbb{R}\) be a function. Also let \(Y:[\alpha_1,\alpha_2]\rightarrow[\omega_1,\omega_2]\),
\(Z:[\alpha_1,\alpha_2]\rightarrow(0,\infty)\)
be two functions such that \(\frac{Y(x)}{Z(x)}\in[\omega_1,\omega_2]\)
for all \(x\in[\alpha_1,\alpha_2]\)
then the Csiszár divergence is defined by
\[D^c(Y,Z)=\int_{\alpha_1}^{\alpha_2}Z(x)\Psi
\left(\frac{Y(x)}{Z(x)}\right)dx.\]
Theorem 17. Let \(\Psi:[\omega_1,\omega_2]\rightarrow \mathbb{R}\) be a function such that \(|\Psi'|^q\) is strongly convex with modulus C, Also \(Y:[\alpha_1,\alpha_2]\rightarrow[\omega_1,\omega_2]\), \(Z:[\alpha_1,\alpha_2]\rightarrow(0,\infty)\) be two functions such that \(\frac{\int_{\alpha_1}^{\alpha_2}Y(x)}{\int_{\alpha_1}^{\alpha_2}Z(x)}, \ \frac{Y(x)}{Z(x)}\in[\omega_1,\omega_2]\), for all \(x\in[\alpha_1,\alpha_2]\) then \[\begin{aligned} \label{h37} & \left|\frac{1}{\int_{\alpha_1}^{\alpha_2}Z(x)dx} D^c(Y,Z)-\Psi\left(\frac{\int_{\alpha_1}^{\alpha_2}Y(x)dx}{\int_{\alpha_1}^{\alpha_2}Z(x)dx}\right) \right|\nonumber\\ &\leq \frac{1}{2^\frac{1}{q}} \frac{1}{\int_{\alpha_1}^{\alpha_2}Z(x)dx}\int_{\alpha_1}^{\alpha_2}\left|Z(x)\left(\frac{\int_{\alpha_1}^{\alpha_2}Y(x)dx}{\int_{\alpha_1}^{\alpha_2}Z(x)dx} -\frac{Y(x)}{Z(x)}\right)\right|\nonumber\\ &\times\left(\left|\Psi'\left(\frac{\int_{\alpha_1}^{\alpha_2}Y(x)dx}{\int_{\alpha_1}^{\alpha_2}Z(x)dx}\right)\right|^q+\left|\Psi' \left(\frac{Y(x)}{Z(x)}\right)\right|^q-C\left(\frac{\int_{\alpha_1}^{\alpha_2}Y(x)dx}{\int_{\alpha_1}^{\alpha_2}Z(x)dx}-\frac{Y(x)}{Z(x)}\right)^2\right)^\frac{1}{q}dx.\nonumber\\ \end{aligned} \tag{28}\]
Proof. As function \(|\Psi'|^q\) is strongly convex with modulus C, on \([\alpha_1,\alpha_2]\). Hence using (13) i.e \[\begin{aligned} &\left|\Psi{(\bar{\beta})}-\frac{1}{\beta_n}\int_{\alpha_1}^{\alpha_2}(\Psi\circ\beta_1)(x)\beta_2(x)dx\right|\\ &\leq\frac{1}{|\beta_n|}\int_{\alpha_1}^{\alpha_2}|\beta_2(x)(\bar{\beta}-\beta_1(x))|\left(\frac{|\Psi'(\bar{\beta})|^q+|\Psi'(\beta_1(x))|^q-C(\bar{\beta}-\beta_1)^2}{2}\right)^\frac{1}{q}dx. \end{aligned}\]
Substituting \(\beta_1=\frac{Y(x)}{Z(x)}\), \(\beta_2=Z(x)\) in the above result, we get the required inequality. ◻
Here. we have another inequality for the aforementioned divergence.
Theorem 18. Let \(\Psi:[\omega_1,\omega_2]\rightarrow \mathbb{R}\) be a function such that \(|\Psi'|^q\) is s-convex. Also \(Y:[\alpha_1,\alpha_2]\rightarrow[\omega_1,\omega_2]\), \(Z:[\alpha_1,\alpha_2]\rightarrow(0,\infty)\) be two functions such that \(\frac{\int_{\alpha_1}^{\alpha_2}Y(x)}{\int_{\alpha_1}^{\alpha_2}Z(x)}, \ \frac{Y(x)}{Z(x)}\in[\omega_1,\omega_2]\), for all \(x\in[\alpha_1,\alpha_2]\) then \[\begin{aligned} \label{h38} \left|\frac{1}{\int_{\alpha_1}^{\alpha_2}Z(x)dx} D^c(Y,Z)-\Psi\left(\frac{\int_{\alpha_1}^{\alpha_2}Y(x)dx}{\int_{\alpha_1}^{\alpha_2}Z(x)dx}\right) \right| &\leq \frac{1}{(s+1)^\frac{1}{q}} \frac{1}{\int_{\alpha_1}^{\alpha_2}Z(x)dx}\int_{\alpha_1}^{\alpha_2}\left|Z(x)\left(\frac{\int_{\alpha_1}^{\alpha_2}Y(x)dx}{\int_{\alpha_1}^{\alpha_2}Z(x)dx}-\frac{Y(x)}{Z(x)}\right)\right|\nonumber\\ &\quad\times\left(\Big|\Psi'\Big(\frac{\int_{\alpha_1}^{\alpha_2}Y(x)dx}{\int_{\alpha_1}^{\alpha_2}Z(x)dx}\Big)\Big|^q+\left|\Psi' \left(\frac{Y(x)}{Z(x)}\right)\right|^q\right)^\frac{1}{q}dx. \end{aligned} \tag{29}\]
Proof. As function \(|\Psi'|^q\) is s-convex, on \([\alpha_1,\alpha_2]\). Thus using (14) i.e \[\begin{aligned} |\Psi{(\bar{\beta})}-\frac{1}{\beta_n}\int_{\alpha_1}^{\alpha_2}(\Psi\circ\beta_1)(x)\beta_2(x)dx|\leq\frac{1}{|\beta_n|}\int_{\alpha_1}^{\alpha_2}|\beta_2(x)(\bar{\beta}-\beta_1(x))| \left(\frac{|\Psi'(\bar{\beta})|^q+|\Psi'(\beta_1(x))|^q}{s+1}\right)^\frac{1}{q}dx. \end{aligned}\]
Substituting \(\beta_1=\frac{Y(x)}{Z(x)}\) and \(\beta_2=Z(x)\) in the above inequality, we get the desired inequality. ◻
Remark 4. Analogously, we can establish other inequalities for the Csiszár divergence as direct applications of Theorem 13–Theorem 16.
Remark 5. Analogously, we can establish inequalities for the Csiszár divergence in discrete form as direct applications of Theorem 5–Theorem 10.
Remark 6. The classical version of Csiszár divergence is presented as: The Csiszár \(f\)-divergence for positive tuples \((r_{1},r_{2},\dots,r_{n})=\mathbf{x}\), and \((w_{1},w_{2},\dots,w_{n})=\mathbf{y}\) and a function \(f:[a,b]\rightarrow\mathbb{R}\) is provided with the following mathematical formula [11]: \[D(\mathbf{c};\mathbf{x},\mathbf{y})=\sum\limits_{k=1}^{n}y_{k}f\left(\frac{x_{k}}{y_{k}}\right).\]
Using the following, we may establish such results for various other divergences, entropies, or distances in discrete form.
Shannon entropy;\[E(\mathbf{s},\mathbf{y})=-\sum\limits_{k=1}^{n}y_{k}\log y_{k};\quad f(x)=-\log x.\]
Renyi-divergence;\[D(\mathbf{re},\mathbf{x},\mathbf{y})=\frac{1}{\mu-1}\log\left(\sum\limits_{k=1}^{m}x_{k}^{\mu}y_{k}^{1-\mu}\right);\quad f(x)=x^\alpha,~\alpha>1.\]
Kullback-Leibler divergence;\[D(\mathbf{kl},\mathbf{x},\mathbf{y})=\sum\limits_{k=1}^{n}x_{k}\log\left(\frac{x_{k}}{y_{k}}\right);\quad f(x)=x\log x.\]
\(\chi^{2}\)-divergence;\[D({\chi^{2}},\mathbf{x},\mathbf{y})=\sum\limits_{k=1}^{m}\frac{(x_{k}-y_{k})^{2}}{y_{k}};~f(x)=(x-1)^{2}.\]
Triangular Discrimination;\[Di({\triangle},\mathbf{x},\mathbf{y})=\sum\limits_{k=1}^{n}\frac{(x_{k}-y_{k})^{2}}{x_{k}+y_{k}};~f(x)=\frac{(x-1)^2}{x+1}.\]
Bhattacharyya-coefficient;\[C(\mathbf{b},\mathbf{x},\mathbf{y})=\sum\limits_{k=1}^{m}\sqrt{x_{k}y_{k}};~f(x)=-\sqrt x.\]
Hellinger Distance;\[Dis(h^{2},\mathbf{x},\mathbf{y})=\frac{1}{2}\sum\limits_{k=1}^{n}(\sqrt{x_{k}}-\sqrt{y_{k}})^{2};~f(x)=\frac{1}{2}\left(1-\sqrt x\right)^2.\]
Similarly, such results can be established for their integral variants as well.
In this work, several generalized upper bounds for the Jensen gap have been successfully established by employing a variety of generalized convex functions, including strongly convex, \(s\)-convex, \(\eta\)-convex, strongly-\(\eta\) convex, \(m-\)convex, and \((a,m)\)-convex functions. These generalizations extend the scope of the classical Jensen inequality and provide new insights into its integral form. Furthermore, the derived inequalities have been effectively utilized to obtain refined versions of the well-known Hölder and Hermite-Hadamard inequalities. As direct applications, novel estimates for the Csiszár divergence were also developed, demonstrating the theoretical and practical relevance of the presented results. Overall, the findings of this study contribute to a deeper understanding of convexity-based inequalities and open avenues for further exploration in mathematical analysis and information theory.
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