Some Hermite-Hadamard type inequalities in the class of hyperbolic p-convex functions

TL;DR

This paper introduces new Hermite-Hadamard type inequalities for p-hyperbolic convex functions using fractional integrals, extending classical inequalities within this specific convexity class.

Contribution

It presents novel Hermite-Hadamard and Hermite-Hadamard-Fejer inequalities for p-hyperbolic convex functions derived through fractional integrals.

Findings

New inequalities for p-hyperbolic convex functions

Simplification of inequalities as consequences of Hermite-Hadamard-Fejer inequality

Extension of classical convexity inequalities

Abstract

In this paper, obtained some new class of Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities via fractional integrals for the p-hyperbolic convex functions. It is shown that such inequalities are simple consequences of Hermite-Hadamard-Fejer inequality for the p-hyperbolic convex function.

Equations

$$u\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}\underset{a}{\overset{b}{\int }}u(y)dy\le \frac{u(a)+u(b)}{2}.$$

$$u(\mu x+(1-\mu )y)\le \mu u(x)+(1-\mu )u(y)$$

$$u\left(\frac{a+b}{2}\right)\underset{a}{\overset{b}{\int }}v(y)dy\le \underset{a}{\overset{b}{\int }}u(y)v(y)dy\le \frac{u(a)+u(b)}{2}\underset{a}{\overset{b}{\int }}v(y)dy$$

$$I_{a+}^{\alpha }\left[f\right]\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}\underset{a}{\overset{t}{\int }}{\left(t-s\right)}^{\alpha -1}f\left(s\right)ds,t\in (a,b],$$

$$I_{b-}^{\alpha }\left[f\right]\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}\underset{t}{\overset{b}{\int }}{\left(s-t\right)}^{\alpha -1}f\left(s\right)ds,t\in [a,b),$$

$${\mathcal{I}}_{a+}^{\alpha }u(x)=\frac{1}{\alpha }\underset{a}{\overset{x}{\int }}\exp \left(-\frac{1-\alpha }{\alpha }(x-s)\right)u(s)ds,x>a$$

$${\mathcal{I}}_{b-}^{\alpha }u(x)=\frac{1}{\alpha }\underset{x}{\overset{b}{\int }}\exp \left(-\frac{1-\alpha }{\alpha }(s-x)\right)u(s)ds,x<b$$

$$u\left(\frac{a+b}{2}\right)\le \frac{\Gamma (\alpha +1)}{2(b-a{)}^{\alpha }}\left[I_a^{\alpha }u(b)+I_b^{\alpha }u(a)\right]\le \frac{u(a)+u(b)}{2}$$

$$\begin{array}{rl}u\left(\frac{a+b}{2}\right)\left[I_a^{\alpha }v(b)+I_b^{\alpha }v(a)\right]& \le \left[I_a^{\alpha }(uv)(b)+I_b^{\alpha }(uv)(a)\right]\\ & \le \frac{u(a)+u(b)}{2}\left[I_a^{\alpha }v(b)+I_b^{\alpha }v(a)\right]\end{array}$$

$$u\left(\frac{a+b}{2}\right)\le \frac{1-\alpha }{2\left(1-\exp \left(-\rho \right)\right)}\left[{\mathcal{I}}_a^{\alpha }u(b)+{\mathcal{I}}_b^{\alpha }u(a)\right]\le \frac{u(a)+u(b)}{2};$$

$$\begin{array}{rl}u\left(\frac{a+b}{2}\right)\left[{\mathcal{I}}_a^{\alpha }w(b)+{\mathcal{I}}_b^{\alpha }w(a)\right]& \le \left[{\mathcal{I}}_a^{\alpha }\left(uw\right)(b)+{\mathcal{I}}_b^{\alpha }\left(uw\right)(a)\right]\\ & \le \frac{u(a)+u(b)}{2}\left[{\mathcal{I}}_a^{\alpha }w(b)+{\mathcal{I}}_b^{\alpha }w(a)\right].\end{array}$$

$$\sinh x:=\frac{e^x-e^{-x}}{2},$$

$$\cosh x:=\frac{e^x+e^{-x}}{2},$$

$$\tanh x:=\frac{\sinh x}{\cosh x},$$

$$\coth x:=\frac{\cosh x}{\sinh x}.$$

$$f(x)\le \frac{\sinh [p(b-x)]}{\sinh [p(b-a)]}f(a)+\frac{\sinh [p(x-a)]}{\sinh [p(b-a)]}f(b)$$

$$H(x)=H(x;a,b,f):=A\cosh (px)+B\sinh (px),$$

$$f((1-t)a+tb)\le \frac{\sinh [p(1-t)(b-a)]}{\sinh [p(b-a)]}f(a)+\frac{\sinh [pt(b-a)]}{\sinh [p(b-a)]}f(b)$$

$$f(y)\ge f(x)\cosh [p(y-x)]+K_{x,f}\sin [p(y-x)]$$

$$\phi (x)=f^{\prime }(x)-p^2\underset{a}{\overset{x}{\int }}f(t)dt$$

$$f^{\prime \prime }(x)-p^{2}f(x)\ge 0.$$

$$f_r^{\prime \prime }(x)-p^2{f}_r(x)=r(r-1)x^{r-2}-p^2{x}^r=p^2{x}^{r-2}\left(\frac{r(r-1)}{p^2}-x^2\right),x>0.$$

$$f_r^{\prime \prime }(x)-p^2{f}_r(x)>0\text{for}x\in \left(0,\frac{\sqrt{r(r-1)}}{|p|}\right)$$

$$f_r^{\prime \prime }(x)-p^2{f}_r(x)<0\text{for}x\in \left(\frac{\sqrt{r(r-1)}}{|p|},\infty \right),$$

$$f_r^{\prime \prime }(x)-p^2{f}_r(x)<0$$

$$f_{\mu }(x)=\exp (\mu x)=e^{\mu x}$$

$$f_{\mu }^{\prime \prime }(x)-p^2{f}_{\mu }(x)={\mu }^2{e}^{\mu x}-p^2{e}^{\mu x}=({\mu }^2-p^2)e^{\mu x},x>0.$$

$$\frac{2}{p}u\left(\frac{a+b}{2}\right)\sinh \left(\frac{p(b-a)}{2}\right)\le \underset{a}{\overset{b}{\int }}f(x)dx\le \frac{u(a)+u(b)}{p}\tanh \left(\frac{p(b-a)}{2}\right).$$

$$\begin{array}{rl}u\left(\frac{a+b}{2}\right)& \underset{a}{\overset{b}{\int }}\cosh \left[p\left(x-\frac{a+b}{2}\right)\right]w(x)dx\\ & \le \underset{a}{\overset{b}{\int }}u(x)w(x)dx\\ & \le \frac{u(a)+u(b)}{2}{\cosh }^{-1}\frac{p(b-a)}{2}\underset{a}{\overset{b}{\int }}\cosh \left[p\left(x-\frac{a+b}{2}\right)\right]w(x)dx.\end{array}$$

$$\begin{array}{rl}& \underset{a}{\overset{b}{\int }}u(x)w(x)dx\\ & \le \frac{u(a)+u(b)}{2}{\cosh }^{-1}\left(\frac{p(b-a)}{2}\right)\underset{a}{\overset{b}{\int }}\cosh \left[p\left(x-\frac{a+b}{2}\right)\right]w(x)dx\\ & +\frac{u(a)-u(b)}{2}{\sinh }^{-1}\left(\frac{p(b-a)}{2}\right)\underset{a}{\overset{b}{\int }}\sinh \left[p\left(x-\frac{a+b}{2}\right)\right]w(x)dx.\end{array}$$

Full text

• Corresponding author. E-mail: torebek@math.kz

1. 1 Introduction

2. 1.1 Definitions of fractional integrals

3. 1.2 Some generalizations of Hermite-Hadamard and Hermite-Hadamard-Fejér inequalities

4. 1.3 Hyperbolic p-convex functions

5. 1.4 Hermite-Hadamard and Hermite-Hadamard-Fejér inequalities for hyperbolic p-convex functions

6. 2 Main results

7. 2.1 Fractional analogues of Hermite-Hadamard inequality for hyperbolic p-convex functions

8. 2.2 Fractional analogues of Hermite-Hadamard-Fejér inequality for hyperbolic p-convex functions

1. Introduction

The inequalities for convex functions due to Hermite and Hadamard are found to be of great importance, for example, see [DP00, PPT92]. According to the inequalities [H93, H83],

• •

if $u:\,I\rightarrow\mathbb{R}$ is a convex function on the interval $I\subset\mathbb{R}$ and $a,b\in I$ with $b>a,$ then

[TABLE]

For a concave function $u$, the inequalities in (1.1) hold in the reversed direction.

Definition 1.1**.**

A function $u:[a,b]\subset\mathbb{R}\rightarrow\mathbb{R}$ is said to be convex if

[TABLE]

for all $x,y\in[a,b]$ and $\mu\in[0,1].$ We call $u$ a concave function if $(-u)$ is convex.

We note that Hadamard’s inequality refines the concept of convexity, and it follows from Jensen’s inequality. The classical Hermite-Hadamard inequality yields estimates for the mean value of a continuous convex function $u:[a,b]\rightarrow\mathbb{R}.$ The well-known inequalities dealing with the integral mean of a convex function $u$ are the Hermite-Hadamard inequalities or its weighted versions. They are also known as Hermite-Hadamard-Fejér inequalities.

In [F06], Fejér obtained the weighted generalization of Hermite-Hadamard inequality (1.1) as follows.

• •

Let $u:\,[a,b]\rightarrow\mathbb{R}$ be a convex function. Then the inequality

[TABLE]

holds for a nonnegative, integrable function $v:\,[a,b]\rightarrow\mathbb{R}$, which is symmetric to $\frac{a+b}{2}.$

Clearly, for $v(x)\equiv 1$ on $[a,b]$ we get (1.1).

1.1. Definitions of fractional integrals

Let us give basic definitions of fractional integrations of the different types.

Definition 1.2**.**

[KST06] The left and right Riemann–Liouville fractional integrals $I_{a+}^{\alpha}$ and $I_{b-}^{\alpha}$ of order $\alpha\in\mathbb{R}$ ($\alpha>0$) are given by

[TABLE]

and

[TABLE]

respectively. Here $\Gamma$ denotes the Euler gamma function.

Definition 1.3**.**

[KT16] Let $f\in L_{1}(a,b).$ The fractional integrals $\mathcal{I}^{\alpha}_{a+}$ and $\mathcal{I}^{\alpha}_{b-}$ of order $\alpha\in(0,1)$ are defined by

[TABLE]

and

[TABLE]

respectively.

1.2. Some generalizations of Hermite-Hadamard and Hermite-Hadamard-Fejér inequalities

Here we present some results on the generalization of the above inequalities.

In [SSYB13], Sarikaya et. al. represented Hermite-Hadamard inequality in Riemann-Liouville fractional integral forms as follows.

• •

Let $u:\,[a,b]\rightarrow\mathbb{R}$ be a positive function and $u\in L^{1}([a,b]).$ If $u$ is a convex function on $[a,b],$ then the following inequalities for fractional integrals hold

[TABLE]

with $\alpha>0.$

In [I16], Işcan gave the following Hermite-Hadamard-Fejér integral inequalities via fractional integrals:

• •

Let $u:\,[a,b]\rightarrow\mathbb{R}$ be convex function with $a<b$ and $u\in L^{1}([a,b]).$ If $v:\,[a,b]\rightarrow\mathbb{R}$ is nonnegative, integrable and symmetric to $(a+b)/2,$ then the following inequalities for fractional integrals hold

[TABLE]

with $\alpha>0.$

In [KT16] the authors obtained the following generalizations of inequality (1.1) and (1.2)

• •

Let $u:[a,b]\rightarrow\mathbb{R}$ and $u\in L^{1}(a,b).$ If $u$ is a convex function on $[a,b],$ then the following inequalities for fractional integrals $\mathcal{I}^{\alpha}_{a+}$ and $\mathcal{I}^{\alpha}_{a+}$ hold:

[TABLE]

• •

Let $u:\,[a,b]\rightarrow\mathbb{R}$ be convex and integrable function with $a<b.$ If $w:\,[a,b]\rightarrow\mathbb{R}$ is nonnegative, integrable and symmetric with respect to $\frac{a+b}{2},$ that is, $w(a+b-x)=w(x),$ then the following inequalities hold

[TABLE]

Many generalizations and extensions of the Hermite-Hadamard and Hermite-Hadamard-Fejér type inequalities were obtained for various classes of functions using fractional integrals; see [C16, CK17, HYT14, I16, JS16, SSYB13, WLFZ12, ZW13] and references therein. In [KS18] Kirane and Samet show that most of those results are particular cases of (or equivalent to) existing inequalities from the literature. These studies motivated us to consider a new class of functional inequalities for hyperbolic p-convex functions generalizing the classical Hermite-Hadamard and Hermite-Hadamard-Fejér inequalities.

1.3. Hyperbolic p-convex functions

We consider the hyperbolic functions of a real argument $x\in\mathbb{R}$ defined by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Definition 1.4**.**

[D18a, D18b] We say that a function $f:I\rightarrow\mathbb{R}$ is hyperbolic p-convex (or sub H-function, according with [A16]) on $I,$ if for any closed subinterval $[a,b]$ of $I$ we have

[TABLE]

for all $x\in[a,b].$

If the inequality (1.9) holds with ”$\geq$”, then the function will be called hyperbolic p-concave on $I.$

Geometrically speaking, this means that the graph of $f$ on $[a,b]$ lies nowhere above the p-hyperbolic function determined by the equation

[TABLE]

where $A$ and $B$ are chosen such that $H(a)=f(a)$ and $H(b)=f(b).$

If we take $x=(1-t)a+tb\in[a,b],t\in[0,1],$ then the condition (1.9) becomes

[TABLE]

for any $t\in[0,1].$ We have the following properties of hyperbolic p-convex function on $I.$

**(i): **

A hyperbolic p-convex function $f:I\rightarrow\mathbb{R}$ has finite right and left derivatives $f^{\prime}_{+}(x)$ and $f^{\prime}_{-}(x)$ at every point $x\in I$ and $f^{\prime}_{-}(x)\leq f^{\prime}_{+}(x).$ The function $f$ is differentiable on $I$ with the exception of an at most countable set.

**(ii): **

A necessary and sufficient condition for the function $f:I\rightarrow\mathbb{R}$ to be hyperbolic p-convex function on $I$ is that it satisfies the gradient inequality

[TABLE]

for any $x,y\in I,$ where $K_{x,f}\in\left[f^{\prime}_{-}(x),f^{\prime}_{+}(x)\right].$ If $f$ is differentiable at the point $x$ then $K_{x,f}=f^{\prime}(x).$

**(iii): **

A necessary and sufficient condition for the function $f$ to be a hyperbolic p-convex in $I,$ is that the function

[TABLE]

is nondecreasing on $I,$ where $a\in I.$

**(iv): **

Let $f:I\rightarrow\mathbb{R}$ be a two times continuously differentiable function on $I.$ Then $f$ is hyperbolic p-convex on $I$ if and only if for all $x\in I$ we have

[TABLE]

For other properties of hyperbolic p-convex functions, see [A16].

Consider the function $f_{r}:(0,\infty)\rightarrow(0,\infty),$ $f_{r}(x)=x^{r}$ with $p\in\mathbb{R}\backslash\{0\}.$ If $r\in(-\infty,0)\cup[1,\infty)$ the function is convex and if $r\in(0,1)$ it is concave. We have for $r\in(-\infty,0)\cup[1,\infty)$

[TABLE]

We observe that

[TABLE]

and

[TABLE]

which shows that the power function $f_{r}$ for $r\in(-\infty,0)\cup[1,\infty)$ is hyperbolic p-convex on $\left(0,\frac{\sqrt{r(r-1)}}{|p|}\right)$ and hyperbolic p-concave on $\left(\frac{\sqrt{r(r-1)}}{|p|},\infty\right).$

If $r\in(0,1),$ then

[TABLE]

for any $x>0,$ which shows that $f_{r}$ is hyperbolic p-concave on $(0,\infty).$

Consider the exponential function

[TABLE]

for $\mu\neq 0$ and $x\in\mathbb{R}.$ Then

[TABLE]

If $|\mu|>|p|,$ then $f_{\mu}$ is hyperbolic p-convex on $\mathbb{R}$ and if $|\mu|<|p|,$ then $f_{\mu}$ is hyperbolic p-concave on $\mathbb{R}.$

1.4. Hermite-Hadamard and Hermite-Hadamard-Fejér inequalities for hyperbolic p-convex functions

In [D18a], Dragomir obtained Hermite-Hadamard type inequality in the class of hyperbolic p-convex functions as follows

Theorem 1.1**.**

Assume that the function $u:\,I\rightarrow\mathbb{R}$ is hyperbolic p-convex function on $I.$ Then for any $a,b\in I$ we have

[TABLE]

Remark 1.1*.*

Note that, if $p\rightarrow 0$ in (1.11) we get the classical Hermite-Hadamard inequality (1.1).

Hermite-Hadamard-Fejér type inequalities in the class of hyperbolic p-convex functions was proven in [D18b]:

Theorem 1.2**.**

Assume that the function $u:\,I\rightarrow\mathbb{R}$ is hyperbolic p-convex on $I$ and $a,b\in I.$ Assume also that $w:\,I\rightarrow\mathbb{R}$ is a positive, symmetric and integrable function on $[a,b],$ then we have

[TABLE]

Remark 1.2*.*

Note that, if $p\rightarrow 0$ in (1.12) we get the classical Hermite-Hadamard-Fejér inequality (1.2).

Theorem 1.3**.**

Assume that the function $u:\,I\rightarrow\mathbb{R}$ is hyperbolic p-convex on $I$ and $a,b\in I.$ Assume also that $w:\,I\rightarrow\mathbb{R}$ is a positive, symmetric and integrable function on $[a,b],$ then

[TABLE]

2. Main results

In this section, we formulate the main results of the paper.

2.1. Fractional analogues of Hermite-Hadamard inequality for hyperbolic p-convex functions

Our first observation is formulated by the following theorem.

Theorem 2.1**.**

Assume that the function $u:\,I\rightarrow\mathbb{R}$ is hyperbolic p-convex function on $I.$ Then for any $a,b\in I$ we have

[TABLE]

where $\mathcal{C}_{\alpha}(1):=\int\limits_{a}^{b}\cosh\left[p\left(x-\frac{a+b}{2}\right)\right]\frac{(b-x)^{\alpha-1}+(x-a)^{\alpha-1}}{\Gamma(\alpha)}dx.$

Proof.

Let us suppose that all assumptions of Theorem are satisfied. Let us define the function $w$ of Theorem 1.2 by

[TABLE]

Clearly, $w$ is a positive, symmetric and integrable function on $[a,b].$ Moreover, for all $x\in(a,b),$ we get

[TABLE]

Moreover, we have

[TABLE]

Therefore, from (1.12) we obtain inequality (2.1). ∎

Remark 2.1*.*

Inequalities (1.1) and (1.5) are special cases of inequality (2.1).

• •

If $p\rightarrow 0$ we have $\lim\limits_{p\rightarrow 0}\mathcal{C}^{1}_{\alpha}(1)=\frac{2(b-a)^{\alpha}}{\Gamma(\alpha+1)}.$ In this case, inequality (2.1) coincide with the inequality (1.5);

• •

If $p\rightarrow 0$ and $\alpha\rightarrow 1$ in (2.1), then we have classical Hermite-Hadamard inequality (1.1).

Theorem 2.2**.**

Assume that the function $u:\,I\rightarrow\mathbb{R}$ is hyperbolic p-convex function on $I.$ Then for any $a,b\in I$ we have

[TABLE]

where $\mathcal{C}^{2}_{\alpha}(1):=\int\limits_{a}^{b}\cosh\left[p\left(x-\frac{a+b}{2}\right)\right]\frac{\exp\left(-\frac{1-\alpha}{\alpha}(b-x)\right)+\exp\left(-\frac{1-\alpha}{\alpha}(x-a)\right)}{\alpha}dx.$

Proof.

Suppose that all assumptions of Theorem are satisfied. Let us define the function $w$ of Theorem 1.2 by

[TABLE]

Clearly, $w$ is a positive, symmetric and integrable function on $[a,b].$ Moreover, for all $x\in(a,b),$ we get

[TABLE]

Moreover, we have

[TABLE]

Therefore, from (1.12) we obtain inequality (2.2). ∎

Remark 2.2*.*

Inequalities (1.1) and (1.5) are special cases of inequality (2.2).

• •

If $p\rightarrow 0$ we have $\lim\limits_{p\rightarrow 0}\mathcal{C}^{2}_{\alpha}(1)=\frac{2\exp\left(-\frac{1-\alpha}{\alpha}(b-a)\right)}{1-\alpha}.$ In this case, inequality (2.2) coincide with the inequality (1.7);

• •

If $p\rightarrow 0$ and $\alpha\rightarrow 1$ in (2.2), then we have classical Hermite-Hadamard inequality (1.1).

2.2. Fractional analogues of Hermite-Hadamard-Fejér inequality for hyperbolic p-convex functions

Theorem 2.3**.**

Assume that the function $u:\,I\rightarrow\mathbb{R}$ is hyperbolic p-convex on $I$ and $a,b\in I.$ Assume also that $v:\,I\rightarrow\mathbb{R}$ is a positive, symmetric and integrable function on $[a,b],$ then we have

[TABLE]

where $\mathcal{C}^{1}_{\alpha}(v):=\int\limits_{a}^{b}\cosh\left[p\left(x-\frac{a+b}{2}\right)\right]\frac{(b-x)^{\alpha-1}+(x-a)^{\alpha-1}}{\Gamma(\alpha)}v(x)dx.$

Proof.

Let us suppose that all assumptions of Theorem are satisfied. Let us define the function $w$ of Theorem 1.2 by

[TABLE]

Clearly, $w$ is a positive, symmetric and integrable function on $[a,b].$ Moreover, for all $x\in(a,b),$ we get $w(a+b-x)=w(x).$ Moreover, we have

[TABLE]

Therefore, from (1.12) we obtain inequality (2.3). ∎

Remark 2.3*.*

Inequalities (1.2) and (1.6) are special cases of inequality (2.3).

• •

If $p\rightarrow 0$ we have $\lim\limits_{p\rightarrow 0}\mathcal{C}^{1}_{\alpha}(v)=I^{\alpha}_{a+}v(b)+I^{\alpha}_{b-}v(a).$ In this case, inequality (2.3) coincide with the inequality (1.6);

• •

If $p\rightarrow 0$ and $\alpha\rightarrow 1$ in (2.3), then we have classical Hermite-Hadamard-Fejér inequality (1.2).

The following theorems are proved similarly.

Theorem 2.4**.**

Assume that the function $u:\,I\rightarrow\mathbb{R}$ is hyperbolic p-convex on $I$ and $a,b\in I.$ Assume also that $v:\,I\rightarrow\mathbb{R}$ is a positive, symmetric and integrable function on $[a,b],$ then we have

[TABLE]

where $\mathcal{C}^{2}_{\alpha}(v):=\int\limits_{a}^{b}\cosh\left[p\left(x-\frac{a+b}{2}\right)\right]\frac{\exp\left(-\frac{1-\alpha}{\alpha}(b-x)\right)+\exp\left(-\frac{1-\alpha}{\alpha}(x-a)\right)}{\alpha}v(x)dx.$

Remark 2.4*.*

Inequalities (1.2) and (1.6) are special cases of inequality (2.4).

• •

If $p\rightarrow 0$ we have $\lim\limits_{p\rightarrow 0}\mathcal{C}^{2}_{\alpha}(v)=\mathcal{I}^{\alpha}_{a+}v(b)+\mathcal{I}^{\alpha}_{b-}v(a).$ In this case, inequality (2.4) coincide with the inequality (1.8);

• •

If $p\rightarrow 0$ and $\alpha\rightarrow 1$ in (2.4), then we have classical Hermite-Hadamard-Fejér inequality (1.2).

Theorem 2.5**.**

Assume that the function $u:\,I\rightarrow\mathbb{R}$ is hyperbolic p-convex on $I$ and $a,b\in I.$ Assume also that $v:\,I\rightarrow\mathbb{R}$ is a positive, symmetric and integrable function on $[a,b],$ then

[TABLE]

where

[TABLE]

Remark 2.5*.*

If $\alpha\rightarrow 1,$ we have

[TABLE]

and

[TABLE]

In this case, inequality (2.5) coincide with the inequality (1.13).

Theorem 2.6**.**

Assume that the function $u:\,I\rightarrow\mathbb{R}$ is hyperbolic p-convex on $I$ and $a,b\in I.$ Assume also that $v:\,I\rightarrow\mathbb{R}$ is a positive, symmetric and integrable function on $[a,b],$ then

[TABLE]

where $\mathcal{S}^{2}_{\alpha}(v):=\int\limits_{a}^{b}\sinh\left[p\left(x-\frac{a+b}{2}\right)\right]\frac{\exp\left(-\frac{1-\alpha}{\alpha}(b-x)\right)+\exp\left(-\frac{1-\alpha}{\alpha}(x-a)\right)}{\alpha}v(x)dx.$

Remark 2.6*.*

If $\alpha\rightarrow 1,$ we have

[TABLE]

and

[TABLE]

In this case, inequality (2.6) coincide with the inequality (1.13).

Acknowledgements

The research of Torebek is financially supported by a grant No.AP05131756 from the Ministry of Science and Education of the Republic of Kazakhstan. No new data was collected or generated during the course of research