A hybrid algorithm based on parareal and Schwarz waveform relaxation

Liping Yang; Hu Li; Liping Yang; Hu Li

doi:10.3934/era.2022207

Electronic Research Archive

2022, Volume 30, Issue 11: 4086-4107. doi: 10.3934/era.2022207

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A hybrid algorithm based on parareal and Schwarz waveform relaxation

Liping Yang ¹,
Hu Li ^{2
,
,}

1.
Department of Mathematics, Zigong Vocational and Technical College, Zigong 643000, China
2.
School of Mathematics, Chengdu Normal University, Chengdu 611130, China

Received: 28 June 2022 Revised: 09 August 2022 Accepted: 11 August 2022 Published: 16 September 2022

In this paper, we present a hybrid algorithm based on parareal and Schwarz waveform relaxation (SWR) for solving time dependent partial differential equations. The parallelism can be simultaneously realized in the time direction by using a parareal and in the space direction via SWR. We give a convergence analysis for the hybrid algorithm for a 1D model problem, the reaction-diffusion equation. Weak scaling of the algorithm in terms of both the number of space subdomains and the number of paralleled time intervals were investigated via theoretical analysis and numerical experiments.

Keywords:

Citation: Liping Yang, Hu Li. A hybrid algorithm based on parareal and Schwarz waveform relaxation[J]. Electronic Research Archive, 2022, 30(11): 4086-4107. doi: 10.3934/era.2022207

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Abstract

1. Introduction

A real-valued function $h: I\rightarrow\mathbb{R}$ is said to be a convex (concave) function of the interval $I\subseteq\mathbb{R}$ if the inequality

$\begin{equation*} h(\lambda\kappa_{1}+(1-\lambda)\kappa_{2})\leq(\geq)\lambda h(\kappa_{1})+(1-\lambda)h(\kappa_{2}) \end{equation*}$

takes place whenever $\kappa_{1}, \kappa_{2}\in I$ and $\lambda\in [0, 1]$ .

It is well known that convex (concave) function plays an important role in mathematics due to convexity (concavity) is widely used in all branches of pure and applied mathematics ^{[1,2,3,4,5,6,7]}. Recently, the generalizations, extensions and invariants for the convex (concave) function have attracted the attention of many researchers, for instance, the quasi-convex function ^[8], harmonic convex function ^[9,10], strongly convex function ^[11,12,13], two-parameter Hölder mean convex function ^[14,15], exponentially convex function ^[16,17], $GG$ and $GA$ convex functions ^[18], and $s$ -convex function ^[19,20]. In particular, many inequalities can be found in the literature ^{[21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]} by use of properties of the convex (concave) function.

Let $h: I\rightarrow\mathbb{R}$ be a convex (concave) function. Then the well known $\mathcal{HH}$ (Hermite-Hadamard) inequality ^[37,38,39] states that the double inequality

$\begin{equation} h\left(\frac{\kappa_{1}+\kappa_{2}}{2}\right) \leq(\geq)\frac{1}{\kappa_{2}-\kappa_{1}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)dx \leq(\geq)\frac{h(\kappa_{1})+h(\kappa_{2})}{2} \end{equation}$

(1.1)

is valid for all $\kappa_{1}, \kappa_{2}\in I$ with $\kappa_{1}\neq \kappa_{2}$ .

Fejér generalized the $\mathcal{HH}$ inequality (1.1) to the Hermite-Hadamard-Fejér inequality (1.2) as follows:

$\begin{equation} \frac{h(\kappa_{1})+h(\kappa_{2})}{2}\int_{\kappa_{1}}^{\kappa_{2}}g(x)dx \geq(\leq)\frac{1}{\kappa_{2}-\kappa_{1}}\int_{\kappa_{1}}^{\kappa_{2}}g(x)h(x)dx \geq(\leq)h\left(\frac{\kappa_{1}+\kappa_{2}}{2}\right)\int_{\kappa_{1}}^{\kappa_{2}}g(x)dx \end{equation}$

(1.2)

if $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ is a convex (concave) function and $g : [\kappa_{1}, \kappa_{2}]\rightarrow \mathbb{R}^{+}$ is symmetric with respect to $(\kappa_{1}+\kappa_{2})/2$ .

The following definition for the $\eta$ -convex function was introduced by Eshaghi Gordji et al. in ^[40].

Definition 1.1. (See ^[40]) Let $\kappa_{1}, \kappa_{2}\in\mathbb{R}$ with $\kappa_{1} < \kappa_{2}$ , $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be a real-valued function and $\eta: h([\kappa_{1}, \kappa_{2}])\times h([\kappa_{1}, \kappa_{2}])\rightarrow \mathbb{R}$ be a two bivariate real-valued function. Then $h$ is said to be $\eta$ -convex (or convex with respect to $\eta$ ) if the inequality

$\begin{equation} h[s\mu_{1}+(1-s)\mu_{2}]\leq h(\mu_{2})+s\eta[h(\mu_{1}), h(\mu_{2})] \end{equation}$

(1.3)

holds for all $\mu_{1}, \mu_{2}\in [\kappa_{1}, \kappa_{2}]$ and $s\in[0, 1]$ .

Let $\eta(\mu_{1}, \mu_{2}) = \mu_{1}-\mu_{2}$ in (1.3). Then Definition 1.1 reduces to the definition of usual convex function.

Eshaghi Gordji et al. ^[40] established a $\mathcal{HH}$ type inequality for the $\eta$ -convex function.

Theorem 1.1. (See ^[40]) Let $\kappa_{1}, \kappa_{2}\in\mathbb{R}$ with $\kappa_{1} < \kappa_{2}$ , $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be a real-valued function and $\eta: h([\kappa_{1}, \kappa_{2}])\times h([\kappa_{1}, \kappa_{2}])\rightarrow \mathbb{R}$ be a two bivariate bounded real-valued function. Then one has

$\begin{equation*} h\left(\frac{\kappa_{1}+\kappa_{2}}{2}\right)-\frac{M_{\eta}}{2} \leq\frac{1}{\kappa_{2}-\kappa_{1}}\int\limits^{\kappa_{2}}_{\kappa_{1}}h(x)dx \end{equation*}$

$\begin{equation*} \leq\frac{h(\kappa_{1})+h(\kappa_{2})}{2}+\frac{\eta(h(\kappa_{1}), h(\kappa_{2}))+\eta(h(\kappa_{2}), h(\kappa_{1}))}{4} \end{equation*}$

$\begin{equation} \leq\frac{h(\kappa_{1})+h(\kappa_{2})}{2}+\frac{M_{\eta}}{2} \end{equation}$

(1.4)

if $h$ is $\eta$ -convex, where $M_{\eta}$ is the upper bound of $\eta$ on $h([\kappa_{1}, \kappa_{2}])\times h([\kappa_{1}, \kappa_{2}])$ .

Let $0 < \beta\leq 1$ , $r > 0$ and $g: [0, \infty)\rightarrow\mathbb{R}$ be a real-valued function. Then the conformable derivative $D_{\beta}(g)(r)$ of order $\beta$ is defined by

$\begin{equation} D_{\beta}(g)(r) = \frac{d_{\beta}g(r)}{d_{\beta}r} = \lim\limits_{\epsilon \rightarrow 0}\frac{g(r+\epsilon r^{1-\beta})-g(r)}{\epsilon}, \end{equation}$

(1.5)

$g$ is said to be conformable differentiable at $r$ if the limit of (1.5) exists and is finite. The conformal derivative at $0$ is defined by $D_{\beta}(g)(0) = \lim\limits_{r \rightarrow 0^{+}}D_{\beta}(g)(r)$ .

Let $\kappa_{1}, \kappa_{2}, \lambda, c\in\mathbb{R}$ be the constants, and $h_{1}$ and $h_{2}$ be differentiable at $r > 0$ . Then the following formulas can be found in the literature ^[41]

$\begin{equation*} \frac{d_{\beta}}{d_{\beta}r}\left(r^{\lambda}\right) = \lambda r^{\lambda-\beta}, \quad \frac{d_{\beta}}{d_{\beta}r}(c) = 0, \end{equation*}$

$\begin{equation*} \frac{d_{\beta}}{d_{\beta}r}\left(\kappa_{1}h_{1}(r)+\kappa_{2}h_{2}(r)\right) = \kappa_{1}\frac{d_{\beta}}{d_{\beta}r}(h_{1}(r)) +\kappa_{2} \frac{d_{\beta}}{d_{\beta}r}(h_{2}(r)), \end{equation*}$

$\begin{equation*} \frac{d_{\beta}}{d_{\beta}r}\left(h_{1}(r)h_{2}(r)\right) = h_{1}(r)\frac{d_{\beta}}{d_{\beta}r}(h_{2}(r))+h_{2}(r)\frac{d_{\beta}}{d_{\beta}r}(h_{1}(r)), \end{equation*}$

$\begin{equation*} \frac{d_{\beta}}{d_{\beta}r}\left(\frac{h_{1}(r)}{h_{2}(r)}\right) = \frac{h_{2}(r)\frac{d_{\beta}}{d_{\beta}r}(h_{1}(r))-h_{1}(r)\frac{d_{\beta}}{d_{\beta}r}(h_{2}(r))}{(h_{2}(r))^{2}}, \end{equation*}$

$\begin{equation*} \frac{d_{\beta}}{d_{\beta}r}\left(h_{1}(h_{2}(r))\right) = h^{\prime}_{1}(h_{2}(r))\frac{d_{\beta}}{d_{\beta}r}(h_{2}(r)) \end{equation*}$

if $h_{1}$ differentiable at $h_{2}(r)$ . In addition,

$\begin{equation*} \frac{d_{\beta}}{d_{\beta}r}\left(h_{1}(r)\right) = r^{1-\beta} \frac{d}{dr}(h_{1}(r)) \end{equation*}$

if $h_{1}$ is differentiable.

Let $\beta\in(0, 1]$ and $0\leq\kappa_{1} < \kappa_{2}$ . Then the function $g: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ is said to be conformable integrable if

$\begin{equation*} \int_{\kappa_{1}}^{\kappa_{2}}g(x)d_{\beta}x = \int_{\kappa_{1}}^{\kappa_{2}}g(x)x^{\beta-1}dx \end{equation*}$

exists and is finite. The set of all conformable integrable functions on $[\kappa_{1}, \kappa_{2}]$ is denoted by $L_{\beta}([\kappa_{1}, \kappa_{2}])$ . Note that

$\begin{equation*} I_{\beta}^{\kappa_{1}}(h_{1})(r) = I_{1}^{\kappa_{1}}(r^{\beta-1}h_{1}) = \int_{\kappa_{1}}^{r}\frac{h_{1}(x)}{x^{1-\beta}}dx \end{equation*}$

for all $\beta\in (0, 1]$ , where the integral is the usual Riemann improper integral.

For the theory and applications of the conformable integrals and derivatives we recommend the readers to refer the literature ^{[42,43,44,45,46,47,48,49,50]}.

Anderson ^[51] established the conformable integral version of the $\mathcal{HH}$ type inequality

$\begin{equation*} \frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x \leq\frac{h(\kappa_{1})+h(\kappa_{2})}{2} \end{equation*}$

if $\beta \in (0, 1]$ and $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ is conformable differentiable such that $D_{\beta}(h)$ is increasing. Moreover, if $h$ is decreasing on $[\kappa_{1}, \kappa_{2}]$ , then

$\begin{equation*} h\left(\frac{\kappa_{1}+\kappa_{2}}{2}\right) \leq\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}} \int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x. \end{equation*}$

It is the aim of the article to establish new Hermite-Hadamard-Fejér type inequalities for the $\eta$ -convex functions via conformable integrals.

2. Hermite-Hadamard-Fejér type inequalities for conformable integrals by using $\eta$ -convex functions

Theorem 2.1. Let $\kappa_{1}, \kappa_{2}\in\mathbb{R^{+}}$ with $\kappa_{1} < \kappa_{2}$ , $h: [\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be an $\eta$ -convex function and symmetric with respect to $\frac{\kappa_{1}+\kappa_{2}}{2}$ , $\xi: [\kappa_{1}, \kappa_{2}]\rightarrow \mathbb{R}$ be a nonnegative integrable function. Then the inequality

$\begin{equation*} h\left(\frac{\kappa_{1}+\kappa_{2}}{2}\right)\int\limits^{\kappa_{2}}_{\kappa_{1}}\xi(x)d_{\beta}x -\frac{M_{\eta}}{2}\int\limits^{\kappa_{2}}_{\kappa_{1}}\xi(x)d_{\beta}x \leq\int\limits^{\kappa_{2}}_{\kappa_{1}}h(x)\xi(x)d_{\beta}x \end{equation*}$

$\begin{equation*} \leq\frac{h(\kappa_{1})+h(\kappa_{2})}{2} \int\limits^{\kappa_{2}}_{\kappa_{1}}\xi(x) d_{\beta}x +\frac{\eta(h(\kappa_{1}), h(\kappa_{2}))+\eta(h(\kappa_{2}), h(\kappa_{1}))}{4} \int\limits^{\kappa_{2}}_{\kappa_{1}} \xi(x)d_{\beta}x \end{equation*}$

$\begin{equation} \leq\frac{h(\kappa_{1})+h(\kappa_{2})}{2}\int\limits^{\kappa_{2}}_{\kappa_{1}}\xi(x) d_{\beta}x +\frac{M_{\eta}}{2}\int\limits^{\kappa_{2}}_{\kappa_{1}}\xi(x)d_{\beta}x \end{equation}$

(2.1)

holds for any $\beta\in (0, 1]$ if $\eta$ is bounded on $h([\kappa_{1}, \kappa_{2}])\times h([\kappa_{1}, \kappa_{2}])$ , where $M_{\eta}$ is the upper bound of $\eta$ on $h([\kappa_{1}, \kappa_{2}])\times h([\kappa_{1}, \kappa_{2}])$ .

Proof. Let $s\in[0, 1]$ . Then it follows from the $\eta$ -convexity and symmetry of $h$ that

$\begin{equation*} h\left(\frac{\kappa_{1}+\kappa_{2}}{2}\right) = h\left(\frac{s\kappa_{1}-s\kappa_{1} +\kappa_{1}+\kappa_{2}+s\kappa_{2}-s\kappa_{2}}{ 2}\right) \end{equation*}$

$\begin{equation*} = h\left(\frac{s\kappa_{1}+(1-s)\kappa_{2}+ s\kappa_{2}+(1-s)\kappa_{1}}{ 2}\right) \end{equation*}$

$\begin{equation*} \leq h(s\kappa_{2}+(1-s)\kappa_{1})+\frac{1}{2}\eta(h(s\kappa_{1}+(1-s)\kappa_{2})), h(s\kappa_{2}+(1-s)\kappa_{1}) \end{equation*}$

$\begin{equation*} \leq h(s\kappa_{2}+(1-s)\kappa_{1})+\frac{1}{2}M_{\eta}. \end{equation*}$

Let $x = s\kappa_{2}+(1-s)\kappa_{1}$ . Then we get

$\begin{equation*} h\left(\frac{\kappa_{1}+\kappa_{2}}{2}\right)\int\limits^{\kappa_{2}}_{\kappa_{1}}\xi(x)d_{\beta}x = (\kappa_{2}-\kappa_{1})h\left(\frac{\kappa_{1}+\kappa_{2}}{2}\right)\int\limits^{1}_{0}\xi(s\kappa_{2} +(1-s)\kappa_{1})(s\kappa_{2}+(1-s)\kappa_{1})^{\beta-1}ds \end{equation*}$

$\begin{equation*} \leq\int\limits^{1}_{0}h(s\kappa_{2}+(1-s)\kappa_{1})\xi(s\kappa_{2} +(1-s)\kappa_{1})(\kappa_{2}-\kappa_{1})(s\kappa_{2}+(1-s)\kappa_{1})^{\beta-1}ds \end{equation*}$

$\begin{equation*} +\frac{M_{\eta}}{2}\int\limits^{1}_{0}\xi(s\kappa_{2}+(1-s)\kappa_{1})(\kappa_{2}-\kappa_{1}) (s\kappa_{2}+(1-s)\kappa_{1})^{\beta-1}ds \end{equation*}$

$\begin{equation*} = \int\limits^{\kappa_{2}}_{\kappa_{1}}h(x)\xi(x)d_{\beta}x +\frac{M_{\eta}}{2}\int\limits^{\kappa_{2}}_{\kappa_{1}}\xi(x)d_{\beta}x, \end{equation*}$

which gives the proof of the first inequality of (2.1).

Next, we prove the second and third inequalities of (2.1). From the $\eta$ -convexity of $h$ we know that

$\begin{equation} h(s\kappa_{1}+(1-s)\kappa_{2})\leq h(\kappa_{2})+s\eta(h(\kappa_{1}), h(\kappa_{2})) \end{equation}$

(2.2)

and

$\begin{equation} h(s\kappa_{2}+(1-s)\kappa_{1})\leq h(\kappa_{1})+s\eta(h(\kappa_{2}), h(\kappa_{1})). \end{equation}$

(2.3)

Let $x = s\kappa_{1}+(1-s)\kappa_{2}$ . Then from the symmetry of $h$ , inequalities (2.2) and (2.3) lead to

$\begin{equation*} \int\limits^{\kappa_{2}}_{\kappa_{1}}h(x)\xi(x)x^{\beta-1}dx = (\kappa_{2}-\kappa_{1})\int\limits^{1}_{0}h(s\kappa_{1}+(1-s)\kappa_{2})\xi(s\kappa_{1} +(1-s)\kappa_{2})(s\kappa_{1}+(1-s)\kappa_{2})^{\beta-1}ds \end{equation*}$

$\begin{equation} \leq(\kappa_{2}-\kappa_{1})\left[h(\kappa_{2})+\frac{1}{2}\eta(h(\kappa_{1}), h(\kappa_{2}))\right] \int\limits^{1}_{0}\xi(s\kappa_{1}+(1-s)\kappa_{2})(s\kappa_{1}+(1-s)\kappa_{2})^{\beta-1}ds \end{equation}$

(2.4)

and

$\begin{equation*} \int\limits^{\kappa_{2}}_{\kappa_{1}}h(x)\xi(x)x^{\beta-1}dx = (\kappa_{2}-\kappa_{1})\int\limits^{1}_{0}h(s\kappa_{1} +(1-s)\kappa_{2})\xi(s\kappa_{1}+(1-s)\kappa_{2})(s\kappa_{1}+(1-s)\kappa_{2})^{\beta-1}ds \end{equation*}$

$\begin{equation*} = (\kappa_{2}-\kappa_{1})\int\limits^{1}_{0}h(s\kappa_{2} +(1-s)\kappa_{1})\xi(s\kappa_{1}+(1-s)\kappa_{2})(s\kappa_{1}+(1-s)\kappa_{2})^{\beta-1}ds \end{equation*}$

$\begin{equation} \leq(\kappa_{2}-\kappa_{1})\left[h(\kappa_{1})+\frac{1}{2}\eta(h(\kappa_{2}), h(\kappa_{1}))\right] \int\limits^{1}_{0}\xi(s\kappa_{1}+(1-s)\kappa_{2})(s\kappa_{1}+(1-s)\kappa_{2})^{\beta-1}ds \end{equation}$

(2.5)

Adding (2.4) and (2.5), and letting $x = s\kappa_{1}+(1-s)\kappa_{2}$ , we obtain

$\begin{equation*} 2\int\limits^{\kappa_{2}}_{\kappa_{1}}h(x)\xi(x)d_{\beta}x \leq(\kappa_{2}-\kappa_{1})(h(\kappa_{1})+h(\kappa_{2})) \int\limits^{1}_{0}\xi(s\kappa_{1}+(1-s)\kappa_{2})(s\kappa_{1}+(1-s)\kappa_{2})^{\beta-1}ds \end{equation*}$

$\begin{equation*} +(\kappa_{2}-\kappa_{1})\frac{(\eta(h(\kappa_{1}), h(\kappa_{2})) +\eta(h(\kappa_{2}), h(\kappa_{1})))}{2}\int\limits^{1}_{0} \xi(s\kappa_{1}+(1-s)\kappa_{2})(s\kappa_{1}+(1-s)\kappa_{2})^{\beta-1}ds \end{equation*}$

and

$\begin{equation*} \int\limits^{\kappa_{2}}_{\kappa_{1}}h(x)\xi(x)d_{\beta}x\leq\frac{h(\kappa_{1})+h(\kappa_{2})}{2} \int\limits^{\kappa_{2}}_{\kappa_{1}}\xi(x) d_{\beta}x +\frac{\eta(h(\kappa_{1}), h(\kappa_{2}))+\eta(h(\kappa_{2}), h(\kappa_{1}))}{4} \int\limits^{\kappa_{2}}_{\kappa_{1}}\xi(x)d_{\beta}x \end{equation*}$

$\begin{equation*} \leq\frac{h(\kappa_{1})+h(\kappa_{2})}{2}\int\limits^{\kappa_{2}}_{\kappa_{1}}\xi(x)d_{\beta}x +\frac{M_{\eta}}{2}\int\limits^{\kappa_{2}}_{\kappa_{1}} \xi(x)d_{\beta}x. \end{equation*}$

Corollary 2.1. Let $\xi(x) = 1$ . Then inequality (2.1) becomes

$\begin{equation*} h\left(\frac{\kappa_{1}+\kappa_{2}}{2}\right)-\frac{M_{\eta}}{2} \leq\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int\limits^{\kappa_{2}}_{\kappa_{1}}h(x)d_{\beta}x \end{equation*}$

$\begin{equation*} \leq\frac{h(\kappa_{1})+h(\kappa_{2})}{2} +\frac{\eta(h(\kappa_{1}), h(\kappa_{2}))+\eta(h(\kappa_{2}), h(\kappa_{1}))}{4} \end{equation*}$

$\begin{equation*} \leq\frac{h(\kappa_{1})+h(\kappa_{2})}{2}+\frac{M_{\eta}}{2}. \end{equation*}$

3. Further generalizations of $\mathcal{HH}$ inequality

In order to establish our main results, we need a key lemma which we present in this section.

Lemma 3.1. Let $\kappa_{1}, \kappa_{2}\in\mathbb{R^{+}}$ with $\kappa_{1} < \kappa_{2}$ , and $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be a differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D _{\beta}(h)\in L_{\beta}([\kappa_{1}, \kappa_{2}])$ . Then the identity

$\begin{equation*} \frac{h(\kappa_{1})+h(\kappa_{2})}{2}-\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x \end{equation*}$

$\begin{equation*} = \frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\int_{0}^{1}\left(((1-s)\kappa_{1}+s\kappa_{2})^{2\beta-1} -\kappa_{1}^{\beta}((1-s)\kappa_{1}+s\kappa_{2})^{\beta-1}\right) \end{equation*}$

$\begin{equation*} \times D_{\beta}(h)((1-s)\kappa_{1}+s\kappa_{2})s^{1-\beta}d_{\beta}s +\int_{0}^{1}\left(((1-s)\kappa_{2}+s\kappa_{1})^{2\beta-1}-a_{2}^{\beta}((1-s)\kappa_{2}+s\kappa_{1})^{\beta-1}\right) \end{equation*}$

$\begin{equation*} \times D_{\beta}(h)((1-s)\kappa_{2}+s\kappa_{1})s^{1-\beta}d_{\beta}s\Bigg]. \end{equation*}$

holds for $\beta\in (0, 1]$ .

Proof. Integration by parts, we have

$\begin{equation*} \int_{0}^{1}\big(((1-s)\kappa_1+s\kappa_2)^{2\beta-1} -\kappa_1^{\beta}((1-s)\kappa_1+s\kappa_2)^{\beta-1}\big)D_{\beta}(h)((1-s)\kappa_1+s\kappa_2)ds \end{equation*}$

$\begin{equation*} +\int_{0}^{1}\left(((1-s)\kappa_2+s\kappa_{1})^{2\beta-1}-\kappa_2^{\beta}((1-s)\kappa_2+s\kappa_{1})^{\beta-1}\right) D_{\beta}(h)((1-s)\kappa_2+s\kappa_{1})ds \end{equation*}$

$\begin{equation*} = \int_{0}^{1}\left(((1-s)\kappa_{1}+s\kappa_2)^{\beta}-\kappa_{1}^{\beta}\right)h^{\prime}((1-s)\kappa_{1}+s\kappa_2)ds \end{equation*}$

$\begin{equation*} +\int_{0}^{1}\left(((1-s)\kappa_2+s\kappa_{1})^{\beta}-\kappa_2^{\beta}\right)h^{\prime}((1-s)\kappa_2+s\kappa_{1})ds \end{equation*}$

$\begin{equation*} = \Bigg[\left(((1-s)\kappa_{1}+s\kappa_2)^{\beta}-\kappa_{1}^{\beta}\right)\frac{h((1-s)\kappa_{1}+s\kappa_2)}{\kappa_2-\kappa_{1}}\bigg|_{0}^{1} \end{equation*}$

$\begin{equation*} -\int_{0}^{1}\beta((1-s)\kappa_{1}+\kappa_2)^{\beta-1}(\kappa_2-a_{1})\frac{h((1-s)\kappa_{1}+s\kappa_2)}{\kappa_2-\kappa_{1}}ds\Bigg] \end{equation*}$

$\begin{equation*} +\Bigg[\left(((1-s)\kappa_2+s\kappa_{1})^{\beta}-\kappa_2^{\beta}\right)\frac{h((1-s)\kappa_2+s\kappa_{1})}{\kappa_{1}-\kappa_2}\bigg|_{0}^{1} \end{equation*}$

$\begin{equation*} -\int_{0}^{1}\beta((1-s)\kappa_2+s\kappa_{1})^{\beta-1}(\kappa_{1}-\kappa_2)\frac{h((1-s)\kappa_2+s\kappa_{1})}{\kappa_{1}-\kappa_2}ds\Bigg] \end{equation*}$

$\begin{equation*} = \left[\frac{\kappa_2^{\beta}-\kappa_{1}^{\beta}}{\kappa_2-\kappa_{1}}h(\kappa_2) -\frac{\beta}{\kappa_2-\kappa_{1}}\int_{\kappa_{1}}^{\kappa_2}h(x)d_{\beta}x\right] \end{equation*}$

$\begin{equation*} +\left[\frac{\kappa_2^{\beta}-\kappa_{1}^{\beta}}{\kappa_2-\kappa_{1}}h(\kappa_{1}) -\frac{\beta}{\kappa_2-\kappa_{1}}\int_{\kappa_{1}}^{\kappa_2}h(x)d_{\beta}x\right] \end{equation*}$

$\begin{equation*} = \frac{\kappa_2^{\beta}-\kappa_{1}^{\beta}}{\kappa_2-\kappa_{1}}(h(\kappa_1)+h(\kappa_2)) -\frac{2\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x. \end{equation*}$

Theorem 3.1. Let $\kappa_{1}, \kappa_{2}\in\mathbb{R}^{+}$ with $\kappa_{1} < \kappa_{2}$ , and $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be a differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D_{\beta}(h)\in L_{\beta}([\kappa_{1}, \kappa_{2}])$ . Then the inequality

$\begin{equation*} \left|\frac{h(\kappa_{1})+h(\kappa_{2})}{2} -\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x\right| \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\left(\frac{\kappa_1^{\beta-1}\kappa_2+\kappa_2^{\beta-1}\kappa_1+2\kappa_2^{\beta}-4\kappa_1^{\beta}}{6}\right) |h^{\prime}(\kappa_{1})|+\eta(|h^{\prime}(\kappa_{2})|, |h^{\prime}(\kappa_{1})|) \end{equation*}$

$\begin{equation} \times\left(\frac{\kappa_1^{\beta-1}\kappa_2+\kappa_2^{\beta-1}\kappa_1+3\kappa_2^{\beta}-5\kappa_1^{\beta}}{12}\right) +|h^{\prime}(\kappa_{2})|\left(\frac{\kappa_2^{\beta}-\kappa_1^{\beta}}{2}\right)+\eta(|h^{\prime}(\kappa_{1})|, |h^{\prime}(\kappa_{2})|) \left(\frac{\kappa_2^{\beta}-\kappa_1^{\beta}}{6}\right)\Bigg] \end{equation}$

(3.1)

holds for $\beta\in (0, 1]$ if $|h^{\prime}|$ is $\eta$ -convex.

Proof. Let $y > 0$ , $\varphi_{1}(y) = y^{\beta-1}$ and $\varphi_{2}(y) = -y^{\beta}$ . Then we clearly see that both the functions $\varphi_{1}$ and $\varphi_{2}$ are convex. It follows from Lemma 3.1 and the convexity of $\varphi_{1}$ and $\varphi_{2}$ together with the $\eta$ -convexity of $|h^{\prime}|$ that

$\begin{equation*} \left|\frac{h(\kappa_{1})+h(\kappa_{2})}{2} -\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x\right| \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\int_{0}^{1}\left(((1-s)\kappa_1+s\kappa_2)^{\beta}-\kappa_1^{\beta}\right) |h^{\prime}((1-s)\kappa_1+s\kappa_2)|ds \end{equation*}$

$\begin{equation*} +\int_{0}^{1}\left(((1-s)\kappa_2+s\kappa_{1})^{\beta}-\kappa_2^{\beta}\right)|h^{\prime}((1-s)\kappa_2+s\kappa_{1})|ds\Bigg] \end{equation*}$

$\begin{equation*} = \frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\int_{0}^{1}\left(((1-s)\kappa_1+s\kappa_2)^{\beta+1-1}-\kappa_1^{\beta}\right) |h^{\prime}((1-s)\kappa_1+s\kappa_2)|ds \end{equation*}$

$\begin{equation*} +\int_{0}^{1}\left(\kappa_2^{\beta}-((1-s)\kappa_2+s\kappa_{1})^{\beta}\right)|h^{\prime}((1-s)\kappa_2+s\kappa_{1})|ds\Bigg] \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\int_{0}^{1}\left(((1-s)\kappa_1+s\kappa_2)^{\beta-1}((1-s)\kappa_1+s\kappa_2)-\kappa_1^{\beta}\right) |h^{\prime}((1-s)\kappa_1+s\kappa_2)|ds \end{equation*}$

$\begin{equation*} +\int_{0}^{1}\left(\kappa_2^{\beta}-\left((1-s)\kappa_2^{\beta}+s\kappa_{1}^{\beta}\right)\right)|h^{\prime}((1-s)\kappa_2+s\kappa_{1})|ds\Bigg] \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\int_{0}^{1}\left(\left((1-s)\kappa_1^{\beta-1}+s\kappa_2^{\beta-1}\right)((1-s)\kappa_1+s\kappa_2)-\kappa_1^{\beta}\right) |h^{\prime}((1-s)\kappa_1+s\kappa_2)|ds \end{equation*}$

$\begin{equation*} +\int_{0}^{1}\left(\kappa_{2}^{\beta}-\left((1-s)\kappa_{2}^{\beta}+s\kappa_{1}^{\beta}\right)\right) |h^{\prime}((1-s)\kappa_{2}+s\kappa_{1})|ds\Bigg] \end{equation*}$

$\begin{equation*} \times\left[|h^{\prime}(\kappa_1)|+s\eta(|h^{\prime}(\kappa_{2})|, |h^{\prime}(\kappa_{1})|)\right]ds +\int_{0}^{1}\left(\kappa_{2}^{\beta}-\left((1-s)\kappa_{2}^{\beta}+s\kappa_{1}^{\beta}\right)\right) \end{equation*}$

$\begin{equation*} \times\left[|h^{\prime}(\kappa_{2})|+s\eta(|h^{\prime}(\kappa_{1})|, |h^{\prime}(\kappa_{2})|)\right]ds\Bigg] \end{equation*}$

and

$\begin{equation*} \left|\frac{h(\kappa_{1})+h(\kappa_{2})}{2} -\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x\right| \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\left(\frac{\kappa_1^{\beta-1}\kappa_2+\kappa_2^{\beta-1}\kappa_1 +2\kappa_2^{\beta}-4\kappa_1^{\beta}}{6}\right)|h^{\prime}(\kappa_{1})| \end{equation*}$

$\begin{equation*} +\eta(|h^{\prime}(\kappa_{2})|, |h^{\prime}(\kappa_{1})|) \left(\frac{\kappa_1^{\beta-1}\kappa_2+\kappa_2^{\beta-1}\kappa_1+3\kappa_2^{\beta}-5\kappa_1^{\beta}}{12}\right) \end{equation*}$

$\begin{equation*} +|h^{\prime}(\kappa_{2})|\left(\frac{\kappa_2^{\beta}-\kappa_1^{\beta}}{2}\right) +\eta(|h^{\prime}(\kappa_{1})|, |h^{\prime}(\kappa_{2})|)\left(\frac{\kappa_2^{\beta}-\kappa_1^{\beta}}{6}\right)\Bigg]. \end{equation*}$

Corollary 3.1. Let $\eta(\kappa_{2}, \kappa_{1}) = \kappa_{2}-\kappa_{1}$ . Then inequality (3.1) reduces to

$\begin{equation*} \left|\frac{h(\kappa_{1})+h(\kappa_{2})}{2} -\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x\right| \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\left(\frac{\kappa_1^{\beta-1}\kappa_2+\kappa_2^{\beta-1}\kappa_1+3a_2^{\beta} -5\kappa_1^{\beta}}{6}\right)|h^{\prime}(\kappa_{1})| \end{equation*}$

$\begin{equation*} +|h^{\prime}(\kappa_{2})|\left(\frac{7\kappa_2^{\beta}-9\kappa_1^{\beta} +\kappa_1^{\beta-1}\kappa_2+\kappa_2^{\beta-1}\kappa_1}{12}\right)\Bigg]. \end{equation*}$

Theorem 3.2. Let $q > 1$ , $\kappa_{1}, \kappa_{2}\in\mathbb{R}^{+}$ with $\kappa_{1} < \kappa_{2}$ , and $h:[\kappa_{1}, \kappa_{2}]\rightarrow\mathbb{R}$ be a differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D_{\beta}(h)\in L_{\beta}([\kappa_{1}, \kappa_{2}])$ . Then the inequality

$\begin{equation*} \left|\frac{h(\kappa_{1})+h(\kappa_{2})}{2} -\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}} \int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x\right| \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\left(A_{1}(\beta)\right)^{1-\frac{1}{q}}\Bigg(|h^{\prime}(\kappa_{1})|^{q} \left(\frac{\kappa_{1}^{\beta-1}\kappa_{2}+\kappa_{2}^{\beta-1}\kappa_{1} +2\kappa_{2}^{\beta}-4\kappa_{1}^{\beta}}{6}\right) \end{equation*}$

$\begin{equation*} +\eta(|h^{\prime}(\kappa_{2})|^{q}, |h^{\prime}(\kappa_1)|^{q}) \left(\frac{\kappa_{1}^{\beta-1}\kappa_{2}+\kappa_{2}^{\beta-1}\kappa_{1} +3\kappa_{2}^{\beta}-5\kappa_{1}^{\beta}}{12}\right)\Bigg) +\left(B_{1}(\beta)\right)^{1-\frac{1}{q}} \end{equation*}$

$\begin{equation*} \times\left(|h^{\prime}(\kappa_{2})|^{q}\left(\frac{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}{2}\right) +\eta(|h^{\prime}(\kappa_{1})|^{q}, |h'(\kappa_{2})|^{q})\left(\frac{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}{3}\right)\right)\Bigg] \end{equation*}$

takes place for $\beta\in (0, 1]$ if $|h^{\prime}|^{q}$ is $\eta$ -convex, where

$\begin{equation*} A_{1}(\beta) = \frac{2\kappa_1^{\beta}+\kappa_1^{\beta-1}\kappa_{2} +\kappa_{2}^{\beta-1}\kappa_1+2\kappa_{2}^{\beta}-6\kappa_1^{\beta}}{6}, \quad B_{1}(\beta) = \frac{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}{2}. \end{equation*}$

Proof. We clearly see that

$\begin{equation*} \left|\frac{h(\kappa_{1})+h(\kappa_{2})}{2} -\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x\right| \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\int_{0}^{1}\left(((1-s)\kappa_1+s\kappa_2)^{\beta-1}((1-s)\kappa_1+s\kappa_2) -\kappa_1^{\beta}\right)|h^{\prime}((1-s)\kappa_1+s\kappa_2)|ds \end{equation*}$

$\begin{equation*} +\int_{0}^{1}\left(\kappa_2^{\beta}-\left((1-s)\kappa_2^{\beta}+s\kappa_{1}^{\beta}\right)\right) |h^{\prime}((1-s)\kappa_2+s\kappa_{1})|ds\Bigg]. \end{equation*}$

It follows from the power-mean inequality that

$\begin{equation*} \int_{0}^{1}\left(\left((1-s)\kappa_1^{\beta-1}+s\kappa_{2}^{\beta-1}\right) \left((1-s)\kappa_1+s\kappa_{2}\right)-\kappa_1^{\beta}\right) |h^{\prime}((1-s)\kappa_1+s\kappa_{2})|ds \end{equation*}$

$\begin{equation*} \leq\left(\int_{0}^{1}\left(\left((1-s)\kappa_1^{\beta-1}+s\kappa_{2}^{\beta-1}\right) \left((1-s)\kappa_1+s\kappa_{2}\right)-\kappa_1^{\beta}\right)ds\right)^{1-\frac{1}{q}} \end{equation*}$

$\begin{equation*} \times\left(\int_{0}^{1}\left(\left((1-s)\kappa_1^{\beta-1}+s\kappa_{2}^{\beta-1}\right) \left((1-s)\kappa_1+s\kappa_{2}\right)-\kappa_1^{\beta}\right) |h^{\prime}\left((1-s)\kappa_1+s\kappa_{2}\right)|^{q}ds\right)^{\frac{1}{q}} \end{equation*}$

and

$\begin{equation*} \int_{0}^{1}\left(\kappa_{2}^{\beta}\left((1-s)\kappa_{2}^{\beta}+s\kappa_{1}^{\beta}\right)\right) \left|h^{\prime}((1-s)\kappa_{2}+s\kappa_{1})\right|ds \end{equation*}$

$\begin{equation*} \leq\left(\int_{0}^{1}\left(\kappa_{2}^{\beta}-\left((1-s)\kappa_{2}^{\beta} +s\kappa_{1}^{\beta}\right)\right)ds\right)^{1-\frac{1}{q}} \end{equation*}$

$\begin{equation*} \times\left(\int_{0}^{1}\left(\kappa_{2}^{\beta}-\left((1-s)\kappa_{2}^{\beta}+s\kappa_{1}^{\beta}\right)\right) \left|h^{\prime}((1-s)\kappa_{2}+s\kappa_{1})\right|^{q}ds\right)^{\frac{1}{q}}. \end{equation*}$

Making use of the $\eta$ -convexity of $|h^{\prime}|^{q}$ and the facts that

$\begin{equation*} \int_{0}^{1}\left(\left((1-s)\kappa_1^{\beta-1}+s\kappa_{2}^{\beta-1}\right) \left((1-s)\kappa_1+s\kappa_{2}\right)-\kappa_{1}^{\beta}\right)dt \end{equation*}$

$\begin{equation*} = A_{1}(\beta) = \frac{2\kappa_1^{\beta}+\kappa_1^{\beta-1}\kappa_{2}+\kappa_{2}^{\beta-1}\kappa_1+2\kappa_{2}^{\beta}-6\kappa_1^{\beta}}{6} \end{equation*}$

and

$\begin{equation*} \int_{0}^{1}\left(\kappa_{2}^{\beta}-\left((1-s)\kappa_{2}^{\beta}+s\kappa_{1}^{\beta}\right)\right)ds = B_{1}(\beta) = \frac{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}{2}, \end{equation*}$

we get

$\begin{equation*} \leq \int_{0}^{1}\left(\left((1-s)\kappa_1^{\beta-1}+s\kappa_{2}^{\beta-1}\right) \left((1-s)\kappa_1+s\kappa_{2}\right)-\kappa_1^{\beta}\right) \left[|h^{\prime}(\kappa_1)|^{q}+s\eta(|h^{\prime}(\kappa_{2})|^{q}, |h^{\prime}(\kappa_1)|^{q})\right]ds \end{equation*}$

$\begin{equation*} = |h^{\prime}(\kappa_{1})|^{q}\int_{0}^{1} \left(\left((1-s)\kappa_1^{\beta-1}+s\kappa_{2}^{\beta-1}\right) \left((1-s)\kappa_1+s\kappa_{2}\right)-\kappa_{1}^{\beta}\right)ds \end{equation*}$

$\begin{equation*} +\eta(|h^{\prime}(\kappa_{2})|^{q}, |h^{\prime}(\kappa_1)|^{q})\int_{0}^{1} \left(\left((1-s)\kappa_1^{\beta-1}+s\kappa_{2}^{\beta-1}\right) \left((1-s)\kappa_1+s\kappa_{2}\right)-\kappa_1^{\beta}\right)sds \end{equation*}$

$\begin{equation*} = |h^{\prime}(\kappa_{1})|^{q}\left(\frac{\kappa_{1}^{\beta-1}\kappa_{2}+\kappa_{2}^{\beta-1}\kappa_{1} +2\kappa_{2}^{\beta}-4\kappa_{1}^{\beta}}{6}\right)+\eta(|h^{\prime}(\kappa_{2})|^{q}, |h^{\prime}(\kappa_1)|^{q}) \end{equation*}$

$\begin{equation*} \times\left(\frac{\kappa_{1}^{\beta-1}\kappa_{2} +\kappa_{2}^{\beta-1}\kappa_{1}+3\kappa_{2}^{\beta}-5\kappa_{1}^{\beta}}{12}\right) \end{equation*}$

and

$\begin{equation*} \int_{0}^{1}\left(\kappa_{2}^{\beta}-\left((1-s)\kappa_{2}^{\beta}+s\kappa_{1}^{\beta}\right)\right) \left|h^{\prime}((1-s)\kappa_{2}+s\kappa_{1})\right|^{q}ds \end{equation*}$

$\begin{equation*} \leq\int_{0}^{1}\left(\kappa_{2}^{\beta}-\left((1-s)\kappa_{2}^{\beta}+s\kappa_{1}^{\beta}\right)\right) \left[|h^{\prime}(\kappa_{2})|^{q}+s\eta\left(|h^{\prime}(\kappa_{1})|^{q}, |h^{\prime}(\kappa_{2})|^{q}\right)\right]ds \end{equation*}$

$\begin{equation*} = |h^{\prime}(\kappa_{2})|^{q}\left(\frac{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}{2}\right) +\eta\left(|h^{\prime}(\kappa_{1})|^{q}, |h^{\prime}(\kappa_{2})|^{q}\right) \left(\frac{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}{3}\right), \end{equation*}$

which completes the proof of Theorem 3.2.

Corollary 3.2. Let $\eta(\kappa_{2}, \kappa_{1}) = \kappa_{2}-\kappa_{1}$ . Then Theorem 3.2 leads to the conclusion that

$\begin{equation*} \left|\frac{h(\kappa_{1})+h(\kappa_{2})}{2} -\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x\right| \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\left(A_{1}(\beta)\right)^{1-\frac{1}{q}} \Bigg(|h^{\prime}(\kappa_{1})|^{q}\left(\frac{\kappa_{1}^{\beta-1}\kappa_{2} +\kappa_{2}^{\beta-1}\kappa_{1}+\kappa_{2}^{\beta}-3\kappa_{1}^{\beta}}{6}\right)+|h^{\prime}(\kappa_{2})|^{q} \end{equation*}$

$\begin{equation*} \times\left(\frac{\kappa_{1}^{\beta-1}\kappa_{2}+\kappa_{2}^{\beta-1}\kappa_{1} +3\kappa_{2}^{\beta}-5\kappa_{1}^{\beta}}{12}\right)\Bigg)+\left(B_{1}(\beta)\right)^{1-\frac{1}{q}} \left(|h^{\prime}(\kappa_{2})|^{q}\left(\frac{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}{6}\right) +|h^{\prime}(\kappa_{1})|^{q}\left(\frac{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}{3}\right)\right)\Bigg]. \end{equation*}$

Theorem 3.3. Let $p, q > 1$ with $1/p+1/q = 1$ , $\kappa_{1}, \kappa_{2}\in \mathbb{R}^{+}$ with $\kappa_{1} < \kappa_{2}$ , and $h: [\kappa_{1}, \kappa_{2}]\rightarrow \mathbb{R}$ be a differentiable function on $(\kappa_{1}, \kappa_{2})$ such that $D_{\beta}(h)\in L_{\beta}([\kappa_{1}, \kappa_{2}])$ . Then the inequality

$\begin{equation*} \left|\frac{h(\kappa_{1})+h(\kappa_{2})}{2} -\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x\right| \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\left(\mathcal{A}_{1}(\beta, p)\right)^{\frac{1}{p}}\left(\frac{2|h^{\prime}(\kappa_1)|^{q} +\eta(|h^{\prime}(\kappa_{2})|^{q}, |h^{\prime}(\kappa_1)|^{q})}{2}\right)^{\frac{1}{q}} \end{equation*}$

$\begin{equation*} +\left(\mathcal{A}_{2}(\beta, p)\right)^{\frac{1}{p}}\left(\frac{2|h^{\prime}(\kappa_2)|^{q} +\eta(|h^{\prime}(\kappa_{1})|^{q}, |h^{\prime}(\kappa_2)|^{q})}{2}\right)^{\frac{1}{q}}\Bigg] \end{equation*}$

is valid for $\beta\in (0, 1]$ if $|h^{\prime}|^{q}$ is $\eta$ -convex, where

$\begin{equation*} \mathcal{A}_{1}(\beta, p) = \int_{0}^{1}\left(\left((1-s)\kappa_1^{\beta-1} +s\kappa_{2}^{\beta-1}\right)((1-s)\kappa_1+s\kappa_{2})-\kappa_1^{\beta}\right)^{p}ds \end{equation*}$

and

$\begin{equation*} \mathcal{A}_{2}(\beta, p) = \int_{0}^{1}\left(\kappa_{2}^{\beta} -\left((1-s)\kappa_{2}^{\beta}+s\kappa_{1}^{\beta}\right)\right)^{p}ds. \end{equation*}$

Proof. We clearly see that

$\begin{equation*} \left|\frac{h(\kappa_{1})+h(\kappa_{2})}{2} -\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x\right| \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \Bigg[\int_{0}^{1}\left(\left((1-s)\kappa_1+s\kappa_2\right)^{\beta-1}((1-s)\kappa_1+s\kappa_2) -\kappa_1^{\beta}\right)|h^{\prime}((1-s)\kappa_1+s\kappa_2)|ds \end{equation*}$

$\begin{equation*} +\int_{0}^{1}\left(\kappa_2^{\beta}-\left((1-s)\kappa_2^{\beta} +s\kappa_{1}^{\beta}\right)\right)|h^{\prime}((1-s)\kappa_2+s\kappa_{1})|ds\Bigg]. \end{equation*}$

Making use of Hölder inequality, we have

$\begin{equation*} \int_{0}^{1}\left(\left((1-s)\kappa_1^{\beta-1}+s\kappa_{2}^{\beta-1}\right) ((1-s)\kappa_1+s\kappa_{2})-\kappa_1^{\beta}\right) |h^{\prime}((1-s)\kappa_1+s\kappa_{2})|ds \end{equation*}$

$\begin{equation*} \times\left(\int_{0}^{1}|h^{\prime}((1-s)\kappa_1+s\kappa_{2})|^{q}ds\right)^{\frac{1}{q}} \end{equation*}$

$\begin{equation*} \leq\left(\int_{0}^{1}\left(\left((1-s)\kappa_1^{\beta-1}+s\kappa_{2}^{\beta-1}\right) ((1-s)\kappa_1+s\kappa_{2})-\kappa_1^{\beta}\right)^pds\right)^{\frac{1}{p}} \end{equation*}$

$\begin{equation*} \times\left(\int_{0}^{1}|h^{\prime}(\kappa_1)|^{q} +s\eta(|h^{\prime}(\kappa_{2})|^{q}, |h^{\prime}(\kappa_1)|^{q})ds\right)^{\frac{1}{q}} \end{equation*}$

$\begin{equation*} = \left(\mathcal{A}_{1}(\beta, p)\right)^{\frac{1}{p}}\left(\frac{2|h^{\prime}(\kappa_1)|^{q} +\eta(|h^{\prime}(\kappa_{2})|^{q}, |h^{\prime}(\kappa_1)|^{q})}{2}\right)^{\frac{1}{q}} \end{equation*}$

and

$\begin{equation*} \int_{0}^{1}\left(\kappa_{2}^{\beta}-\left((1-s)\kappa_{2}^{\beta} +s\kappa_{1}^{\beta}\right)\right)\left|h^{\prime}((1-s)\kappa_{2}+s\kappa_{1})\right|ds \end{equation*}$

$\begin{equation*} \leq\left(\int_{0}^{1}\left(\kappa_{2}^{\beta}-\left((1-s)\kappa_{2}^{\beta}+s\kappa_{1}^{\beta}\right)\right)^pds\right)^{\frac{1}{p}} \left(\left|h^{\prime}((1-s)\kappa_{2}+s\kappa_{1})\right|^{q}ds\right)^{\frac{1}{q}} \end{equation*}$

$\begin{equation*} \leq\left(\int_{0}^{1}\left(\kappa_{2}^{\beta}-\left((1-s)\kappa_{2}^{\beta}+s\kappa_{1}^{\beta}\right)\right)^pds\right)^{\frac{1}{p}} \left(\int_{0}^{1}\left(|h^{\prime}(\kappa_{2})|^{q}+s\eta\left(|h^{\prime}(\kappa_{1})|^{q}, |h^{\prime}(\kappa_{2})|^{q}\right)\right)ds\right)^{\frac{1}{q}} \end{equation*}$

$\begin{equation*} = \left(\mathcal{A}_{2}(\beta, p)\right)^{\frac{1}{p}}\left(\frac{2|h^{\prime}(\kappa_2)|^{q} +\eta(|h^{\prime}(\kappa_{1})|^{q}, |h^{\prime}(\kappa_2)|^{q})}{2}\right)^{\frac{1}{q}}. \end{equation*}$

Corollary 3.3. Let $\eta(\kappa_{2}, \kappa_{1}) = \kappa_{2}-\kappa_{1}$ . Then Theorem 3.3 leads to

$\begin{equation*} \left|\frac{h(\kappa_{1})+h(\kappa_{2})}{2} -\frac{\beta}{\kappa_{2}^{\beta}-\kappa_{1}^{\beta}}\int_{\kappa_{1}}^{\kappa_{2}}h(x)d_{\beta}x\right| \end{equation*}$

$\begin{equation*} \leq\frac{(\kappa_{2}-\kappa_{1})}{2(\kappa_{2}^{\beta}-\kappa_{1}^{\beta})} \left[\left(\mathcal{A}_{1}(\beta, p)\right)^{\frac{1}{p}}\left(\frac{|h^{\prime}(\kappa_1)|^{q} +|h^{\prime}(\kappa_{2})|^{q}}{2}\right)^{\frac{1}{q}}+\left(\mathcal{A}_{2}(\beta, p)\right)^{\frac{1}{p}} \left(\frac{|h^{\prime}(\kappa_2)|^{q}+|h^{\prime}(\kappa_{1})|^{q}}{2}\right)^{\frac{1}{q}}\right]. \end{equation*}$

4. Conclusion

We have generalized the Hermite-Hadamard and Hermite-Hadamard-Fejér inequalities for convex functions to the $\eta$ -convex functions via the conformable integral. Our obtained results are the improvements and generalizations of some previous known results, our ideas and approach may lead to a lot of follow-up research.

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to considerable improvement of the article.

The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11971142, 11701176, 11626101, 11601485).

Conflict of interest

The authors declare no conflict of interest.

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