Some Companions of Fejér-Type Inequalities for Harmonically Convex Functions

Abstract

In this paper, we present some mappings defined over $\left[0,1\right]$ related to the Fejér-type inequalities that have been established for harmonically convex functions. As a consequence, we obtain companions of Fejér-type inequalities for harmonically convex functions by using these mappings. Properties of these mappings are discussed, and consequently, we obtain refinement inequalities of some known results.

1. Introduction

The following double inequality has significant importance in the literature of mathematical inequalities for convex functions and is recognized as Hermite–Hadamard’s inequality (see [1,2]):

Let $\phi :I⟶\mathbb{R}$, $\varnothing \ne I\subseteq \mathbb{R}$, ${\nu }_1,{\nu }_2\in I$ with ${\nu }_1<{\nu }_2$ being a convex function, then

$$\phi \left(\frac{{\nu }_1+{\nu }_2}{2}\right)\le \frac{1}{{\nu }_2-{\nu }_1}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\phi \left(\sigma \right)d\sigma \le \frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2},$$

(1)

the inequality holds in reversed direction if $\phi$ is a concave function.

Fejér [3] established the following double inequality as a weighted generalization of (1):

$$\phi \left(\frac{{\nu }_1+{\nu }_2}{2}\right){\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\kappa \left(\sigma \right)d\sigma \le {\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\phi \left(\sigma \right)\kappa \left(\sigma \right)d\sigma \le \frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\kappa \left(\sigma \right)d\sigma ,$$

(2)

where $\phi :I⟶\mathbb{R}$, $\varnothing \ne I\subseteq \mathbb{R}$, ${\nu }_1,{\nu }_2\in I$ with ${\nu }_1<{\nu }_2$ being any convex function and $\kappa :\left[{\nu }_1,{\nu }_2\right]\to \mathbb{R}$ being non-negative integrable and symmetric about $\sigma =\frac{{\nu }_1+{\nu }_2}{2}$.

There are several variants and generalizations of these inequalities, as illustrated in [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].

The following mappings on $\left[0,1\right]$ are of interest:

$$\mathcal{G}\left(\vartheta \right)=\frac{1}{2}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\left[\phi \left(\vartheta {\nu }_1+\left(1-\vartheta \right)\frac{{\nu }_1+{\nu }_2}{2}\right)+\phi \left(\vartheta {\nu }_2+\left(1-\vartheta \right)\frac{{\nu }_1+{\nu }_2}{2}\right)\right]d\sigma ,$$

$$\mathcal{H}\left(\vartheta \right)=\frac{1}{{\nu }_2-{\nu }_1}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\phi \left(\vartheta \sigma +\left(1-\vartheta \right)\frac{{\nu }_1+{\nu }_2}{2}\right)d\sigma ,$$

$$\mathcal{L}\left(\vartheta \right)=\frac{1}{2\left({\nu }_2-{\nu }_1\right)}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\left[\phi \left(\vartheta {\nu }_1+\left(1-\vartheta \right)\sigma \right)+\phi \left(\vartheta {\nu }_2+\left(1-\vartheta \right)\sigma \right)\right]d\sigma ,$$

$$I\left(\vartheta \right)=\frac{1}{2}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\left[\phi \left(\vartheta \frac{{\nu }_1+\sigma }{2}+\left(1-\vartheta \right)\frac{{\nu }_1+{\nu }_2}{2}\right)+\phi \left(\vartheta \frac{{\nu }_2+\sigma }{2}+\left(1-\vartheta \right)\frac{{\nu }_1+{\nu }_2}{2}\right)\right]\kappa \left(\sigma \right)d\sigma$$

and

$${\mathcal{Z}}_{\kappa }\left(\vartheta \right)=\frac{1}{2}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\left[\phi \left(\vartheta {\nu }_1+\left(1-\vartheta \right)\sigma \right)+\phi \left(\vartheta {\nu }_2+\left(1-\vartheta \right)\sigma \right)\right]\kappa \left(\sigma \right)d\sigma ,$$

where $\phi :\left[{\nu }_1,{\nu }_2\right]\to \mathbb{R}$ is a convex function and $\kappa :\left[{\nu }_1,{\nu }_2\right]\to \mathbb{R}$ is non-negative integrable and symmetric about $\sigma =\frac{{\nu }_1+{\nu }_2}{2}$.

The important results that characterize the properties of the above mappings and inequalities are discussed by a number of mathematicians.

In [4], Dragomir established a result which refines the first inequality of (1). In another paper [9], Dragomir et al. obtained further refinements of (1). In [18], Hwang et al. proved the following useful result in order to establish some results for coordinated convex functions on a rectangle from the plane $\mathbb{R}$.

Lemma 1.

[18] Let $\phi :$$\left[{\nu }_1,{\nu }_2\right]\to \mathbb{R}$ be a convex function and let ${\nu }_1\le y_1\le {\sigma }_1\le {\sigma }_2\le y_2\le {\nu }_2$ with ${\sigma }_1+{\sigma }_2=y_1+y_2$. Then,

$$\phi \left({\sigma }_1\right)+\phi \left({\sigma }_2\right)\le \phi \left(y_1\right)+\phi \left(y_2\right).$$

Yang and Tseng [28] used Lemma 1 to refine the first inequality of (2) and generalized a result proved by Dragomir in [4]. In [24], Tseng et al. established Hermite–Hadamard and Fejér-type inequalities by using the mapping I. Furthermore, Fejér-type inequalities have also been proven by Tseng et al. in [25] by using the mapping I. Hwang et al. [25] obtained some Fejér-type inequalities in connection to the mappings $\mathcal{G}$, $\mathcal{H}$, I, ${\mathcal{Z}}_{\kappa }$.

One of the generalizations of the convex functions is considered as a harmonically convex function:

Definition 1.

[32] Define $I\subseteq \mathbb{R}\backslash \left\{0\right\}$ as an interval of real numbers. A function φ from I to the real numbers is considered to be harmonically convex if

$$\phi \left(\frac{\sigma y}{\vartheta \sigma +\left(1-\vartheta \right)y}\right)\le \vartheta \phi \left(y\right)+\left(1-\vartheta \right)\phi \left(\sigma \right)$$

(3)

for all $\sigma ,y\in I$ and $\vartheta \in \left[0,1\right]$. The function φ is defined to be harmonically concave if the inequality in (3) is reversed.

İşcan [32] used harmonically convex functions to develop an inequality of Hermite–Hadamard type.

Theorem 1.

[32] Let $\phi :I\subseteq \mathbb{R}\backslash \left\{0\right\}\to \mathbb{R}$ be a harmonically convex function and ${\nu }_1,{\nu }_2\in I$ with ${\nu }_1<{\nu }_2$. If $\phi \in \mathcal{L}\left(\left[{\nu }_1,{\nu }_2\right]\right)$, then the following inequalities hold:

$$\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)\le \frac{{\nu }_1{\nu }_2}{{\nu }_2-{\nu }_1}{\int }_{{\nu }_2}^{{\nu }_1}\frac{\phi \left(\sigma \right)}{{\sigma }^2}d\sigma \le \frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}.$$

(4)

Some important facts which relate harmonically convex and convex functions are given in the results below.

Theorem 2.

[10,11] If $\left[{\nu }_1,{\nu }_2\right]\subset I\subset \left(0,\infty \right)$, and if we consider the function $w:\left[\frac{1}{{\nu }_2},\frac{1}{{\nu }_1}\right]\to \mathbb{R}$ defined by $w\left(\vartheta \right)=\phi \left(\frac{1}{\vartheta }\right)$, then φ is harmonically convex on $\left[{\nu }_1,{\nu }_2\right]$ if, and only if, w is convex in the usual sense on $\left[\frac{1}{{\nu }_2},\frac{1}{{\nu }_1}\right]$.

Theorem 3.

[10,11] If $I\subset \left(0,\infty \right)$ and φ is a convex and non-decreasing function, then φ is HA-convex (harmonically convex), and if φ is HA-convex (harmonically convex) and a non-increasing function, then φ is convex.

Let $\phi :\left[{\nu }_1,{\nu }_2\right]\subset \left(0,\infty \right)\to \mathbb{R}$ be a harmonically convex mapping and let $\mathcal{S},V:\left[0,1\right]\to \mathbb{R}$ be defined by

$$\mathcal{S}\left(\vartheta \right)=\frac{{\nu }_1{\nu }_2}{{\nu }_2-{\nu }_1}{\int }_{{\nu }_1}^{{\nu }_2}\frac{1}{{\sigma }^2}\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2\vartheta +\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)d\sigma$$

(5)

and

$$V\left(\vartheta \right)=\frac{{\nu }_1{\nu }_2}{2\left({\nu }_2-{\nu }_1\right)}{\int }_{{\nu }_1}^{{\nu }_2}\frac{1}{{\sigma }^2}\left[\phi \left(\frac{2{\nu }_2\sigma }{\left(1+\vartheta \right)\sigma +\left(1-\vartheta \right){\nu }_2}\right)+\phi \left(\frac{2{\nu }_1\sigma }{\left(1+\vartheta \right)\sigma +\left(1-\vartheta \right){\nu }_1}\right)\right]d\sigma .$$

(6)

The author obtained the refinement inequalities for (4) related to the above mappings:

Theorem 4.

[20] Let $\phi :\left[{\nu }_1,{\nu }_2\right]\subset \left(0,\infty \right)\to \mathbb{R}$ be a harmonically convex function on $\left[{\nu }_1,{\nu }_2\right]$. Then,

(i) 

$\mathcal{S}$ is harmonically convex $\left(0,1\right]$ and increases monotonically on $\left[0,1\right]$.

(ii) 

The following holds:

$$\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)=\mathcal{S}\left(0\right)\le \mathcal{S}\left(\vartheta \right)\le \mathcal{S}\left(1\right)=\frac{{\nu }_1{\nu }_2}{{\nu }_2-{\nu }_1}{\int }_{{\nu }_1}^{{\nu }_2}\frac{\phi \left(\sigma \right)}{{\sigma }^2}d\sigma .$$

Theorem 5.

[20] Let $\phi :\left[{\nu }_1,{\nu }_2\right]\subset \left(0,\infty \right)\to \mathbb{R}$ be a harmonically convex function on $\left[{\nu }_1,{\nu }_2\right]$. Then,

(i) 

V is harmonically convex $\left(0,1\right]$ and increases monotonically on $\left[0,1\right]$.

(ii) 

The following holds:

$$\frac{{\nu }_1{\nu }_2}{{\nu }_2-{\nu }_1}{\int }_{{\nu }_1}^{{\nu }_2}\frac{\phi \left(\sigma \right)}{{\sigma }^2}d\sigma =V\left(0\right)\le V\left(\vartheta \right)\le V\left(1\right)=\frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}.$$

A harmonically symmetric function is defined in the definition below.

Definition 2.

[22] A function $\kappa :\left[{\nu }_1,{\nu }_2\right]\subseteq \mathbb{R}\backslash \left\{0\right\}\to \mathbb{R}$ is harmonically symmetric with respect to $\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}$ if

$$\kappa \left(\sigma \right)=\kappa \left(\frac{1}{\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }}\right)$$

holds for all $\sigma \in \left[{\nu }_1,{\nu }_2\right]$.

Fejér-type inequalities using harmonically convex functions and the notion of harmonically symmetric functions were presented in Chan and Wu [33].

Theorem 6.

[33] Let $\phi :I\subseteq \mathbb{R}\backslash \left\{0\right\}\to \mathbb{R}$ be a harmonically convex function and ${\nu }_1,{\nu }_2\in I$ with ${\nu }_1<{\nu }_2$. If $\phi \in \mathcal{L}\left(\left[{\nu }_1,{\nu }_2\right]\right)$ and $\kappa :\left[{\nu }_1,{\nu }_2\right]\subseteq \mathbb{R}\backslash \left\{0\right\}\to \mathbb{R}$ are non-negative, integrable and harmonically symmetric with respect to $\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}$, then

$$\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right){\int }_{{\nu }_2}^{{\nu }_1}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \le {\int }_{{\nu }_2}^{{\nu }_1}\frac{\phi \left(\sigma \right)\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \le \frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}{\int }_{{\nu }_2}^{{\nu }_1}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma .$$

(7)

Chan and Wu [33] also defined some mappings related to (7) and discussed important properties of these mappings.

Let us now define the following mappings on $\left[0,1\right]$ related to (7):

$${\mathcal{G}}_1\left(\vartheta \right)=\frac{1}{2}\left[\phi \left(\frac{2{\nu }_1{\nu }_2}{2{\nu }_1\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\right)+\phi \left(\frac{2{\nu }_1{\nu }_2}{2{\nu }_2\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\right)\right]d\sigma ,$$

$$\mathcal{S}\left(\vartheta \right)=\frac{{\nu }_1{\nu }_2}{{\nu }_2-{\nu }_1}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\frac{1}{{\sigma }^2}\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma }\right)d\sigma ,$$

$${\mathcal{S}}_{\kappa }\left(\vartheta \right)={\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma }\right)\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma ,$$

$$\begin{array}{c}{I}_1\left(\vartheta \right)=\frac{1}{2}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta {\nu }_2\left({\nu }_1+\sigma \right)+\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)\right.\hfill \\ \hfill \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta {\nu }_1\left(\sigma +{\nu }_2\right)+\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma ,\end{array}$$

$$T\left(\vartheta \right)=\frac{{\nu }_1{\nu }_2}{2\left({\nu }_2-{\nu }_1\right)}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\frac{1}{{\sigma }^2}\left[\phi \left(\frac{{\nu }_2\sigma }{\vartheta \sigma +\left(1-\vartheta \right){\nu }_2}\right)+\phi \left(\frac{{\nu }_1\sigma }{\vartheta \sigma +\left(1-\vartheta \right){\nu }_1}\right)\right]d\sigma ,$$

$$T_{\kappa }\left(\vartheta \right)=\frac{1}{2}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\left[\phi \left(\frac{{\nu }_2\sigma }{\vartheta \sigma +\left(1-\vartheta \right){\nu }_2}\right)+\phi \left(\frac{{\nu }_1\sigma }{\vartheta \sigma +\left(1-\vartheta \right){\nu }_1}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma ,$$

$$\begin{array}{c}{\mathcal{R}}_1\left(\vartheta \right)=\frac{1}{2}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\left[\phi \left(\frac{2{\nu }_2\sigma }{2\vartheta \sigma +\left(1-\vartheta \right)\left({\nu }_2+\sigma \right)}\right)+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_1\sigma \vartheta +\left(1-\vartheta \right){\nu }_2\left({\nu }_1+\sigma \right)}\right)\right.\hfill \\ \hfill \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_2\sigma \vartheta +\left(1-\vartheta \right){\nu }_1\left({\nu }_2+\sigma \right)}\right)+\phi \left(\frac{2{\nu }_1\sigma }{2\vartheta \sigma +\left(1-\vartheta \right)\left({\nu }_1+\sigma \right)}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma ,\end{array}$$

and

$$\begin{array}{c}{\overline{\mathcal{R}}}_1\left(\vartheta \right)=\frac{{\nu }_1{\nu }_2}{2\left({\nu }_2-{\nu }_1\right)}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\left[\phi \left(\frac{2{\nu }_2\sigma }{2\vartheta \sigma +\left(1-\vartheta \right)\left({\nu }_2+\sigma \right)}\right)+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_1\sigma \vartheta +\left(1-\vartheta \right){\nu }_2\left({\nu }_1+\sigma \right)}\right)\right.\hfill \\ \hfill \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_2\sigma \vartheta +\left(1-\vartheta \right){\nu }_1\left({\nu }_2+\sigma \right)}\right)+\phi \left(\frac{2{\nu }_1\sigma }{2\vartheta \sigma +\left(1-\vartheta \right)\left({\nu }_1+\sigma \right)}\right)\right]\frac{d\sigma }{{\sigma }^2},\end{array}$$

where $\phi :\left[{\nu }_1,{\nu }_2\right]\to \mathbb{R}$ is a harmonically convex function and $\kappa :\left[{\nu }_1,{\nu }_2\right]\to \mathbb{R}$ is non-negative integrable and symmetric about $\sigma =\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}$.

Here, we point out an important lemma to be used in the sequel of the paper.

Lemma 2.

[21] Let $\phi :$$\left[{\nu }_1,{\nu }_2\right]\subset \left(0,\infty \right)\to \mathbb{R}$ be a harmonically convex function and let ${\nu }_1\le y_1\le {\sigma }_1\le {\sigma }_2\le y_2\le {\nu }_2$ with $\frac{{\sigma }_1{\sigma }_2}{{\sigma }_1+{\sigma }_2}=\frac{y_1{y}_2}{y_1+y_2}$. Then,

$$\phi \left({\sigma }_1\right)+\phi \left({\sigma }_2\right)\le \phi \left(y_1\right)+\phi \left(y_2\right).$$

We refer to the paper [21] for further important Fejér-type inequalities connected with the mappings ${\mathcal{G}}_1$, $\mathcal{S}$, $I_1$, ${\mathcal{S}}_{\kappa }$, T and $T_{\kappa }$.

Inspired by the studies conducted in [4,20,21,24,25,27], we consider the above mappings in connection to (7) and prove new Féjer-type inequalities which are variants of results proven in [25] using harmonically convex functions, Lemma 2 and novel techniques.

2. Main Results

We begin this section with the following result.

Theorem 7.

Let φ, κ, I be defined as above. Then, the following Fejér-type inequalities hold:

(i) 

The inequalities

$$\begin{array}{c}\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \hfill \\ \le 2\left[{\int }_{\frac{4{\nu }_1{\nu }_2}{{\nu }_1+3{\nu }_2}}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}\phi \left(\sigma \right)\frac{\kappa \left(\frac{{\nu }_1{\nu }_2\sigma }{4{\nu }_1{\nu }_2-\left({\nu }_1+2{\nu }_2\right)\sigma }\right)}{{\sigma }^2}d\sigma +{\int }_{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}^{\frac{4{\nu }_1{\nu }_2}{3{\nu }_1+{\nu }_2}}\phi \left(\sigma \right)\frac{\kappa \left(\frac{{\nu }_1{\nu }_2\sigma }{\left(2{\nu }_1+3{\nu }_2\right)\sigma -4{\nu }_1{\nu }_2}\right)}{{\sigma }^2}d\sigma \right]\\ \le {\int }_0^1{I}_1\left(\vartheta \right)d\vartheta \le \frac{1}{2}\left[\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \right.\\ \hfill \left.+\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}\left[\phi \left(\frac{2{\nu }_1\sigma }{{\nu }_1+\sigma }\right)+\phi \left(\frac{2\sigma {\nu }_2}{\sigma +{\nu }_2}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \right]\end{array}$$

(8)

hold.

(ii) 

If φ is differentiable on $\left[{\nu }_1,{\nu }_2\right]$ and κ is bounded on $\left[{\nu }_1,{\nu }_2\right]$, then the inequalities

$$\begin{array}{c}0\le \frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}\left[\phi \left(\frac{2{\nu }_1\sigma }{{\nu }_1+\sigma }\right)+\phi \left(\frac{2\sigma {\nu }_2}{\sigma +{\nu }_2}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma -I_1\left(\vartheta \right)\hfill \\ \hfill \le \left(1-\vartheta \right)\left[\left(\frac{{\nu }_2-{\nu }_1}{{\nu }_1{\nu }_2}\right)\left[\frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}\right]-{\int }_{{\nu }_1}^{{\nu }_2}\frac{\phi \left(\sigma \right)}{{\sigma }^2}d\sigma \right]{∥\kappa ∥}_{\infty },\end{array}$$

(9)

hold for all $\vartheta \in \left[0,1\right]$, where ${∥\kappa ∥}_{\infty }=\underset{\sigma \in \left[{\nu }_1,{\nu }_2\right]}{sup}\kappa \left(\sigma \right)$.

(iii) 

If φ is differentiable on $\left[{\nu }_1,{\nu }_2\right]$, then the inequalities

$$\begin{array}{c}0\le \frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}{\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma -I_1\left(\vartheta \right)\hfill \\ \hfill \le \frac{\left({\nu }_2-{\nu }_1\right)\left({\nu }_2^2{\phi }^{{}^{\prime }}\left({\nu }_2\right)-{\nu }_1^2{\phi }^{{}^{\prime }}\left({\nu }_1\right)\right)}{4{\nu }_1{\nu }_2}{\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \end{array}$$

(10)

for all $\vartheta \in \left[0,1\right]$.

Proof. 

(i)

We can obtain the following identities using integration tools and the postulates of $\kappa$:

$$\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma =\left(2{\int }_0^{\frac{1}{2}}d\vartheta \right)\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)\left({\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \right).$$

(11)

By making use of the substitution $\sigma =\frac{{\nu }_{1}u}{2{\nu }_1-u}$ on the right hand side of (11), we obtain

$$\begin{array}{c}\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma =\left(2{\int }_0^{\frac{1}{2}}d\vartheta \right)\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)\hfill \\ \hfill \times \left(2{\int }_{{\nu }_1}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}d\sigma \right)=4\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right){\int }_{{\nu }_1}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}{\int }_0^{\frac{1}{2}}\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}d\vartheta d\sigma .\end{array}$$

(12)

$$\begin{array}{c}2\left[{\int }_{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}^{\frac{4{\nu }_1{\nu }_2}{3{\nu }_1+{\nu }_2}}\phi \left(\sigma \right)\frac{\kappa \left(\frac{{\nu }_1{\nu }_2\sigma }{4{\nu }_1{\nu }_2-\left(2{\nu }_1+{\nu }_2\right)\sigma }\right)}{{\sigma }^2}d\sigma +{\int }_{\frac{4{\nu }_1{\nu }_2}{{\nu }_1+3{\nu }_2}}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}\phi \left(\sigma \right)\frac{\kappa \left(\frac{{\nu }_1{\nu }_2\sigma }{\left(2{\nu }_1+3{\nu }_2\right)\sigma -4{\nu }_1{\nu }_2}\right)}{{\sigma }^2}d\sigma \right]\hfill \\ =2{\int }_{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}^{\frac{4{\nu }_1{\nu }_2}{3{\nu }_1+{\nu }_2}}\left[\phi \left(\sigma \right)+\phi \left(\frac{1}{\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }}\right)\right]\frac{\kappa \left(\frac{{\nu }_1{\nu }_2\sigma }{4{\nu }_1{\nu }_2-\left(2{\nu }_1+{\nu }_2\right)\sigma }\right)}{{\sigma }^2}d\sigma \\ =2{\int }_{{\nu }_1}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}{\int }_0^{\frac{1}{2}}\left[\phi \left(\frac{4{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2+\left({\nu }_1+{\nu }_2\right)\sigma }\right)\right.\\ \hfill \left.+\phi \left(\frac{4{\nu }_1{\nu }_2\sigma }{3\left({\nu }_1+{\nu }_2\right)\sigma -2{\nu }_1{\nu }_2}\right)\right]\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}d\vartheta d\sigma ,\end{array}$$

(13)

$$\begin{array}{c}{\int }_0^1{I}_1\left(\vartheta \right)d\vartheta =\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}{\int }_0^{\frac{1}{2}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{{\nu }_1\left(\sigma +{\nu }_2\right)\vartheta +\sigma \left({\nu }_1+{\nu }_2\right)\left(1-\vartheta \right)}\right)\right.\hfill \\ \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{{\nu }_1\left(\sigma +{\nu }_2\right)\left(1-\vartheta \right)+\sigma \left({\nu }_1+{\nu }_2\right)\vartheta }\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\vartheta d\sigma \\ +\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}{\int }_0^{\frac{1}{2}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{{\nu }_2\left(\sigma +{\nu }_1\right)\vartheta +\sigma \left({\nu }_1+{\nu }_2\right)\left(1-\vartheta \right)}\right)\right.\\ \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{{\nu }_2\left(\sigma +{\nu }_1\right)\left(1-\vartheta \right)+\sigma \left({\nu }_1+{\nu }_2\right)\vartheta }\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\vartheta d\sigma \\ =\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}{\int }_0^{\frac{1}{2}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{{\nu }_2\left(\sigma +{\nu }_1\right)\vartheta +\sigma \left({\nu }_1+{\nu }_2\right)\left(1-\vartheta \right)}\right)\right.\\ \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{{\nu }_2\left(\sigma +{\nu }_1\right)\left(1-\vartheta \right)+\sigma \left({\nu }_1+{\nu }_2\right)\vartheta }\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\vartheta d\sigma \\ +\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}{\int }_0^{\frac{1}{2}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\left(\left({\nu }_2+2{\nu }_1\right)\sigma -{\nu }_1{\nu }_2\right)\vartheta +\sigma \left({\nu }_1+{\nu }_2\right)\left(1-\vartheta \right)}\right)\right.\\ \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\left(\left({\nu }_1+2{\nu }_2\right)\sigma -{\nu }_1{\nu }_2\right)\left(1-\vartheta \right)+\sigma \left({\nu }_1+{\nu }_2\right)\vartheta }\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\vartheta d\sigma \\ =\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}{\int }_0^{\frac{1}{2}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta \left({\nu }_1+{\nu }_2\right)\sigma +2\left(1-\vartheta \right){\nu }_1{\nu }_2}\right)\right.\\ \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma +2\vartheta {\nu }_1{\nu }_2}\right)\right]\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}d\vartheta d\sigma \\ +\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}{\int }_0^{\frac{1}{2}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta \left({\nu }_1+{\nu }_2\right)\sigma +2\left(1-\vartheta \right)\left(\left({\nu }_1+{\nu }_2\right)\sigma -{\nu }_1{\nu }_2\right)}\right)\right.\\ \hfill \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma +2\vartheta \left(\left({\nu }_1+{\nu }_2\right)\sigma -{\nu }_1{\nu }_2\right)}\right)\right]\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}d\vartheta d\sigma \end{array}$$

(14)

and

$$\begin{array}{c}\frac{1}{2}\left[\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma +\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}\left[\phi \left(\frac{2{\nu }_1\sigma }{{\nu }_1+\sigma }\right)+\phi \left(\frac{2\sigma {\nu }_2}{\sigma +{\nu }_2}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \right]\hfill \\ =\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}{\int }_0^{\frac{1}{2}}\left[2\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)+\phi \left(\frac{2{\nu }_1\sigma }{{\nu }_1+\sigma }\right)+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\left({\nu }_1+2{\nu }_2\right)\sigma -{\nu }_1{\nu }_2}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\vartheta d\sigma \\ ={\int }_{{\nu }_1}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}{\int }_0^{\frac{1}{2}}\left[\phi \left(\sigma \right)+\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)\right]\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}d\vartheta d\sigma \\ \hfill +{\int }_{{\nu }_1}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}{\int }_0^{\frac{1}{2}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)+\phi \left(\frac{1}{\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }}\right)\right]\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}d\vartheta d\sigma .\end{array}$$

(15)

By using Lemma 2, we observe that the following inequality holds for all $\vartheta \in \left[0,\frac{1}{2}\right]$ and $\sigma \in \left[{\nu }_1,\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right]$:

The inequality

$$4\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)\le 2\left[\phi \left(\frac{4{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2+\left({\nu }_1+{\nu }_2\right)\sigma }\right)+\phi \left(\frac{4{\nu }_1{\nu }_2\sigma }{3\left({\nu }_1+{\nu }_2\right)\sigma -2{\nu }_1{\nu }_2}\right)\right]$$

(16)

holds for

$${\sigma }_1={\sigma }_2=\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2},y_1=\frac{4{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2+\left({\nu }_1+{\nu }_2\right)\sigma }\mathrm{and}{y}_2=\frac{4{\nu }_1{\nu }_2\sigma }{3\left({\nu }_1+{\nu }_2\right)\sigma -2{\nu }_1{\nu }_2}.$$

The inequality

$$\begin{array}{c}2\phi \left(\frac{4{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2+\left({\nu }_1+{\nu }_2\right)\sigma }\right)\hfill \\ \hfill \le \phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta \left({\nu }_1+{\nu }_2\right)\sigma +2\left(1-\vartheta \right){\nu }_1{\nu }_2}\right)+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma +2\vartheta {\nu }_1{\nu }_2}\right)\end{array}$$

(17)

holds for

$$\begin{array}{cc}\hfill {\sigma }_1& ={\sigma }_2=\frac{4{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2+\left({\nu }_1+{\nu }_2\right)\sigma },\hfill \\ \hfill y_1& =\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta \left({\nu }_1+{\nu }_2\right)\sigma +2\left(1-\vartheta \right){\nu }_1{\nu }_2}\hfill \\ \hfill \mathrm{and}{y}_2& =\frac{2{\nu }_1{\nu }_2\sigma }{\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma +2\vartheta {\nu }_1{\nu }_2}.\hfill \end{array}$$

The inequality

$$\begin{array}{c}2\phi \left(\frac{4{\nu }_1{\nu }_2\sigma }{3\left({\nu }_1+{\nu }_2\right)\sigma -2{\nu }_1{\nu }_2}\right)\hfill \\ \le \phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta \left({\nu }_1+{\nu }_2\right)\sigma +2\left(1-\vartheta \right)\left(\left({\nu }_1+{\nu }_2\right)\sigma -{\nu }_1{\nu }_2\right)}\right)\\ \hfill +\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma +2\vartheta \left(\left({\nu }_1+{\nu }_2\right)\sigma -{\nu }_1{\nu }_2\right)}\right)\end{array}$$

(18)

holds for

$$\begin{array}{cc}\hfill {\sigma }_1& ={\sigma }_2=\frac{4{\nu }_1{\nu }_2\sigma }{3\left({\nu }_1+{\nu }_2\right)\sigma -2{\nu }_1{\nu }_2},\hfill \\ \hfill y_1& =\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta \left({\nu }_1+{\nu }_2\right)\sigma +2\left(1-\vartheta \right)\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)}\hfill \\ \hfill \mathrm{and}{y}_2& =\frac{2{\nu }_1{\nu }_2\sigma }{\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma +2\vartheta \left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)}.\hfill \end{array}$$

The inequality

$$\begin{array}{c}\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta \left({\nu }_1+{\nu }_2\right)\sigma +2\left(1-\vartheta \right){\nu }_1{\nu }_2}\right)+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma +2\vartheta {\nu }_1{\nu }_2}\right)\hfill \\ \hfill \le \phi \left(\sigma \right)+\phi \left(\frac{1}{\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }}\right)\end{array}$$

(19)

holds for

$$\begin{array}{cc}\hfill {\sigma }_1& =\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta \left({\nu }_1+{\nu }_2\right)\sigma +2\left(1-\vartheta \right){\nu }_1{\nu }_2},\hfill \\ \hfill {\sigma }_2& =\frac{2{\nu }_1{\nu }_2\sigma }{\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma +2\vartheta {\nu }_1{\nu }_2},\hfill \\ \hfill y_1& =\sigma \mathrm{and}{y}_2=\frac{1}{\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }}.\hfill \end{array}$$

Finally, the inequality

$$\begin{array}{c}\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta \left({\nu }_1+{\nu }_2\right)\sigma +2\left(1-\vartheta \right)\left(\left({\nu }_1+{\nu }_2\right)\sigma -{\nu }_1{\nu }_2\right)}\right)\hfill \\ +\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma +2\vartheta \left(\left({\nu }_1+{\nu }_2\right)\sigma -{\nu }_1{\nu }_2\right)}\right)\\ \hfill \le \phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)+\phi \left(\frac{1}{\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }}\right)\end{array}$$

(20)

holds for

$$\begin{array}{cc}\hfill {\sigma }_1& =\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta \left({\nu }_1+{\nu }_2\right)\sigma +2\left(1-\vartheta \right)\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)},\hfill \\ \hfill {\sigma }_2& =\frac{2{\nu }_1{\nu }_2\sigma }{\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)\sigma +2\vartheta \left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)},\hfill \\ \hfill y_1& =\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\mathrm{and}{y}_2=\frac{1}{\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }}.\hfill \end{array}$$

Multiplying the inequalities (16)–(20) by $\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}$ and integrating them over $\vartheta$ on $\left[0,\frac{1}{2}\right]$, over $\sigma$ on $\left[{\nu }_1,\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right]$ and using identities (11)–(15), we derive (8).

(ii)

Since $\phi :\left[{\nu }_1,{\nu }_2\right]\to \mathbb{R}$ is harmonically convex on $\left[{\nu }_1,{\nu }_2\right]$, hence $w:\left[\frac{1}{{\nu }_2},\frac{1}{{\nu }_1}\right]\to \mathbb{R}$ defined by $w\left(\sigma \right)=\phi \left(\frac{1}{\sigma }\right)$ is convex on $\left[\frac{1}{{\nu }_2},\frac{1}{{\nu }_1}\right]$. Thus, by integration by parts, we obtain that the following identity holds:

$$\begin{array}{c}{\int }_{\frac{1}{{\nu }_2}}^{\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}}\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}-\sigma \right)\left[w^{{}^{\prime }}\left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\sigma \right)-w^{{}^{\prime }}\left(\sigma \right)\right]d\sigma \hfill \\ \hfill =\left(\frac{{\nu }_2-{\nu }_1}{2{\nu }_1{\nu }_2}\right)\left[w\left(\frac{1}{{\nu }_1}\right)+w\left(\frac{1}{{\nu }_2}\right)\right]-{\int }_{\frac{1}{{\nu }_2}}^{\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}}\left[w\left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\sigma \right)+w\left(\sigma \right)\right]d\sigma .\end{array}$$

(21)

The equality (21) is equivalent to the following equality:

$$\begin{array}{c}{\int }_{{\nu }_1}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}\frac{1}{{\sigma }^2}\left(\frac{1}{\sigma }-\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\left[\frac{{\phi }^{{}^{\prime }}\left(\frac{1}{\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }}\right)}{{\left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }\right)}^2}-{\sigma }^2{\phi }^{{}^{\prime }}\left(\sigma \right)\right]d\sigma \hfill \\ \hfill =\left(\frac{{\nu }_2-{\nu }_1}{{\nu }_1{\nu }_2}\right)\left[\frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}\right]-{\int }_{{\nu }_1}^{{\nu }_2}\frac{\phi \left(\sigma \right)}{{\sigma }^2}d\sigma .\end{array}$$

(22)

Under the propositions of $\kappa$, we have the following identities using substitution rules for integration:

$$\begin{array}{c}\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}\left[\phi \left(\frac{2{\nu }_1\sigma }{{\nu }_1+\sigma }\right)+\phi \left(\frac{2{\nu }_2\sigma }{\sigma +{\nu }_2}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \hfill \\ =\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}\left[\phi \left(\frac{2{\nu }_1\sigma }{{\nu }_1+\sigma }\right)+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\sigma \left({\nu }_1+2{\nu }_2\right)-{\nu }_1{\nu }_2}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \\ \hfill ={\int }_{{\nu }_1}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}\left[\phi \left(\sigma \right)+\phi \left(\frac{1}{\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }}\right)\right]\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}d\sigma \end{array}$$

(23)

and

$$\begin{array}{c}{I}_1\left(\vartheta \right)=\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}\left[\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta {\nu }_1\left(\sigma +{\nu }_2\right)+\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)\right.\hfill \\ \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta \left(\sigma \left({\nu }_1+2{\nu }_2\right)-{\nu }_1{\nu }_2\right)+\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \\ =\frac{1}{2}{\int }_{{\nu }_1}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2\vartheta {\nu }_1{\nu }_2+\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)\right.\\ \hfill \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2\vartheta \left(\sigma {\nu }_1+{\nu }_2\sigma -{\nu }_1{\nu }_2\right)+\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)\right]\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}d\sigma \end{array}$$

(24)

for all $\vartheta \in \left[0,1\right]$.

Using the convexity of w and the hypothesis of $\kappa$, the inequality holds for all $\vartheta \in \left[0,1\right]$ and $\sigma \in \left[\frac{1}{{\nu }_2},\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right]$:

$$\begin{array}{c}\left[w\left(\sigma \right)-w\left(\vartheta \sigma +\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\right]\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right)\hfill \\ +\left[w\left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\sigma \right)-w\left(\vartheta \left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\sigma \right)+\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\right]\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right)\\ \le \left(1-\vartheta \right)\left(\sigma -\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)w^{{}^{\prime }}\left(\sigma \right)\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right)\\ +\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}-\sigma \right)w^{{}^{\prime }}\left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\sigma \right)\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right)\\ \hfill =\left(1-\vartheta \right)\left(\sigma -\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\left[w^{{}^{\prime }}\left(\sigma \right)-w^{{}^{\prime }}\left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\sigma \right)\right]\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right).\end{array}$$

(25)

The inequality (25) is equivalent to

$$\begin{array}{c}\left[\phi \left(\sigma \right)-\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2\vartheta {\nu }_1{\nu }_2+\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)\right]\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}\hfill \\ +\left[\phi \left(\frac{1}{\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }}\right)-\phi \left(\frac{1}{\vartheta \left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }\right)+\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)}\right)\right]\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}\\ \hfill \le \left(1-\vartheta \right)\frac{1}{{\sigma }^2}\left(\frac{1}{\sigma }-\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\left[\frac{{\phi }^{{}^{\prime }}\left(\frac{1}{\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }}\right)}{{\left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\frac{1}{\sigma }\right)}^2}-{\sigma }^2{\phi }^{{}^{\prime }}\left(\sigma \right)\right]{∥\kappa ∥}_{\infty }.\end{array}$$

(26)

Integrating the above inequalities over $\sigma$ on $\left[\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2},{\nu }_2\right]$ and using (21)–(24) and (26).

(iii)

Using the convexity of w, we have

$$\frac{w\left(\frac{1}{{\nu }_2}\right)-w\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)}{2}\le \frac{1}{2}\left(\frac{{\nu }_2-{\nu }_1}{2{\nu }_1{\nu }_2}\right)w^{{}^{\prime }}\left(\frac{1}{{\nu }_2}\right)$$

and

$$\frac{w\left(\frac{1}{{\nu }_1}\right)-w\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)}{2}\le \frac{1}{2}\left(\frac{{\nu }_2-{\nu }_1}{2{\nu }_1{\nu }_2}\right)w^{{}^{\prime }}\left(\frac{1}{{\nu }_1}\right).$$

The above inequalities are equivalent to the following inequalities

$$\frac{\phi \left({\nu }_2\right)-\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)}{2}\le \frac{1}{2}\left(\frac{{\nu }_2-{\nu }_1}{2{\nu }_1{\nu }_2}\right){\nu }_2^2{\phi }^{{}^{\prime }}\left({\nu }_2\right)$$

(27)

and

$$\frac{\phi \left({\nu }_1\right)-\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)}{2}\le -\frac{1}{2}\left(\frac{{\nu }_2-{\nu }_1}{2{\nu }_1{\nu }_2}\right){\nu }_1^2{\phi }^{{}^{\prime }}\left({\nu }_1\right).$$

(28)

Adding (27) and (28)

$$\frac{\phi \left({\nu }_2\right)+\phi \left({\nu }_1\right)}{2}-\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)\le \frac{1}{2}\left(\frac{{\nu }_2-{\nu }_1}{2{\nu }_1{\nu }_2}\right)\left[{\nu }_2^2{\phi }^{{}^{\prime }}\left({\nu }_2\right)-{\nu }_1^2{\phi }^{{}^{\prime }}\left({\nu }_1\right)\right].$$

(29)

Hence, we obtain from (29) that

$$\begin{array}{c}\frac{\phi \left({\nu }_2\right)+\phi \left({\nu }_1\right)}{2}{\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma -\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \hfill \\ \hfill \le \frac{\left({\nu }_2-{\nu }_1\right)\left[{\nu }_2^2{\phi }^{{}^{\prime }}\left({\nu }_2\right)-{\nu }_1^2{\phi }^{{}^{\prime }}\left({\nu }_1\right)\right]}{4{\nu }_1{\nu }_2}{\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma .\end{array}$$

(30)

The inequality (10) follows from (19), (20) and (30). □

Corollary 1.

Let $\kappa \left(\sigma \right)=\frac{{\nu }_1{\nu }_2}{{\nu }_2-{\nu }_1}$, $\sigma \in \left[{\nu }_1,{\nu }_2\right]$ in Theorem 7. Then, $I_1\left(\vartheta \right)=V\left(\vartheta \right)$, $\vartheta \in \left[0,1\right]$, and therefore we observe that

(i) 

The inequalities

$$\begin{array}{c}\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)\le \frac{2{\nu }_1{\nu }_2}{{\nu }_2-{\nu }_1}\left[{\int }_{\frac{4{\nu }_1{\nu }_2}{{\nu }_1+3{\nu }_2}}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}\frac{\phi \left(\sigma \right)}{{\sigma }^2}d\sigma +{\int }_{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}^{\frac{4{\nu }_1{\nu }_2}{3{\nu }_1+{\nu }_2}}\frac{\phi \left(\sigma \right)}{{\sigma }^2}d\sigma \right]\\ \le {\int }_0^{1}V\left(\vartheta \right)d\vartheta \le \frac{1}{2}\left[\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)+\frac{{\nu }_1{\nu }_2}{2\left({\nu }_2-{\nu }_1\right)}{\int }_{{\nu }_1}^{{\nu }_2}\frac{1}{{\sigma }^2}\left[\phi \left(\frac{2{\nu }_1\sigma }{{\nu }_1+\sigma }\right)+\phi \left(\frac{2\sigma {\nu }_2}{\sigma +{\nu }_2}\right)\right]d\sigma \right]\end{array}$$

(31)

hold.

(ii) 

If φ is differentiable on $\left[{\nu }_1,{\nu }_2\right]$ and κ is bounded on $\left[{\nu }_1,{\nu }_2\right]$, then the inequalities

$$\begin{array}{c}0\le \frac{{\nu }_1{\nu }_2}{2\left({\nu }_2-{\nu }_1\right)}{\int }_{{\nu }_1}^{{\nu }_2}\frac{1}{{\sigma }^2}\left[\phi \left(\frac{2{\nu }_1\sigma }{{\nu }_1+\sigma }\right)+\phi \left(\frac{2\sigma {\nu }_2}{\sigma +{\nu }_2}\right)\right]d\sigma -V\left(\vartheta \right)\hfill \\ \hfill \le \left(1-\vartheta \right)\left[\frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}-\frac{{\nu }_1{\nu }_2}{{\nu }_2-{\nu }_1}{\int }_{{\nu }_1}^{{\nu }_2}\frac{\phi \left(\sigma \right)}{{\sigma }^2}d\sigma \right],\end{array}$$

(32)

hold for all $\vartheta \in \left[0,1\right]$.

(iii) 

If φ is differentiable on $\left[{\nu }_1,{\nu }_2\right]$, then, for all $\vartheta \in \left[0,1\right]$, we have the inequality

$$0\le \frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}-V\left(\vartheta \right)\le \frac{\left({\nu }_2-{\nu }_1\right)\left[{\nu }_2^2{\phi }^{{}^{\prime }}\left({\nu }_2\right)-{\nu }_1^2{\phi }^{{}^{\prime }}\left({\nu }_1\right)\right]}{4{\nu }_1{\nu }_2}.$$

(33)

In the following theorems, we point out some inequalities for the mappings ${\mathcal{G}}_1$, $\mathcal{S}$, $I_1$, ${\mathcal{S}}_{\kappa }$ as considered above:

Theorem 8.

Let φ, κ, ${\mathcal{G}}_1$, $I_1$ be defined as above. Then:

(i) 

The inequality

$$I_1\left(\vartheta \right)\le {\mathcal{G}}_1\left(\vartheta \right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma$$

(34)

holds for all $\vartheta \in \left[0,1\right]$.

(ii) 

If φ is differentiable on $\left[{\nu }_1,{\nu }_2\right]$ and κ is bounded on $\left[{\nu }_1,{\nu }_2\right]$, then the inequalities

$$0\le I_1\left(\vartheta \right)-\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \le \left(\frac{{\nu }_2-{\nu }_1}{{\nu }_1{\nu }_2}\right)\left[{\mathcal{G}}_1\left(\vartheta \right)-\mathcal{S}\left(\vartheta \right)\right]{∥\kappa ∥}_{\infty },$$

(35)

hold for all $\vartheta \in \left[0,1\right]$, where ${∥\kappa ∥}_{\infty }=\underset{\sigma \in \left[{\nu }_1,{\nu }_2\right]}{sup}\kappa \left(\sigma \right)$.

Proof. 

(i)

Using integration techniques and the hypothesis of $\kappa$, we find that the following identity holds on $\left[0,1\right]$:

$$\begin{array}{c}{\mathcal{G}}_1\left(\vartheta \right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma ={\int }_{{\nu }_1}^{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2}{2{\nu }_1\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\right)\right.\hfill \\ \hfill \left.+\phi \left(\frac{2{\nu }_1{\nu }_2}{2{\nu }_2\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\right)\frac{\kappa \left(\frac{{\nu }_2\sigma }{2{\nu }_2-\sigma }\right)}{{\sigma }^2}d\sigma \right].\end{array}$$

(36)

By Lemma 2, the following inequality holds for all $\vartheta \in \left[0,1\right]$ and $\sigma \in \left[{\nu }_1,\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right]$:

$$\begin{array}{c}\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2\vartheta +\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_2\right)\vartheta +\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)\hfill \\ \hfill \le \phi \left(\frac{2{\nu }_1{\nu }_2}{2{\nu }_1\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\right)+\phi \left(\frac{2{\nu }_1{\nu }_2}{2{\nu }_2\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\right)\end{array}$$

(37)

holds for

$$\begin{array}{cc}\hfill {\sigma }_1& =\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2\vartheta +\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)},{\sigma }_2=\frac{2{\nu }_1{\nu }_2\sigma }{2\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_2\right)\vartheta +\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)},\hfill \\ \hfill y_1& =\frac{2{\nu }_1{\nu }_2}{2{\nu }_1\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\mathrm{and}{y}_2=\frac{2{\nu }_1{\nu }_2}{2{\nu }_2\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}.\hfill \end{array}$$

Multiplying the inequality (37) by $\frac{\kappa \left(\frac{{\nu }_2\sigma }{2{\nu }_2-\sigma }\right)}{{\sigma }^2}$, integrating both sides over $\sigma$ on $\left[{\nu }_1,\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right]$ and using identities (24) and (36), we derive (34).

(ii)

Since $\phi :\left[{\nu }_1,{\nu }_2\right]\to \mathbb{R}$ is harmonically convex on $\left[{\nu }_1,{\nu }_2\right]$, hence $w:\left[\frac{1}{{\nu }_2},\frac{1}{{\nu }_1}\right]\to \mathbb{R}$ defined by $w\left(\sigma \right)=\phi \left(\frac{1}{\sigma }\right)$ is convex on $\left[\frac{1}{{\nu }_2},\frac{1}{{\nu }_1}\right]$. Thus, by integration by parts, we obtain that the following identity holds:

$$\begin{array}{c}\vartheta {\int }_{\frac{1}{{\nu }_2}}^{\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}}\left[\left(\sigma -\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)w^{{}^{\prime }}\left(\vartheta \sigma +\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\right.\hfill \\ \left.+\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}-\sigma \right)w^{{}^{\prime }}\left(\vartheta \left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\sigma \right)+\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\right]d\sigma \\ =\vartheta {\int }_{\frac{1}{{\nu }_2}}^{\frac{1}{{\nu }_1}}\left(\sigma -\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)w^{{}^{\prime }}\left(\vartheta \sigma +\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)d\sigma \\ =\frac{{\nu }_2-{\nu }_1}{2{\nu }_1{\nu }_2}\left[\phi \left(\frac{2{\nu }_1{\nu }_2}{2{\nu }_2\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\right)+\phi \left(\frac{2{\nu }_1{\nu }_2}{2{\nu }_1\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\right)\right]\\ \hfill -{\int }_{{\nu }_1}^{{\nu }_2}\frac{1}{{\sigma }^2}\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\right)d\sigma =\left(\frac{{\nu }_2-{\nu }_1}{{\nu }_1{\nu }_2}\right)\left[{\mathcal{G}}_1\left(\vartheta \right)-\mathcal{S}\left(\vartheta \right)\right].\end{array}$$

(38)

Using the convexity of w and the hypothesis of $\kappa$, the inequality holds for all $\vartheta \in \left[0,1\right]$ and $\sigma \in \left[\frac{1}{{\nu }_2},\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right]$:

$$\begin{array}{c}\left[w\left(\vartheta \sigma +\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)-w\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right]\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right)\\ +\left[w\left(\vartheta \left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\sigma \right)+\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)-w\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right]\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right)\\ \le \vartheta \left(\sigma -\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)w^{{}^{\prime }}\left(\vartheta \sigma +\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right)\\ +\vartheta \left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}-\sigma \right)w^{{}^{\prime }}\left(\vartheta \left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\sigma \right)+\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right)\\ =\vartheta \left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}-\sigma \right)\left[w^{{}^{\prime }}\left(\vartheta \left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\sigma \right)+\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\right.\\ \left.-w^{{}^{\prime }}\left(\vartheta \sigma +\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\right]\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right)\le \vartheta \left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}-\sigma \right)\\ \times \left[w^{{}^{\prime }}\left(\vartheta \left(\frac{1}{{\nu }_1}+\frac{1}{{\nu }_2}-\sigma \right)+\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)-w^{{}^{\prime }}\left(\vartheta \sigma +\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\right]{∥\kappa ∥}_{\infty }.\end{array}$$

(39)

Integrating (39) over $\sigma$ on $\left[\frac{1}{{\nu }_2},\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right]$ and using (38), we obtain

$$\begin{array}{c}{\int }_{\frac{1}{{\nu }_2}}^{\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}}\left[w\left(\vartheta \sigma +\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)+w\left(\vartheta \sigma +\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\right]\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right)d\sigma \\ -2w\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right){\int }_{\frac{1}{{\nu }_2}}^{\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}}\kappa \left(\frac{1}{2\sigma -\frac{1}{{\nu }_2}}\right)d\sigma \le \left(\frac{{\nu }_2-{\nu }_1}{{\nu }_1{\nu }_2}\right)\left[{\mathcal{G}}_1\left(\vartheta \right)-\mathcal{S}\left(\vartheta \right)\right].\end{array}$$

(40)

The last inequality is equivalent to

$$I_1\left(\vartheta \right)-\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \le \left(\frac{{\nu }_2-{\nu }_1}{{\nu }_1{\nu }_2}\right)\left[{\mathcal{G}}_1\left(\vartheta \right)-\mathcal{S}\left(\vartheta \right)\right].$$

(41)

Since $\phi :\left[{\nu }_1,{\nu }_2\right]\to \mathbb{R}$ is harmonically convex on $\left[{\nu }_1,{\nu }_2\right]$, hence $w:\left[\frac{1}{{\nu }_2},\frac{1}{{\nu }_1}\right]\to \mathbb{R}$ defined by $w\left(\sigma \right)=\phi \left(\frac{1}{\sigma }\right)$ is convex on $\left[\frac{1}{{\nu }_2},\frac{1}{{\nu }_1}\right]$. We can prove that the mapping ${\overline{I}}_1:\left[0,1\right]\to \mathbb{R}$ defined by

$$\begin{array}{c}{\overline{I}}_1\left(\vartheta \right)=\frac{1}{2}{\displaystyle \underset{\frac{1}{{\nu }_2}}{\overset{\frac{1}{{\nu }_1}}{\int }}}\left[w\left(\vartheta \left(\frac{\sigma +\frac{1}{{\nu }_1}}{2}\right)+\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\right.\hfill \\ \hfill \left.+w\left(\vartheta \left(\frac{\sigma +\frac{1}{{\nu }_2}}{2}\right)+\left(1-\vartheta \right)\left(\frac{{\nu }_1+{\nu }_2}{2{\nu }_1{\nu }_2}\right)\right)\right]\kappa \left(\frac{1}{\sigma }\right)d\sigma \end{array}$$

(42)

is convex and monotonically increasing on $\left[0,1\right]$. Thus, the mapping $I_1$ is harmonically convex on $\left(0,1\right]$ and monotonically increasing on $\left[0,1\right]$. Consequently, we have

$$\begin{array}{c}\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma =I_1\left(0\right)\le I_1\left(\vartheta \right)\le I_1\left(1\right)\hfill \\ \hfill =\frac{1}{2}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\left[\phi \left(\frac{2{\nu }_1\sigma }{{\nu }_1+\sigma }\right)+\phi \left(\frac{2\sigma {\nu }_2}{\sigma +{\nu }_2}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma .\end{array}$$

(43)

The inequality (35) follows from the inequalities (41) and (43). □

Corollary 2.

Let $\kappa \left(\sigma \right)=\frac{{\nu }_1{\nu }_2}{{\nu }_2-{\nu }_1}$, $\sigma \in \left[{\nu }_1,{\nu }_2\right]$ in Theorem 8. Then, for all $\vartheta \in \left[0,1\right]$

$$\begin{array}{c}{I}_1\left(\vartheta \right)={\mathcal{H}}_1\left(\vartheta \right)=\frac{{\nu }_1{\nu }_2}{2\left({\nu }_2-{\nu }_1\right)}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta {\nu }_2\left({\nu }_1+\sigma \right)+\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)\right.\hfill \\ \hfill \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{\vartheta {\nu }_1\left(\sigma +{\nu }_2\right)+\left(1-\vartheta \right)\sigma \left({\nu }_1+{\nu }_2\right)}\right)\right]\frac{d\sigma }{{\sigma }^2}\end{array}$$

(44)

and the inequalities (34) and (35) reduce to the given inequalities:

(i) 

The inequality

$${\mathcal{H}}_1\left(\vartheta \right)\le {\mathcal{G}}_1\left(\vartheta \right)$$

holds for all $\vartheta \in \left[0,1\right]$.

(ii) 

If φ is differentiable on $\left[{\nu }_1,{\nu }_2\right]$ and κ is bounded on $\left[{\nu }_1,{\nu }_2\right]$, then, the inequalities

$$0\le {\mathcal{H}}_1\left(\vartheta \right)-\phi \left(\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right)\le {\mathcal{G}}_1\left(\vartheta \right)-\mathcal{S}\left(\vartheta \right)$$

(45)

hold for all $\vartheta \in \left[0,1\right]$.

Theorem 9.

Let φ, κ, ${\mathcal{G}}_1$, $I_1$, ${\mathcal{R}}_1$ be defined as above. Then

(i) 

${\mathcal{R}}_1$ is convex on $\left[0,1\right]$.

(ii) 

The inequalities

$$\begin{gathered} {\mathcal{G}}_1\left(\vartheta \right){\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \le {\mathcal{R}}_1\left(\vartheta \right)\le \left(1-\vartheta \right)·\frac{1}{2}{\int }_{{\nu }_1}^{{\nu }_2}\left[\phi \left(\frac{2\sigma {\nu }_2}{\sigma +{\nu }_2}\right)+\phi \left(\frac{2{\nu }_1\sigma }{{\nu }_1+\sigma }\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \\ +\vartheta ·\frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}{\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma \le \frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}{\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma , \end{gathered}$$

(46)

$$I_1\left(1-\vartheta \right)\le {\mathcal{R}}_1\left(\vartheta \right)$$

(47)

and

$$\frac{I_1\left(\vartheta \right)+I_1\left(1-\vartheta \right)}{2}\le {\mathcal{R}}_1\left(\vartheta \right)$$

(48)

hold for all $\vartheta \in \left[0,1\right]$.

(iii) 

The identity

$$\underset{\vartheta \in \left[0,1\right]}{sup}{\mathcal{R}}_1\left(\vartheta \right)=\frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}{\int }_{{\nu }_1}^{{\nu }_2}\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma$$

(49)

holds.

Proof. 

(i)

Since $\phi :\left[{\nu }_1,{\nu }_2\right]\to \mathbb{R}$ is harmonically convex on $\left[{\nu }_1,{\nu }_2\right]$, hence $w:\left[\frac{1}{{\nu }_2},\frac{1}{{\nu }_1}\right]\to \mathbb{R}$ defined by $w\left(\sigma \right)=\phi \left(\frac{1}{\sigma }\right)$ is convex on $\left[\frac{1}{{\nu }_2},\frac{1}{{\nu }_1}\right]$. This shows that the mapping ${\mathcal{R}}_1^{{}^{\prime }}:\left[0,1\right]\to \mathbb{R}$ defined by

$$\begin{array}{c}{\mathcal{R}}_1^{{}^{\prime }}\left(\vartheta \right)=\frac{1}{2}{\displaystyle \underset{\frac{1}{{\nu }_2}}{\overset{\frac{1}{{\nu }_1}}{\int }}}\left[w\left(\vartheta \frac{1}{{\nu }_2}+\left(1-\vartheta \right)\left(\frac{\sigma +\frac{1}{{\nu }_2}}{2}\right)\right)+w\left(\vartheta \frac{1}{{\nu }_2}+\left(1-\vartheta \right)\left(\frac{\frac{1}{{\nu }_1}+\sigma }{2}\right)\right)\right.\hfill \\ \hfill \left.+w\left(\vartheta \frac{1}{{\nu }_1}+\left(1-\vartheta \right)\left(\frac{\sigma +\frac{1}{{\nu }_2}}{2}\right)\right)+w\left(\vartheta \frac{1}{{\nu }_1}+\left(1-\vartheta \right)\left(\frac{\sigma +\frac{1}{{\nu }_1}}{2}\right)\right)\right]\kappa \left(\frac{1}{\sigma }\right)d\sigma ,\end{array}$$

(50)

is convex on $\left[0,1\right]$.

This conclude that the mapping ${\mathcal{R}}_1:\left[0,1\right]\to \mathbb{R}$ defined by

$$\begin{array}{c}{\mathcal{R}}_1\left(\vartheta \right)=\frac{1}{2}{\displaystyle \underset{{\nu }_1}{\overset{{\nu }_2}{\int }}}\left[\phi \left(\frac{2{\nu }_2\sigma }{2\vartheta \sigma +\left(1-\vartheta \right)\left({\nu }_2+\sigma \right)}\right)+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_1\sigma \vartheta +\left(1-\vartheta \right){\nu }_2\left({\nu }_1+\sigma \right)}\right)\right.\hfill \\ \hfill \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_2\sigma \vartheta +\left(1-\vartheta \right){\nu }_1\left({\nu }_2+\sigma \right)}\right)+\phi \left(\frac{2{\nu }_1\sigma }{2\vartheta \sigma +\left(1-\vartheta \right)\left({\nu }_1+\sigma \right)}\right)\right]\frac{\kappa \left(\sigma \right)}{{\sigma }^2}d\sigma ,\end{array}$$

(51)

is harmonically convex on $\left(0,1\right]$.

(ii)

We observe that the following identity holds for all $\vartheta \in \left[0,1\right]$:

$$\begin{array}{c}{\mathcal{R}}_1\left(\vartheta \right)=\frac{1}{2}{\displaystyle \underset{{\nu }_1}{\overset{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}{\int }}}\left[\phi \left(\frac{{\nu }_2\sigma }{\vartheta \sigma +\left(1-\vartheta \right){\nu }_2}\right)+\phi \left(\frac{{\nu }_1{\nu }_2\sigma }{{\nu }_1\sigma \vartheta +\left(1-\vartheta \right)\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)}\right)\right.\hfill \\ \hfill \left.+\phi \left(\frac{{\nu }_1\sigma }{\vartheta \sigma +\left(1-\vartheta \right){\nu }_1}\right)+\phi \left(\frac{{\nu }_1{\nu }_2\sigma }{{\nu }_2\sigma \vartheta +\left(1-\vartheta \right)\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)}\right)\right]\frac{\kappa \left(\frac{{\nu }_1\sigma }{2{\nu }_1-\sigma }\right)}{{\sigma }^2}d\sigma .\end{array}$$

(52)

By Lemma 2, the following inequalities hold for all $\vartheta \in \left[0,1\right]$ and $\sigma \in \left[{\nu }_1,\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right]$:

The inequality

$$\begin{array}{c}2\phi \left(\frac{2{\nu }_1{\nu }_2}{2{\nu }_1\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\right)\hfill \\ \hfill \le \phi \left(\frac{{\nu }_1{\nu }_2\sigma }{{\nu }_1\sigma \vartheta +\left(1-\vartheta \right)\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)}\right)+\phi \left(\frac{{\nu }_1\sigma }{\vartheta \sigma +\left(1-\vartheta \right){\nu }_1}\right)\end{array}$$

(53)

holds for

$${\sigma }_1={\sigma }_2=\frac{2{\nu }_1{\nu }_2}{2{\nu }_1\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)},y_1=\frac{{\nu }_1\sigma }{\vartheta \sigma +\left(1-\vartheta \right){\nu }_1}\mathrm{and}{y}_2=\frac{{\nu }_1{\nu }_2\sigma }{{\nu }_1\sigma \vartheta +\left(1-\vartheta \right)\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)}$$

in Lemma 2.

The inequality

$$\begin{array}{c}2\phi \left(\frac{2{\nu }_1{\nu }_2}{2{\nu }_1\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)}\right)\hfill \\ \hfill \le \phi \left(\frac{{\nu }_1{\nu }_2\sigma }{{\nu }_1\sigma \vartheta +\left(1-\vartheta \right)\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)}\right)+\phi \left(\frac{{\nu }_1\sigma }{\vartheta \sigma +\left(1-\vartheta \right){\nu }_1}\right)\end{array}$$

(54)

holds for

$${\sigma }_1={\sigma }_2=\frac{2{\nu }_1{\nu }_2}{2{\nu }_2\vartheta +\left(1-\vartheta \right)\left({\nu }_1+{\nu }_2\right)},y_1=\frac{{\nu }_2\sigma }{\vartheta \sigma +\left(1-\vartheta \right){\nu }_2}\mathrm{and}{y}_2=\frac{{\nu }_1{\nu }_2\sigma }{{\nu }_1{\nu }_2\vartheta +\left(1-\vartheta \right)\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)}$$

in Lemma 2.

Multiplying the inequalities (53), (54) by $\frac{\kappa \left(\frac{{\nu }_2\sigma }{2{\nu }_2-\sigma }\right)}{{\sigma }^2}$, adding the resulting inequalities, integrating the resulting inequality over $\sigma$ on $\left[{\nu }_1,\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}\right]$ and using identities (36) and (52), we derive the first inequality of (2). Using the harmonic convexity of $\phi$ and the inequality (52), the last part of (2) holds.

Next, by the harmonic convexity of $\phi$ and the identity (52), we obtain

$$\begin{array}{c}{I}_1\left(1-\vartheta \right)={\displaystyle \underset{{\nu }_1}{\overset{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}{\int }}}\left[\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2{\nu }_1{\nu }_2\left(1-\vartheta \right)+\vartheta \left({\nu }_1+{\nu }_2\right)\sigma }\right)\right.\hfill \\ \left.+\phi \left(\frac{2{\nu }_1{\nu }_2\sigma }{2\left(1-\vartheta \right)\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)+\vartheta \sigma \left({\nu }_1+{\nu }_2\right)}\right)\right]\frac{\kappa \left(\frac{{\nu }_2\sigma }{2{\nu }_2-\sigma }\right)}{{\sigma }^2}d\sigma \\ ={\displaystyle \underset{{\nu }_1}{\overset{\frac{2{\nu }_1{\nu }_2}{{\nu }_1+{\nu }_2}}{\int }}}\left[\phi \left(\frac{1}{\frac{1}{2}\left(\frac{{\nu }_1\sigma \vartheta +\left(1-\vartheta \right)\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)}{{\nu }_1{\nu }_2\sigma }\right)+\frac{1}{2}\left(\frac{{\nu }_2\sigma \vartheta +\left(1-\vartheta \right)\left({\nu }_1\sigma +{\nu }_2\sigma -{\nu }_1{\nu }_2\right)}{{\nu }_1{\nu }_2\sigma }\right)}\right)\right.\\ \hfill \left.+\phi \left(\frac{1}{\frac{1}{2}\left(\frac{\vartheta \sigma +\left(1-\vartheta \right){\nu }_2}{{\nu }_2\sigma }\right)+\frac{1}{2}\left(\frac{\vartheta \sigma +\left(1-\vartheta \right){\nu }_1}{{\nu }_1\sigma }\right)}\right)\right]\frac{\kappa \left(\frac{{\nu }_2\sigma }{2{\nu }_2-\sigma }\right)}{{\sigma }^2}d\sigma \le {\mathcal{R}}_1\left(\vartheta \right).\end{array}$$

(55)

Thus, the inequality (47) is proved. From (34), (2) and (55), we obtain (48).

(iii)

The identity (49) holds by using (2).

□

Corollary 3.

Let $\kappa \left(\sigma \right)=\frac{{\nu }_1{\nu }_2}{{\nu }_2-{\nu }_1}$, $\sigma \in \left[{\nu }_1,{\nu }_2\right]$ in Theorem 9. Then, for all $\vartheta \in \left[0,1\right]$

(i) 

${\overline{\mathcal{R}}}_1$ is harmonically convex on $\left(0,1\right]$.

(ii) 

The inequalities

$$\begin{array}{c}{\mathcal{G}}_1\left(\vartheta \right)\le {\overline{\mathcal{R}}}_1\left(\vartheta \right)\le \left(1-\vartheta \right)·\frac{1}{2}\frac{{\nu }_1{\nu }_2}{{\nu }_2-{\nu }_1}{\int }_{{\nu }_1}^{{\nu }_2}\left[\phi \left(\frac{2\sigma {\nu }_2}{\sigma +{\nu }_2}\right)+\phi \left(\frac{2{\nu }_1\sigma }{{\nu }_1+\sigma }\right)\right]\frac{d\sigma }{{\sigma }^2}\hfill \\ \hfill +\vartheta ·\frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}\le \frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2},\end{array}$$

(56)

$${\mathcal{H}}_1\left(1-\vartheta \right)\le {\overline{\mathcal{R}}}_1\left(\vartheta \right)$$

(57)

and

$$\frac{{\mathcal{H}}_1\left(\vartheta \right)+{\mathcal{H}}_1\left(1-\vartheta \right)}{2}\le {\overline{\mathcal{R}}}_1\left(\vartheta \right)$$

(58)

hold for all $\vartheta \in \left[0,1\right]$.

(iii) 

The identity

$$\underset{\vartheta \in \left[0,1\right]}{sup}{\overline{\mathcal{R}}}_1\left(\vartheta \right)=\frac{\phi \left({\nu }_1\right)+\phi \left({\nu }_2\right)}{2}$$

(59)

holds.

3. Conclusions

Mathematical inequalities utilizing convex functions are a relatively new topic. New conclusions are being added to the theory of inequalities as scholars endeavor to generalize convex functions. We employed harmonically convex functions to generalize convex function results. In this work, we defined new mappings over $[0,1]$. We examined intriguing aspects of these mappings and refined Hermite–Hadamard and Fejér-type inequalities for harmonically convex functions. We hope the outcomes of this publication will inspire mathematicians and young researchers to enter this subject.