Some weighted integral inequalities for differentiable preinvex and prequasiinvex functions with applications

Abstract

In this paper, we present weighted integral inequalities of Hermite-Hadamard type for differentiable preinvex and prequasiinvex functions. Our results, on the one hand, give a weighted generalization of recent results for preinvex functions and, on the other hand, extend several results connected with the Hermite-Hadamard type integral inequalities. Applications of the obtained results are provided as well.

MSC: 26D15, 26D20, 26D07.

1 Introduction

Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be a convex mapping and $a,b\in I$ with $a<b$. Then

$$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_a^{b}f(x)dx\le \frac{f(a)+f(b)}{2}.$$

(1.1)

Both the inequalities in (1.1) hold in reversed direction if f is concave. Inequalities (1.1) are famous in mathematical literature due to their rich geometrical significance and applications and are known as the Hermite-Hadamard inequalities (see [1]).

For several results which generalize, improve and extend inequalities (1.1), we refer the interested reader to [2–18].

In [3], Dragomir and Agarwal obtained the following inequalities for differentiable functions which estimate the difference between the middle and the rightmost terms in (1.1).

Theorem 1 [3]

Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, where $a,b\in I$ with $a<b$, and $f^{\prime }\in L([a,b])$. If $|f^{\prime }|$ is a convex function on $[a,b]$, the following inequality holds:

$$|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)dx|\le \frac{b-a}{8}\left[\right|f^{\prime }(a)|+|f^{\prime }(b)\left|\right].$$

(1.2)

Theorem 2 [3]

Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, where $a,b\in I$ with $a<b$, and $f^{\prime }\in L([a,b])$. If $|f^{\prime }{|}^{\frac{p}{p-1}}$ is a convex function on $[a,b]$, the following inequality holds:

$$|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)dx|\le \frac{b-a}{2{(p+1)}^{\frac{1}{p}}}\left[\right|f^{\prime }(a){|}^{\frac{p}{p-1}}+|f^{\prime }(b){|}^{\frac{p}{p-1}}],$$

(1.3)

where $p>1$ and $\frac{1}{p}+\frac{1}{q}=1$.

In [11], Pearce and Pečarić gave an improvement and simplification of the constant in Theorem 2 and consolidated these results with Theorem 1. The following is the main result from [11].

Theorem 3 [11]

Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, where $a,b\in I$ with $a<b$, and $f^{\prime }\in L([a,b])$. If $|f^{\prime }{|}^q$ is a convex function on $[a,b]$, for some $q\ge 1$, the following inequality holds:

$$|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)dx|\le \frac{b-a}{4}{\left[\frac{|f^{\prime }(a){|}^q+|f^{\prime }(b){|}^q}{2}\right]}^{\frac{1}{q}}.$$

(1.4)

If $|f^{\prime }{|}^q$ is concave on $[a,b]$ for some $q\ge 1$, then

$$|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)dx|\le \frac{b-a}{4}\left|f^{\prime }\left(\frac{a+b}{2}\right)\right|.$$

(1.5)

Now, we recall that the notion of quasi-convex functions generalizes the notion of convex functions. More exactly, a function $f:[a,b]\to \mathbb{R}$ is said to be quasi-convex on $[a,b]$ if

$$f(tx+(1-t)y)\le max\{f(x),f(y)\}$$

for all $x,y\in [a,b]$ and $t\in [0,1]$. Clearly, any convex function is a quasi-convex function. Furthermore, there exist quasi-convex functions which are not convex (see [6]).

Recently, Ion [6] introduced two inequalities of the right-hand side of Hadamard type for quasi-convex functions, as follows.

Theorem 4 [6]

Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, where $a,b\in I^{\circ }$ with $a<b$. If $|f^{\prime }|$ is a quasi-convex function on $[a,b]$, the following inequality holds:

$$|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)dx|\le \frac{b-a}{4}max\left\{\right|f^{\prime }(a)|,|f^{\prime }(b)\left|\right\}.$$

(1.6)

Theorem 5 [6]

Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, where $a,b\in I^{\circ }$ with $a<b$. If $|f^{\prime }{|}^p$ is a quasi-convex function on $[a,b]$, for some $p>1$, the following inequality holds:

$$\begin{array}{c}|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)dx|\hfill \\ \le \frac{b-a}{2{(p+1)}^{\frac{1}{p}}}{[max\left\{\right|f^{\prime }(a){|}^{\frac{p}{p-1}},|f^{\prime }(b){|}^{\frac{p}{p-1}}\}]}^{\frac{p-1}{p}},\hfill \end{array}$$

(1.7)

where $\frac{1}{p}+\frac{1}{q}=1$.

In [2], Alomari et al. established Hermite-Hadamard-type inequalities for quasi-convex functions which give refinements of those given above in Theorem 4 and Theorem 5.

Theorem 6 [2]

Let $f:I\subseteq [0,\mathrm{\infty })\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$ such that $f^{\prime }\in L([a,b])$, where $a,b\in I^{\circ }$ with $a<b$. If the mapping $|f^{\prime }|$ is a quasi-convex function on $[a,b]$, the following inequality holds:

$$\begin{array}{c}|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)dx|\hfill \\ \le \frac{b-a}{8}[max\left\{\right|f^{\prime }(a)|,|f^{\prime }\left(\frac{a+b}{2}\right)\left|\right\}+max\left\{\right|f^{\prime }(b)|,|f^{\prime }\left(\frac{a+b}{2}\right)\left|\right\}].\hfill \end{array}$$

(1.8)

Theorem 7 [2]

Let $f:I\subseteq [0,\mathrm{\infty })\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$ such that $f^{\prime }\in L([a,b])$, where $a,b\in I^{\circ }$ with $a<b$. If $|f^{\prime }{|}^{\frac{p}{p-1}}$ is a quasi-convex function on $[a,b]$, for $p>1$, the following inequality holds:

$$\begin{array}{c}|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)dx|\hfill \\ \le \frac{b-a}{4{(p+1)}^{\frac{1}{p}}}[{(max\left\{\right|f^{\prime }(a){|}^{\frac{p}{p-1}},\left|f^{\prime }\left(\frac{a+b}{2}\right){|}^{\frac{p}{p-1}}\right\})}^{\frac{p-1}{p}}\hfill \\ +{(max\left\{\right|f^{\prime }(b){|}^{\frac{p}{p-1}},\left|f^{\prime }\left(\frac{a+b}{2}\right){|}^{\frac{p}{p-1}}\right\})}^{\frac{p-1}{p}}].\hfill \end{array}$$

(1.9)

Theorem 8 [2]

Let $f:I\subseteq [0,\mathrm{\infty })\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$ such that $f^{\prime }\in L([a,b])$, where $a,b\in I^{\circ }$ with $a<b$. If $|f^{\prime }{|}^q$ is a quasi-convex function on $[a,b]$, for $q\ge 1$, the following inequality holds:

$$\begin{array}{c}|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)dx|\hfill \\ \le \frac{b-a}{8}[{(max\left\{\right|f^{\prime }(a){|}^q,\left|f^{\prime }\left(\frac{a+b}{2}\right){|}^q\right\})}^{\frac{1}{q}}\hfill \\ +{(max\left\{\right|f^{\prime }(b){|}^q,\left|f^{\prime }\left(\frac{a+b}{2}\right){|}^q\right\})}^{\frac{1}{q}}].\hfill \end{array}$$

(1.10)

In [5], Hwang established the following results for convex and quasi-convex functions; those results provide a weighted generalization of the results given in Theorem 1, Theorem 3, Theorem 6 and Theorem 8.

Theorem 9 [5]

Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, where $a,b\in I^{\circ }$ with $a<b$, and let $g:[a,b]\to [0,\mathrm{\infty })$ be a continuous positive mapping and symmetric to $\frac{a+b}{2}$. If $|f^{\prime }|$ is a convex function on $[a,b]$, the following inequality holds:

$$\begin{array}{c}|\left[\frac{f(a)+f(b)}{2}\right]{\int }_a^{b}g(x)dx-{\int }_a^{b}f(x)g(x)dx|\hfill \\ \le \frac{b-a}{4}\left[\right|f^{\prime }(a)|+|f^{\prime }(b)\left|\right]{\int }_0^1{\int }_{L(a,b,t)}^{U(a,b,t)}g(x)dxdt,\hfill \end{array}$$

(1.11)

where $U(a,b,t)=\frac{1-t}{2}a+\frac{1+t}{2}b$ and $L(a,b,t)=\frac{1+t}{2}a+\frac{1-t}{2}b$.

Theorem 10 [5]

Suppose that the assumptions of Theorem  9 are satisfied and $q\ge 1$. If $|f^{\prime }{|}^q$ is a convex function on $[a,b]$, the following inequality holds:

$$\begin{array}{c}|\left[\frac{f(a)+f(b)}{2}\right]{\int }_a^{b}g(x)dx-{\int }_a^{b}f(x)g(x)dx|\hfill \\ \le \frac{b-a}{2}{\left[\frac{|f^{\prime }(a){|}^q+|f^{\prime }(b){|}^q}{2}\right]}^{\frac{1}{q}}{\int }_0^1{\int }_{L(a,b,t)}^{U(a,b,t)}g(x)dxdt,\hfill \end{array}$$

(1.12)

where $U(a,b,t)$ and $L(a,b,t)$ are as defined in Theorem  9.

Theorem 11 [5]

Suppose that the assumptions of Theorem  9 are satisfied. If $|f^{\prime }|$ is a quasi-convex function on $[a,b]$, the following inequality holds:

$$\begin{array}{c}|\left[\frac{f(a)+f(b)}{2}\right]{\int }_a^{b}g(x)dx-{\int }_a^{b}f(x)g(x)dx|\hfill \\ \le \frac{b-a}{4}[max\left\{\right|f^{\prime }(a)|,|f^{\prime }\left(\frac{a+b}{2}\right)\left|\right\}+max\left\{\right|f^{\prime }(b)|,|f^{\prime }\left(\frac{a+b}{2}\right)\left|\right\}]\hfill \\ \times {\int }_0^1{\int }_{L(a,b,t)}^{U(a,b,t)}g(x)dxdt,\hfill \end{array}$$

(1.13)

where $U(a,b,t)$ and $L(a,b,t)$ are as defined in Theorem  9.

Theorem 12 [5]

Suppose that the assumptions of Theorem  9 are satisfied and $q\ge 1$. If $|f^{\prime }{|}^q$ is a quasi-convex function on $[a,b]$, the following inequality holds:

$$\begin{array}{c}|\left[\frac{f(a)+f(b)}{2}\right]{\int }_a^{b}g(x)dx-{\int }_a^{b}f(x)g(x)dx|\hfill \\ \le \frac{b-a}{4}[{(max\left\{\right|f^{\prime }(a){|}^q,\left|f^{\prime }\left(\frac{a+b}{2}\right){|}^q\right\})}^{\frac{1}{q}}\hfill \\ +{(max\left\{\right|f^{\prime }(b){|}^q,\left|f^{\prime }\left(\frac{a+b}{2}\right){|}^q\right\})}^{\frac{1}{q}}]{\int }_0^1{\int }_{L(a,b,t)}^{U(a,b,t)}g(x)dxdt,\hfill \end{array}$$

(1.14)

where $U(a,b,t)$ and $L(a,b,t)$ are as defined in Theorem  9.

In recent years, a lot of efforts have been made by many mathematicians to generalize the classical convexity. These studies include, among others, the work of Hanson [19], Ben-Israel and Mond [20], Pini [21], Noor [22, 23], Yang and Li [24] and Weir and Mond [25]. Ben-Israel and Mond [20], Weir and Mond [25] and Noor [22, 23] have studied the basic properties of the preinvex functions and their role in optimization, variational inequalities and equilibrium problems. Hanson [19] introduced invex functions as a significant generalization of the convex functions. Ben-Israel and Mond [20] gave the concept of preinvex functions which is a special case of invexity. Pini [21] introduced the concept of prequasiinvex functions as a generalization of invex functions.

Let us recall some known results concerning preinvexity and prequasiinvexity.

Let K be a subset in ${\mathbb{R}}^n$ and let $f:K\to \mathbb{R}$ and $\eta :K\times K\to {\mathbb{R}}^n$ be continuous functions. Let $x\in K$, then the set K is said to be invex at x with respect to $\eta (\cdot ,\cdot )$ if

$$x+t\eta (y,x)\in K,\mathrm{\forall }x,y\in K,t\in [0,1].$$

K is said to be an invex set with respect to η if K is invex at each $x\in K$. The invex set K is also called an η-connected set.

Definition 1 [25]

The function f on the invex set K is said to be preinvex with respect to η if

$$f(u+t\eta (v,u))\le (1-t)f(u)+tf(v),\mathrm{\forall }u,v\in K,t\in [0,1].$$

The function f is said to be preconcave if and only if −f is preinvex.

It is to be noted that every convex function is preinvex with respect to the map $\eta (x,y)=x-y$, but the converse is not true; see, for instance, [18].

Definition 2 [26]

The function f on the invex set K is said to be prequasiinvex with respect to η if

$$f(u+t\eta (v,u))\le max\{f(u),f(v)\},\mathrm{\forall }u,v\in K,t\in [0,1].$$

Also every quasi-convex function is prequasiinvex with respect to the map $\eta (v,u)=v-u$, but the converse does not hold; see, for example, [27].

In the recent paper, Noor [28] obtained the following Hermite-Hadamard inequalities for the preinvex functions.

Theorem 13 [28]

Let $f:[a,a+\eta (b,a)]\to (0,\mathrm{\infty })$ be a preinvex function on the interval of the real numbers $K^{\circ }$ (the interior of K) and $a,b\in K^{\circ }$ with $\eta (b,a)>0$. Then the following inequalities hold:

$$f\left(\frac{2a+\eta (b,a)}{2}\right)\le \frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx\le \frac{f(a)+f(b)}{2}.$$

(1.15)

Barani et al. in [29] presented the following estimates of the right-hand side of a Hermite-Hadamard-type inequality in which some preinvex functions are involved.

Theorem 14 [29]

Let $K\subseteq \mathbb{R}$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$. Suppose that $f:K\to \mathbb{R}$ is a differentiable function. If $|f^{\prime }|$ is preinvex on K, for every $a,b\in K$ with $\eta (b,a)\ne 0$, the following inequality holds:

$$|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\le \frac{|\eta (b,a)|}{8}\left(\right|f^{\prime }(a)|+|f^{\prime }(b)\left|\right).$$

(1.16)

Theorem 15 [29]

Let $K\subseteq \mathbb{R}$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$. Suppose that $f:K\to \mathbb{R}$ is a differentiable function. Assume $p\in \mathbb{R}$ with $p>1$. If $|f^{\prime }{|}^{\frac{p}{p-1}}$ is preinvex on K, for every $a,b\in K$ with $\eta (b,a)\ne 0$, the following inequality holds:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{|\eta (b,a)|}{2{(1+p)}^{\frac{1}{p}}}{\left[\frac{|f^{\prime }(a){|}^{\frac{p}{p-1}}+|f^{\prime }(b){|}^{\frac{p}{p-1}}}{2}\right]}^{\frac{p-1}{p}}.\hfill \end{array}$$

(1.17)

In [30], Barani et al. gave similar results for prequasiinvex functions as follows.

Theorem 16 [30]

Let $K\subseteq \mathbb{R}$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$. Suppose that $f:K\to \mathbb{R}$ is a differentiable function. If $|f^{\prime }|$ is prequasiinvex on K, for every $a,b\in K$ with $\eta (b,a)\ne 0$, the following inequality holds:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{|\eta (b,a)|}{8}max\left\{\right|f^{\prime }(a)|,|f^{\prime }(b)\left|\right\}.\hfill \end{array}$$

(1.18)

Theorem 17 [30]

Let $K\subseteq \mathbb{R}$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$. Suppose that $f:K\to \mathbb{R}$ is a differentiable function. Assume $p\in \mathbb{R}$ with $p>1$. If $|f^{\prime }{|}^{\frac{p}{p-1}}$ is prequasiinvex on K, for every $a,b\in K$ with $\eta (b,a)\ne 0$, the following inequality holds:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{|\eta (b,a)|}{2{(1+p)}^{\frac{1}{p}}}{(max\left\{\right|f^{\prime }(a){|}^{\frac{p}{p-1}},|f^{\prime }(b){|}^{\frac{p}{p-1}}\})}^{\frac{p-1}{p}}.\hfill \end{array}$$

(1.19)

Latif [31] proved the following results which give a refinement of the results given in Theorems 14-17.

Theorem 18 [31]

Let $K\subseteq [0,\mathrm{\infty })$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$. Suppose that $f:K\to \mathbb{R}$ is a differentiable mapping on K such that $f^{\prime }\in L([a,a+\eta (b,a)])$. If $|f^{\prime }|$ is prequasiinvex on K, then for every $a,b\in K$ with $\eta (b,a)>0$, we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{\eta (b,a)}{8}[max\left\{\right|f^{\prime }(a)|,|f^{\prime }(a+\frac{1}{2}\eta (b,a))\left|\right\}\hfill \\ +max\left\{\right|f^{\prime }(a+\frac{1}{2}\eta (b,a))|,|f^{\prime }(a+\eta (b,a))\left|\right\}].\hfill \end{array}$$

(1.20)

Theorem 19 [31]

Let $K\subseteq [0,\mathrm{\infty })$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$. Suppose that $f:K\to \mathbb{R}$ is a differentiable mapping on K such that $f^{\prime }\in L([a,a+\eta (b,a)])$. If $|f^{\prime }{|}^p$ is prequasiinvex on K for some $p>1$, then for every $a,b\in K$ with $\eta (b,a)>0$, we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4{(p+1)}^{\frac{1}{p}}}[{(max\left\{\right|f^{\prime }(a){|}^{\frac{p}{p-1}},\left|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^{\frac{p}{p-1}}\right\})}^{\frac{p-1}{p}}\hfill \\ +{(max\left\{\right|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^{\frac{p}{p-1}},\left|f^{\prime }(a+\eta (b,a)){|}^{\frac{p}{p-1}}\right\})}^{\frac{p-1}{p}}].\hfill \end{array}$$

(1.21)

Theorem 20 [31]

Let $K\subseteq \mathbb{R}$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$. Suppose that $f:K\to \mathbb{R}$ is a differentiable mapping on K such that $f^{\prime }\in L([a,a+\eta (b,a)])$. If $|f^{\prime }{|}^q$ for $q\ge 1$ is prequasiinvex on K, then for every $a,b\in K$ with $\eta (b,a)>0$, we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{\eta (b,a)}{8}[{(max\left\{\right|f^{\prime }(a){|}^q,\left|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^q\right\})}^{\frac{1}{q}}\hfill \\ +{(max\left\{\right|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^q,\left|f^{\prime }(a+\eta (b,a)){|}^q\right\})}^{\frac{1}{q}}].\hfill \end{array}$$

(1.22)

For several new results on inequalities for preinvex and prequasiinvex functions, we refer the interested reader to [26, 29, 32] and the references therein.

In the present paper we give new inequalities of Hermite-Hadamard for functions whose derivatives in absolute value are preinvex and prequasiinvex. Our results extend those results presented in very recent results from [2, 3, 5, 6] and [12] and generalize those results from [29, 30] and [33].

2 Main results

The following lemma is essential in establishing our main results in this section.

Lemma 1 Let $K\subseteq \mathbb{R}$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$ and $a,b\in K$ with $\eta (b,a)>0$. Suppose that $f:K\to \mathbb{R}$ is a differentiable mapping on K such that $f^{\prime }\in L([a,a+\eta (b,a)])$. If $h:[a,a+\eta (b,a)]\to [0,\mathrm{\infty })$ is a differentiable mapping, then the following equality holds:

$$\begin{array}{c}\frac{1}{2}[(h(a+\eta (b,a))-2h(a))f(a)+h(a+\eta (b,a))f(a+\eta (b,a))]-{\int }_a^{a+\eta (b,a)}f(x)h^{\prime }(x)dx\hfill \\ =\frac{\eta (b,a)}{4}\{{\int }_0^1[2h(a+\left(\frac{1-t}{2}\right)\eta (b,a))-h(a+\eta (b,a))]f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a))dt\hfill \\ +{\int }_0^1[2h(a+\left(\frac{1+t}{2}\right)\eta (b,a))-h(a+\eta (b,a))]\hfill \\ \times f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a))dt\}.\hfill \end{array}$$

(2.1)

Proof It suffices to note that

$$\begin{array}{rcl}{I}_1& =& {\int }_0^1[2h(a+\left(\frac{1-t}{2}\right)\eta (b,a))-h(a+\eta (b,a))]f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a))dt\\ =& -2\frac{[2h(a+(\frac{1-t}{2})\eta (b,a))-h(a+\eta (b,a))]f(a+(\frac{1-t}{2})\eta (b,a))}{\eta (b,a)}{|}_0^1\\ -2{\int }_0^1{h}^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a))f(a+\left(\frac{1-t}{2}\right)\eta (b,a))dt\\ =& \frac{-2[2h(a)-h(a+\eta (b,a))]f(a)}{\eta (b,a)}\\ +\frac{2[2h(a+\frac{1}{2}\eta (b,a))-h(a+\eta (b,a))]f(a+\frac{1}{2}\eta (b,a))}{\eta (b,a)}\\ -2{\int }_0^1{h}^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a))f(a+\left(\frac{1-t}{2}\right)\eta (b,a))dt.\end{array}$$

Setting $x=a+(\frac{1-t}{2})\eta (b,a)$ and $dx=-\frac{\eta (b,a)}{2}dt$, which gives

$$\begin{array}{rcl}{I}_1& =& \frac{2[h(a+\eta (b,a))-2h(a)]f(a)}{\eta (b,a)}-\frac{4}{\eta (b,a)}{\int }_a^{a+\frac{1}{2}\eta (b,a)}{h}^{\prime }(x)f(x)dx\\ +\frac{2[2h(a+\frac{1}{2}\eta (b,a))-h(a+\eta (b,a))]f(a+\frac{1}{2}\eta (b,a))}{\eta (b,a)}.\end{array}$$

(2.2)

Similarly, we also have

$$\begin{array}{rcl}{I}_2& =& {\int }_0^1[2h(a+\left(\frac{1+t}{2}\right)\eta (b,a))-h(a+\eta (b,a))]f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a))dt\\ =& \frac{2h(a+\eta (b,a))f(a+\eta (b,a))}{\eta (b,a)}-\frac{4}{\eta (b,a)}{\int }_{a+\frac{1}{2}\eta (b,a)}^{a+\eta (b,a)}{h}^{\prime }(x)f(x)dx\\ -\frac{2[2h(a+\frac{1}{2}\eta (b,a))-h(a+\eta (b,a))]f(a+\frac{1}{2}\eta (b,a))}{\eta (b,a)}.\end{array}$$

(2.3)

Thus, from (2.2) and (2.3), we have

$$\begin{array}{rcl}\frac{\eta (b,a)}{4}[I_1+I_2]& =& \frac{1}{2}[(h(a+\eta (b,a))-2h(a))f(a)+h(a+\eta (b,a))f(a+\eta (b,a))]\\ -{\int }_a^{a+\eta (b,a)}f(x)h^{\prime }(x)dx,\end{array}$$

which is the required result. □

Remark 1 If we take $\eta (b,a)=b-a$, then Lemma 1 reduces to Lemma 2.1 from [5].

Now using Lemma 1, we shall propose some new upper bounds for the difference between the rightmost and middle terms of a weighted version of the Hadamard inequality (1.15) using preinvex and prequasiinvex mappings. Our results provide a weighted generalization of those results given in [29, 30] and [31].

In what follows we use the notations $L^{\prime }(a,b,t)=a+(\frac{1-t}{2})\eta (b,a)$ and $U^{\prime }(a,b,t)=a+(\frac{1+t}{2})\eta (b,a)$.

Theorem 21 Let $K\subseteq \mathbb{R}$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$ and $a,b\in K$ with $\eta (b,a)>0$. Suppose that $f:K\to \mathbb{R}$ is a differentiable mapping on K and $w:[a,a+\eta (b,a)]\to [0,\mathrm{\infty })$ is continuous and symmetric to $a+\frac{1}{2}\eta (b,a)$. If $|f^{\prime }|$ is preinvex on K, we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}\left[\right|f^{\prime }(a)|+|f^{\prime }(b)\left|\right]{\int }_0^1{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dxdt.\hfill \end{array}$$

(2.4)

Proof Let $h(t)={\int }_a^{t}w(t)dt$ for all $t\in [a,a+\eta (b,a)]$ in Lemma 1, we obtain

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(t)dt-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}\left\{{\int }_0^1\right|2h(a+\left(\frac{1-t}{2}\right)\eta (b,a))-h(a+\eta (b,a))|\hfill \\ \times \left|f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a))\right|dt\hfill \\ +{\int }_0^1|2h(a+\left(\frac{1+t}{2}\right)\eta (b,a))-h(a+\eta (b,a))|\hfill \\ \times \left|f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a))\right|dt\}.\hfill \end{array}$$

(2.5)

Since $w(x)$ is symmetric to $a+\frac{1}{2}\eta (b,a)$, we have

$$|2h(a+\left(\frac{1-t}{2}\right)\eta (b,a))-h(a+\eta (b,a))|={\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx$$

(2.6)

and

$$|2h(a+\left(\frac{1+t}{2}\right)\eta (b,a))-h(a+\eta (b,a))|={\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx$$

(2.7)

for all $t\in [0,1]$. Using (2.6) and (2.7) in (2.5), we have

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(t)dt-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}{\int }_0^1({\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx)\left[\right|f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a))|\hfill \\ +\left|f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a))\right|]dt.\hfill \end{array}$$

(2.8)

Since $|f^{\prime }|$ is preinvex on K, hence for every $a,b\in K$ with $\eta (b,a)>0$, we have

$$\begin{array}{c}\left|f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a))\right|+\left|f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a))\right|\hfill \\ \le \left(\frac{1+t}{2}\right)|f^{\prime }(a)|+\left(\frac{1-t}{2}\right)|f^{\prime }(b)|+\left(\frac{1-t}{2}\right)|f^{\prime }(a)|+\left(\frac{1+t}{2}\right)|f^{\prime }(b)|\hfill \\ =|f^{\prime }(a)|+|f^{\prime }(b)|.\hfill \end{array}$$

(2.9)

Using (2.9) in (2.8), we get the required inequality. This completes the proof of the theorem. □

Remark 2 In Theorem 21, if we take $w(x)=\frac{1}{\eta (b,a)}$ for all $x\in [a,a+\eta (b,a)]$, then (2.4) becomes inequality (1.16).

Remark 3 If $\eta (b,a)=b-a$ in Theorem 21, then (2.4) reduces to inequality (1.11) from [5].

Theorem 22 Let $K\subseteq \mathbb{R}$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$ and $a,b\in K$ with $\eta (b,a)>0$. Suppose that $f:K\to \mathbb{R}$ is a differentiable mapping on K and $w:[a,a+\eta (b,a)]\to [0,\mathrm{\infty })$ is continuous and symmetric to $a+\frac{1}{2}\eta (b,a)$. If $|f^{\prime }{|}^q$ is preinvex on K for $q>1$, we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{2}{\left[\frac{|f^{\prime }(a){|}^q+|f^{\prime }(b){|}^q}{2}\right]}^{\frac{1}{q}}{\left({\int }_0^1{[{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx]}^{p}dt\right)}^{\frac{1}{p}},\hfill \end{array}$$

(2.10)

where $\frac{1}{p}+\frac{1}{q}=1$.

Proof Continuing from inequality (2.8) in the proof of Theorem 21 and using the well-known Hölder integral inequality, we have

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(t)dt-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}{\left({\int }_0^1{[{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx]}^{p}dt\right)}^{\frac{1}{p}}\hfill \\ \times [{\left({\int }_0^1\right|f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a)){|}^{q}dt)}^{\frac{1}{q}}\hfill \\ +{\left({\int }_0^1\right|f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a)){|}^{q}dt)}^{\frac{1}{q}}].\hfill \end{array}$$

(2.11)

By the power-mean inequality $t^r+s^r<2^{1-r}{(t+s)}^r$ for $t>0$, $s>0$ and $r<1$, and by the preinvexity of $|f^{\prime }{|}^q$ on K for $q>1$, we have, for every $a,b\in K$ with $\eta (b,a)>0$, the following inequality:

$$\begin{array}{c}{\left({\int }_0^1\right|f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a)){|}^{q}dt)}^{\frac{1}{q}}+{\left({\int }_0^1\right|f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a)){|}^{q}dt)}^{\frac{1}{q}}\hfill \\ \le 2^{1-\frac{1}{q}}{\left[{\int }_0^1\right|f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a)){|}^{q}dt+{\int }_0^1\left|f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a)){|}^{q}dt\right]}^{\frac{1}{q}}\hfill \\ \le 2^{1-\frac{1}{q}}\left[{\int }_0^1\right\{\left(\frac{1+t}{2}\right)|f^{\prime }(a){|}^q+\left(\frac{1-t}{2}\right)|f^{\prime }(b){|}^q\hfill \\ {+\left(\frac{1-t}{2}\right)|f^{\prime }(a){|}^q+\left(\frac{1+t}{2}\right)|f^{\prime }(b){|}^q\}dt]}^{\frac{1}{q}}\hfill \\ =2^{1-\frac{1}{q}}{\left[\right|f^{\prime }(a){|}^q+|f^{\prime }(b){|}^q]}^{\frac{1}{q}}.\hfill \end{array}$$

(2.12)

Using the last inequality (2.12) in (2.11), we get the desired inequality. This completes the proof of the theorem as well. □

Remark 4 In Theorem 22 if we take $w(x)=\frac{1}{\eta (b,a)}$ for all $x\in [a,a+\eta (b,a)]$ with $\eta (b,a)>0$, then (2.10) reduces to inequality (1.17).

Remark 5 If we take $\eta (b,a)=b-a$ in Theorem 22, then (2.10) reduces to the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(b)}{2}{\int }_a^{b}w(x)dx-{\int }_a^{b}f(x)w(x)dx|\hfill \\ \le \frac{b-a}{2}{\left[\frac{|f^{\prime }(a){|}^q+|f^{\prime }(b){|}^q}{2}\right]}^{\frac{1}{q}}{\left({\int }_0^1{[{\int }_{L(a,b,t)}^{U(a,b,t)}w(x)dx]}^{p}dt\right)}^{\frac{1}{p}},\hfill \end{array}$$

(2.13)

where $\frac{1}{p}+\frac{1}{q}=1$, $L(a,b,t)=(\frac{1+t}{2})a+(\frac{1-t}{2})b$, $U(a,b,t)=(\frac{1-t}{2})a+(\frac{1+t}{2})b$, $t\in [a,b]$.

A similar result may be stated as follows.

Theorem 23 Let $K\subseteq \mathbb{R}$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$. Suppose that $f:K\to \mathbb{R}$ is a differentiable mapping on K and $w:[a,a+\eta (b,a)]\to [0,\mathrm{\infty })$ is continuous and symmetric to $a+\frac{1}{2}\eta (b,a)$. If $|f^{\prime }{|}^q$ is preinvex on K for $q\ge 1$, then for every $a,b\in K$ with $\eta (b,a)>0$, we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{2}{\left[\frac{|f^{\prime }(a){|}^q+|f^{\prime }(b){|}^q}{2}\right]}^{\frac{1}{q}}{\int }_0^1{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dxdt.\hfill \end{array}$$

(2.14)

Proof Continuing from inequality (2.8) in the proof of Theorem 21 and using the well-known Hölder integral inequality, we have

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(t)dt-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}{\left[{\int }_0^1({\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx)dt\right]}^{1-\frac{1}{q}}\hfill \\ \times [{\left\{\left({\int }_0^1({\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx)\right|f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a)){|}^{q}dt)\right\}}^{\frac{1}{q}}\hfill \\ +{\left\{\left({\int }_0^1({\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx)\right|f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a)){|}^{q}dt)\right\}}^{\frac{1}{q}}].\hfill \end{array}$$

(2.15)

By the power-mean inequality $t^r+s^r<2^{1-r}{(t+s)}^r$ for $t>0$, $s>0$ and $r<1$, and by the preinvexity of $|f^{\prime }{|}^q$ on K for $q>1$, we have, for every $a,b\in K$ with $\eta (b,a)>0$, the following inequality:

$$\begin{array}{c}{\left\{\left({\int }_0^1({\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx)\right|f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a)){|}^{q}dt)\right\}}^{\frac{1}{q}}\hfill \\ +{\left\{\left({\int }_0^1({\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx)\right|f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a)){|}^{q}dt)\right\}}^{\frac{1}{q}}\hfill \\ \le 2^{1-\frac{1}{q}}{\left[{\int }_0^1({\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx)dt\right]}^{\frac{1}{q}}{\left[\right|f^{\prime }(a){|}^q+|f^{\prime }(b){|}^q]}^{\frac{1}{q}}.\hfill \end{array}$$

(2.16)

Utilizing inequality (2.16) in (2.15), we get inequality (2.14). This completes the proof of the theorem. □

Corollary 1 Suppose that all the assumptions of Theorem  23 are satisfied and if $w(x)=\frac{1}{\eta (b,a)}$ for all $x\in [a,a+\eta (b,a)]$ with $\eta (b,a)>0$, then we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}{\left[\frac{|f^{\prime }(a){|}^q+|f^{\prime }(b){|}^q}{2}\right]}^{\frac{1}{q}}.\hfill \end{array}$$

(2.17)

Remark 6 If we take $\eta (b,a)=b-a$ in Theorem 23, then the inequality reduces to inequality (1.12) from [5].

Remark 7 For $q=1$, (2.17) reduces to the inequality proved in Theorem 14. If $q=\frac{p}{p-1}$ ($p>1$), we have $2^p>p+1$ for $p>1$ and, accordingly,

$$\frac{1}{4}<\frac{1}{2{(p+1)}^{\frac{1}{p}}}.$$

This reveals that inequality (2.17) is better than the one given by (1.17) in Theorem 15 from [29].

Now we give our results for prequasiinvex functions.

Theorem 24 Let $K\subseteq [0,\mathrm{\infty })$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$. Suppose that $f:K\to \mathbb{R}$ is a differentiable mapping on K and $w:[a,a+\eta (b,a)]\to [0,\mathrm{\infty })$ is continuous and symmetric to $a+\frac{1}{2}\eta (b,a)$. If $|f^{\prime }|$ is prequasiinvex on K, then for every $a,b\in K$ with $\eta (b,a)>0$, we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}[max\left\{\right|f^{\prime }(a)|,|f^{\prime }(a+\frac{1}{2}\eta (b,a))\left|\right\}\hfill \\ +max\left\{\right|f^{\prime }(a+\frac{1}{2}\eta (b,a))|,|f^{\prime }(a+\eta (b,a))\left|\right\}]{\int }_0^1{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dxdt.\hfill \end{array}$$

(2.18)

Proof We continue inequality (2.8) in the proof of Theorem 21. Since $|f^{\prime }|$ is prequasiinvex on K, hence for every $t\in [0,1]$, we obtain

$$\left|f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a))\right|\le max\left\{\right|f^{\prime }(a)|,|f^{\prime }(a+\frac{1}{2}\eta (b,a))\left|\right\}$$

(2.19)

and

$$\left|f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a))\right|\le max\left\{\right|f^{\prime }(a+\frac{1}{2}\eta (b,a))|,|f^{\prime }(a+\eta (b,a))\left|\right\}.$$

(2.20)

A combination of (2.8), (2.19) and (2.20) gives the required inequality (2.18). □

Corollary 2 Suppose that all the conditions of Theorem  24 are satisfied. Moreover,

• (1) if $|f^{\prime }|$ is non-decreasing, then the following inequality holds:

  $$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}\left[\right|f^{\prime }(a+\frac{1}{2}\eta (b,a))|+|f^{\prime }(a+\eta (b,a))\left|\right]\hfill \\ \times {\int }_0^1{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dxdt;\hfill \end{array}$$

  (2.21)

• (2) if $|f^{\prime }|$ is non-increasing, then the following inequality holds:

  $$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}\left[\right|f^{\prime }(a)|+|f^{\prime }(a+\frac{1}{2}\eta (b,a))\left|\right]{\int }_0^1{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dxdt.\hfill \end{array}$$

  (2.22)

Remark 8 [31]

If in Theorem 24 we take $w(x)=\frac{1}{\eta (b,a)}$ for all $x\in [a,a+\eta (b,a)]$ with $\eta (b,a)>0$, then we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{\eta (b,a)}{8}[max\left\{\right|f^{\prime }(a)|,|f^{\prime }(a+\frac{1}{2}\eta (b,a))\left|\right\}\hfill \\ +max\left\{\right|f^{\prime }(a+\frac{1}{2}\eta (b,a))|,|f^{\prime }(a+\eta (b,a))\left|\right\}].\hfill \end{array}$$

(2.23)

Inequality (2.23) represents a new refinement of inequality (1.16) for prequasiinvex functions and hence for preinvex functions. Moreover,

• (1) if $|f^{\prime }|$ is non-decreasing, then the following inequality holds:

  $$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{\eta (b,a)}{8}\left[\right|f^{\prime }(a+\frac{1}{2}\eta (b,a))|+|f^{\prime }(a+\eta (b,a))\left|\right];\hfill \end{array}$$

  (2.24)

• (2) if $|f^{\prime }|$ is non-increasing, then the following inequality holds:

  $$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{\eta (b,a)}{8}\left[\right|f^{\prime }(a)|+|f^{\prime }(a+\frac{1}{2}\eta (b,a))\left|\right].\hfill \end{array}$$

  (2.25)

Remark 9 If $\eta (b,a)=b-a$ in Theorem 24, then (2.18) reduces to inequality (1.13) established in Theorem 11 from [5], and inequalities (2.24) and (2.25) recapture the related inequalities given in the corollary of Theorem 11.

Remark 10 If $\eta (b,a)=b-a$ in Remark 8, then (2.23) becomes inequality (1.8) of Theorem 6 from [2], and inequalities (2.24) and (2.25) recapture the related inequalities of the corollary of Theorem 6.

Theorem 25 Let $K\subseteq [0,\mathrm{\infty })$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$. Suppose that $f:K\to \mathbb{R}$ is a differentiable mapping on K and $w:[a,a+\eta (b,a)]\to [0,\mathrm{\infty })$ is continuous and symmetric to $a+\frac{1}{2}\eta (b,a)$. If $|f^{\prime }{|}^q$ is prequasiinvex on K for $q>1$, then for every $a,b\in K$ with $\eta (b,a)>0$, we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}{\left({\int }_0^1{[{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx]}^{p}dt\right)}^{\frac{1}{p}}\hfill \\ \times [{(max\left\{\right|f^{\prime }(a){|}^q,\left|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^q\right\})}^{\frac{1}{q}}\hfill \\ +{(max\left\{\right|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^q,\left|f^{\prime }(a+\eta (b,a)){|}^q\right\})}^{\frac{1}{q}}],\hfill \end{array}$$

(2.26)

where $\frac{1}{p}+\frac{1}{q}=1$.

Proof We continue inequality (2.11) in the proof of Theorem 22. By the prequasiinvexity of $|f^{\prime }{|}^q$ on K for $q>1$, we have, for every $t\in [0,1]$,

$$|f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a)){|}^q\le max\left\{\right|f^{\prime }(a){|}^q,\left|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^q\right\}$$

(2.27)

and

$$\begin{array}{c}|f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a)){|}^q\hfill \\ \le max\left\{\right|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^q,\left|f^{\prime }(a+\eta (b,a)){|}^q\right\}.\hfill \end{array}$$

(2.28)

A combination of (2.11), (2.27) and (2.28) gives us the required inequality (2.26). This completes the proof of the theorem. □

Corollary 3 Suppose that all the conditions of Theorem  25 are satisfied. Moreover,

• (1) if $|f^{\prime }{|}^q$ is non-decreasing for $q>1$, then the following inequality holds:

  $$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}\left[\right|f^{\prime }(a+\frac{1}{2}\eta (b,a))|+|f^{\prime }(a+\eta (b,a))\left|\right]\hfill \\ \times {\left({\int }_0^1{[{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx]}^{p}dt\right)}^{\frac{1}{p}};\hfill \end{array}$$

  (2.29)

• (2) if $|f^{\prime }{|}^q$ is non-increasing for $q>1$, then the following inequality holds:

  $$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}\left[\right|f^{\prime }(a)|+|f^{\prime }(a+\frac{1}{2}\eta (b,a))\left|\right]\hfill \\ \times {\left({\int }_0^1{[{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx]}^{p}dt\right)}^{\frac{1}{p}},\hfill \end{array}$$

  (2.30)

where $\frac{1}{p}+\frac{1}{q}=1$.

Remark 11 [31]

If in Theorem 25 we take $w(x)=\frac{1}{\eta (b,a)}$ for all $x\in [a,a+\eta (b,a)]$ with $\eta (b,a)>0$, then we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4{(p+1)}^{\frac{1}{p}}}[max\left\{\right|f^{\prime }(a)|,|f^{\prime }(a+\frac{1}{2}\eta (b,a))\left|\right\}\hfill \\ +max\left\{\right|f^{\prime }(a+\frac{1}{2}\eta (b,a))|,|f^{\prime }(a+\eta (b,a))\left|\right\}].\hfill \end{array}$$

(2.31)

Inequality (2.31) represents a new refinement of inequality (1.19) for prequasiinvex functions and hence for preinvex functions. Moreover,

• (1) if $|f^{\prime }|$ is non-decreasing, then the following inequality holds:

  $$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4{(p+1)}^{\frac{1}{p}}}\left[\right|f^{\prime }(a+\frac{1}{2}\eta (b,a))|+|f^{\prime }(a+\eta (b,a))\left|\right];\hfill \end{array}$$

  (2.32)

• (2) if $|f^{\prime }|$ is non-increasing, then the following inequality holds:

  $$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}-\frac{1}{\eta (b,a)}{\int }_a^{a+\eta (b,a)}f(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4{(p+1)}^{\frac{1}{p}}}\left[\right|f^{\prime }(a)|+|f^{\prime }(a+\frac{1}{2}\eta (b,a))\left|\right],\hfill \end{array}$$

  (2.33)

where $\frac{1}{p}+\frac{1}{q}=1$.

Remark 12 If we take $\eta (b,a)=b-a$ in Remark 11, then (2.31) becomes inequality (1.9) of Theorem 7 from [2], and inequalities (2.32) and (2.33) become the related inequalities given in the corollary of Theorem 7.

Theorem 26 Let $K\subseteq [0,\mathrm{\infty })$ be an open invex subset with respect to $\eta :K\times K\to \mathbb{R}$. Suppose that $f:K\to \mathbb{R}$ is a differentiable mapping on K and $w:[a,a+\eta (b,a)]\to [0,\mathrm{\infty })$ is continuous and symmetric to $a+\frac{1}{2}\eta (b,a)$. If $|f^{\prime }{|}^q$ is prequasiinvex on K for $q\ge 1$, then for every $a,b\in K$ with $\eta (b,a)>0$, we have the following inequality:

$$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}({\int }_0^1{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dxdt)[{(max\left\{\right|f^{\prime }(a){|}^q,\left|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^q\right\})}^{\frac{1}{q}}\hfill \\ +{(max\left\{\right|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^q,\left|f^{\prime }(a+\eta (b,a)){|}^q\right\})}^{\frac{1}{q}}].\hfill \end{array}$$

(2.34)

Proof We continue inequality (2.15) in the proof of Theorem 23. By the prequasiinvexity of $|f^{\prime }{|}^q$ on K for $q\ge 1$, we have, for every $t\in [0,1]$,

$$|f^{\prime }(a+\left(\frac{1-t}{2}\right)\eta (b,a)){|}^q\le max\left\{\right|f^{\prime }(a){|}^q,\left|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^q\right\}$$

(2.35)

and

$$|f^{\prime }(a+\left(\frac{1+t}{2}\right)\eta (b,a)){|}^q\le max\left\{\right|f^{\prime }(a+\frac{1}{2}\eta (b,a)){|}^q,\left|f^{\prime }(a+\eta (b,a)){|}^q\right\}.$$

(2.36)

A combination of (2.15), (2.35) and (2.36) gives us the required inequality (2.34). This completes the proof of the theorem. □

Corollary 4 Suppose that all the conditions of Theorem  26 are satisfied. Moreover,

• (1) if $|f^{\prime }{|}^q$ is non-decreasing for $q\ge 1$, then the following inequality holds:

  $$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}({\int }_0^1{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dxdt)\hfill \\ \times \left[\right|f^{\prime }(a+\frac{1}{2}\eta (b,a))|+|f^{\prime }(a+\eta (b,a))\left|\right];\hfill \end{array}$$

  (2.37)

• (2) if $|f^{\prime }{|}^q$ is non-increasing for $q\ge 1$, then the following inequality holds:

  $$\begin{array}{c}|\frac{f(a)+f(a+\eta (b,a))}{2}{\int }_a^{a+\eta (b,a)}w(x)dx-{\int }_a^{a+\eta (b,a)}f(x)w(x)dx|\hfill \\ \le \frac{\eta (b,a)}{4}({\int }_0^1{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dxdt)\left[\right|f^{\prime }(a)|+|f^{\prime }(a+\frac{1}{2}\eta (b,a))\left|\right].\hfill \end{array}$$

  (2.38)

Remark 13 [31]

If in Theorem 26 we take $w(x)=\frac{1}{\eta (b,a)}$ for all $x\in [a,a+\eta (b,a)]$ with $\eta (b,a)>0$, then we have inequality (1.22). Moreover,

• (1) if $|f^{\prime }{|}^q$ is non-decreasing, then inequality (2.24) holds,

• (2) if $|f^{\prime }{|}^q$ is non-increasing, then inequality (2.25) holds.

Remark 14 If $\eta (b,a)=b-a$ in Theorem 26, then (2.34) reduces to inequality (1.14) established in Theorem 12 from [5], and inequalities (2.37) and (2.38) recapture the related inequalities established in the corollary of Theorem 12.

Remark 15 If $\eta (b,a)=b-a$ in Remark 13, then (1.22) becomes inequality (1.10) of Theorem 8 from [2], and inequalities (2.24) and (2.25) recapture the related inequalities of the corollary of Theorem 8.

3 Applications to special means

In what follows we give certain generalizations of some notions for a positive valued function of a positive variable.

Definition 3 [34]

A function $M:{\mathbb{R}}_{+}^2\to {\mathbb{R}}_{+}$ is called a mean function if it has the following properties:

• (1) Homogeneity: $M(ax,ay)=aM(x,y)$ for all $a>0$;

• (2) Symmetry: $M(x,y)=M(y,x)$;

• (3) Reflexivity: $M(x,x)=x$;

• (4) Monotonicity: If $x\le x^{\prime }$ and $y\le y^{\prime }$, then $M(x,y)\le M(x^{\prime },y^{\prime })$;

• (5) Internality: $min\{x,y\}\le M(x,y)\le max\{x,y\}$.

We consider some means for arbitrary positive real numbers α, β (see, for instance, [34]).

• (1) The arithmetic mean:

  $$A:=A(\alpha ,\beta )=\frac{\alpha +\beta }{2}.$$

• (2) The geometric mean:

  $$G:=G(\alpha ,\beta )=\sqrt{\alpha \beta }.$$

• (3) The harmonic mean:

  $$H:=H(\alpha ,\beta )=\frac{2}{\frac{1}{\alpha }+\frac{1}{\beta }}.$$

• (4) The power mean:

  $$P_r:=P_r(\alpha ,\beta )={\left(\frac{{\alpha }^r+{\beta }^r}{2}\right)}^{\frac{1}{r}},r\ge 1.$$

• (5) The identric mean:

  $$I:=I(\alpha ,\beta )=\{\begin{array}{ll}\frac{1}{e}(\frac{{\beta }^{\beta }}{{\alpha }^{\alpha }}),& \alpha \ne \beta ,\\ \alpha ,& \alpha =\beta .\end{array}$$

• (6) The logarithmic mean:

  $$L:=L(\alpha ,\beta )=\frac{\alpha -\beta }{ln|\alpha |-ln|\beta |},|\alpha |\ne |\beta |.$$

• (7) The generalized log-mean:

  $$L_p:=L_p(\alpha ,\beta )=\left[\frac{{\beta }^{p+1}-{\alpha }^{p+1}}{(p+1)(\beta -\alpha )}\right],\alpha \ne \beta ,p\in \mathbb{R}\mathrm{\setminus }\{-1,0\}.$$

It is well known that $L_p$ is monotonic nondecreasing over $p\in \mathbb{R}$, with $L_{-1}:=L$ and $L_0:=I$. In particular, we have the inequality $H\le G\le L\le I\le A$.

Now, let a and b be positive real numbers such that $a<b$. Consider the function $M:=M(a,b):[a,a+\eta (b,a)]\times [a,a+\eta (b,a)]\to {\mathbb{R}}^{+}$, which is one of the above mentioned means, therefore one can obtain variant inequalities for these means as follows.

Setting $\eta (b,a)=M(b,a)$ in (2.4), (2.10) and (2.14), one can obtain the following interesting inequalities involving means:

$$\begin{array}{c}|\frac{f(a)+f(a+M(a,b))}{2}{\int }_a^{a+M(a,b)}w(x)dx-{\int }_a^{a+M(a,b)}f(x)w(x)dx|\hfill \\ \le \frac{M(a,b)}{4}\left[\right|f^{\prime }(a)|+|f^{\prime }(b)\left|\right]{\int }_0^1{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dxdt,\hfill \end{array}$$

(3.1)

$$\begin{array}{c}|\frac{f(a)+f(a+M(a,b))}{2}{\int }_a^{a+M(a,b)}w(x)dx-{\int }_a^{a+M(a,b)}f(x)w(x)dx|\hfill \\ \le \frac{M(a,b)}{2}{\left[\frac{|f^{\prime }(a){|}^q+|f^{\prime }(b){|}^q}{2}\right]}^{\frac{1}{q}}{\left({\int }_0^1{[{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dx]}^{p}dt\right)}^{\frac{1}{p}}\hfill \end{array}$$

(3.2)

for $q>1$, $\frac{1}{p}+\frac{1}{q}=1$ and

$$\begin{array}{c}|\frac{f(a)+f(a+M(a,b))}{2}{\int }_a^{a+M(a,b)}w(x)dx-{\int }_a^{a+M(a,b)}f(x)w(x)dx|\hfill \\ \le \frac{M(a,b)}{2}{\left[\frac{|f^{\prime }(a){|}^q+|f^{\prime }(b){|}^q}{2}\right]}^{\frac{1}{q}}{\int }_0^1{\int }_{L^{\prime }(a,b,t)}^{U^{\prime }(a,b,t)}w(x)dxdt\hfill \end{array}$$

(3.3)

for $q\ge 1$, where $U^{\prime }(a,b,t)=a+(\frac{1+t}{2})M(a,b)$, $L^{\prime }(a,b,t)=a+(\frac{1-t}{2})M(a,b)$. Letting $M=A,G,H,P_r,I,L,L_p$ in (3.1), (3.2) and (3.3), we can get the required inequalities for a different weight function $w(x)$, and the details are left to the interested reader.