Integral inequalities of Hermite-Hadamard type for GA-$F$-convex functions

1.   Motivations

In this paper, we let $I$ denote an interval on $\mathbb{R}$ and $I^\circ$ be the interior of $I$.

The following definitions are well known in the literature.

Definition 1.1. A function $f:I\subseteq\mathbb{R} = (-\infty, \infty)\to\mathbb{R}$ is said to be convex function if the inequality

holds for all $x, y\in I$ and $t\in[0, 1].$

Definition 1.2. Let $f:I\subseteq\mathbb{R}_+ = (0, \infty)\to\mathbb{R}$. If for any $x, y\in I$ and $t\in[0, 1]$, the inequality

is valid, then we call $f$ a geometric-arithmetic convex function on $I$. For simplicity, we call it a GA-convex function.

In the paper ^[11], the notion of $s$-GA-convex functions was defined.

Definition 1.3 ([11,Definition 2.1]). Let $f:I\subseteq\mathbb{R}_+\to\mathbb{R}$ and $s\in(0, 1]$. If the inequality

is valid for any $x, y\in I$ and $t\in[0, 1]$, then we call $f$ an $s$-geometric-arithmetic convex function on $I$. For shortness, we call it an $s$-GA-convex function.

Now we recall several Hermite-Hadamard type inequalities for GA-convex functions in the form of theorems.

Theorem 1.1. ([6,Theorem 1]). Let $f:[a, b]\subseteq\mathbb{R}_+\to\mathbb{R}$ be GA-convex. Then

Theorem 1.2. ([17,Theorem 3.1]). Let $f:I\subseteq\mathbb{R}_+\to\mathbb{R}$ be differentiable on $I^\circ$, $a, b\in I$ with $a < b$, and $f'\in L([a, b])$. If $|f'|^q$ is GA-convex on $[a, b]$ for $q\ge1$, then

where $A(u, v) = \frac{u+v}{2}$ and

for $u, v > 0$ are the arithmetic mean and logarithmic mean respectively.

Theorem 1.3 ([11,Theorem 3.3]). Let $f:I^\circ\subseteq\mathbb{R}\to\mathbb{R}$ be differentiable and decreasing on $I^\circ$, $a, b\in I^\circ$ with $a < b$, and $f\in L_1([a, b])$. If $|f'|^p$ is $s$-GA-convex on $[a, b]$ for $s\in(0, 1]$ and $p > 1$, then

For recent developments related to integral inequalities of the Hermite-Hadamard type for convex functions, please refer to ^{[3,4,5,7,8,9,10,12,13,14,16]} and closely related references therein.

In this paper, we will define a new notion of GA-$F$-convex functions and establish several integral inequalities of the Hermite-Hadamard type for GA-$F$-convex functions.

2.   A new notion and two lemmas

In the papers ^[1,2], as a generalization of strongly convex functions, the notion of $F$-convex functions was defined as follows.

Definition 2.1 ([2,p. 868,Definition 1]). Let $F:\mathbb{R}\to \mathbb{R}$ be a given function. A function $f:I\to\mathbb{R}$ is called to be $F$-convex if

for all $x, y\in I$ and $t\in[0, 1].$

Motivated by Definition 2.1, we now define a new notion of GA-$F$-convex functions as follows.

Definition 2.2. Let $F:\mathbb{R}\to \mathbb{R}$ be a given function. A function $f:I\subseteq\mathbb{R}_+\to\mathbb{R}$ is called GA-$F$-convex if

for all $x, y\in I$ and $t\in[0, 1].$

Remark 1. It is easy to see that, if $f:I\subseteq\mathbb{R}_+\to\mathbb{R}$ is a decreasing and GA-$F$-convex function, then $f$ is an $F$-convex function on $I$.

For establishing integral inequalities of the Hermite-Hadamard type for GA-$F$-convex functions, we need the following lemmas.

Lemma 2.1 (^[15]). Let $f:I\subseteq \mathbb{R}_+\to \mathbb{R}$ be a differentiable function on $I^\circ$ and let $a, b\in I^\circ$ with $a < b$. If $f'\in L[a, b]$, then

Lemma 2.2. Let $u, v\in\mathbb{R}_+$. Then

where $L(u, v)$ is defined by $(1.1)$.

Proof. The proof is straightforward.

3.   Some new integral inequalities of Hermite-Hadamard type

In this section, we establish several integral inequalities of the Hermite-Hadamard type for GA-$F$-convex functions.

Theorem 3.1. Let $F:\mathbb{R}\to \mathbb{R}$ be a given function. Suppose that $f:I\subseteq\mathbb{R}_+\to\mathbb{R}$ is differentiable, that $a, b \in I$ with $a < b$, and that $x\in[a, b]$. If $|f'|^q$ is GA-$F$-convex on $I$ for $q\ge1$, then

where $G_1(u, v)$, $G_2(u, v)$, and $G_3(u, v)$ are defined as in Lemma 2.2.

Proof. By Lemma 2.1 and Hölder's integral inequality, we have

Since $|f'|^q$ is a GA-$F$-convex function on $[a, b]$, using Lemma 2.2, we obtain

and

Substituting inequalities (3.3) –(3.5) into the inequality (3.2) leads to the inequality (3.1). The proof of Theorem 3.1 is complete.

Corollary 3.1.1. Let $F:\mathbb{R}\to \mathbb{R}$ be a given function. Suppose that $f:I\subseteq\mathbb{R}_+\to\mathbb{R}$ is differentiable, $a, b \in I$ with $a < b$, and $x\in[a, b]$. If $|f'|$ is GA-$F$-convex on $I$, then

where $G_1(u, v)$, $G_2(u, v)$, and $G_3(u, v)$ are defined as in Lemma 2.2.

Theorem 3.2. Let $F:\mathbb{R}\to \mathbb{R}$ be a given function. Suppose that $f:I\subseteq\mathbb{R}_+\to\mathbb{R}$ is differentiable, $a, b \in I$ with $a < b$, and $x\in [a, b]$. If $|f'|^q$ is GA-$F$-convex on $I$ for $q > 1$, then

where $L(u, v)$, $G_1(u, v)$, and $G_2(u, v)$ are defined as in Lemma 2.2.

Proof. By Lemma 2.1 and Hölder's integral inequality, we have

Using the GA-$F$-convexity of $|f'|^q$ and Lemma 2.2, we obtain

and

Substituting inequalities (3.8)–(3.10) into the inequality (3.7) results in the inequality (3.6). The proof of Theorem 3.2 is complete.

Theorem 3.3. Let $F:\mathbb{R}\to \mathbb{R}$ be a given function. Suppose that $f:I\subseteq\mathbb{R}_+\to\mathbb{R}$ is differentiable, $a, b \in I$ with $a < b$, and $x\in [a, b]$. If $|f'|^q$ is GA-$F$-convex on $I$ for $q > 1$, then

where $L(u, v)$ is defined as in Lemma 2.2.

Proof. By Lemma 2.1 and Hölder's integral inequality, we have

Since $|f'|^q$ is GA-$F$-convex on $[a, b]$, using Lemma 2.2, we obtain

Substituting (3.13)–(3.16) into the inequality (3.12) arrives at the inequality (3.11). The proof of Theorem 3.3 is thus complete.

4.   Conclusions

In this paper, we defined the notion of geometric-arithmetic-$F$-convex functions in Definition 2.2 and, via the integral identity (2.2) and other analytic techniques such as those in Lemma 2.2, establish those integral inequalities (3.1) in Theorem 3.1, (3.6) in Theorem 3.2, and (3.11) in Theorem 3.3 of the Hermite-Hadamard type for geometric-arithmetic-$F$-convex functions defined by the inequality (2.1) in Definition 2.2.

Acknowledgements

The authors appreciate anonymous referees for their careful corrections to, helpful suggestions to, and valuable comments on the original version of this paper.

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11901322), by the Natural Science Foundation of Inner Mongolia (Grant No. 2018LH01002), by the Fostering Project for Successfully Applying for the National Natural Science Foundation of China at the Inner Mongolia University for Nationalities (Grant No. NMDGP17104), and by the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (Grant No. NJZY19157), China.

Authors' contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Conflict of interest

The authors declare that they have no conflict of interest.