Integral inequalities of Hermite-Hadamard type for quasi-convex functions with applications

1.   Introduction

Many research papers have studied the properties of convex functions that make this concept interesting in mathematical analysis ^[1,2,3,4]. In recent years, important generalizations have been made in the context of convexity: quasi-convex ^[5], pseudo-convex ^[6], invex and preinvex ^[7], strongly convex ^[8], approximately convex ^[9], $MT$-convex ^[10], $(\alpha, m)$-convex ^[11], and strongly $(s, m)$-convex ^{[12,13,14,15]}. Here, we recall the notion of convexity: A function $g:[\beta_1, \beta_2]\subset \mathbb{R}\to \mathbb{R}$ is said to be convex if the following inequality holds

Now, we recall our basic definition, the so-called quasi-convex function.

Definition 1.1 (^[16]). A function $g:[\beta_1, \beta_2]\to \mathbb{R}$ is said quasi-convex on $[\beta_1, \beta_2]$ if

for any $x, y\in [\beta_1, \beta_2]$ and $t \in [0, 1]$.

It is important to note that, any convex function is a quasi-convex but the reverse is not true. In the following example we explain that fact.

Example 1.1 (^[5]). The function $h:[-2, 2]\to \mathbb{R}$, defined by

is not convex on $[-2, 2]$ but it is easy to see that the function is quasi-convex on $[-2, 2]$.

Notice that $h$ is quasi-convex if and only if all the level sets of $h$ are intervals (convex sets of the line).

The use of the convex function to study the integral inequalities have been deeply investigated, especially for the well-known inequality of Hermite-Hadamard type (HH-type inequality). The HH-type inequalities are one of the most important type inequalities and have a strong relationship to convex functions. In 1893 Hermite and Hadamard ^[17] found independently that for any convex function $g:[\beta_1, \beta_2]\to \mathbb{R}$, the inequality

holds.

In the field of mathematical analysis, many scholars have focused on defining new convexity and implementing of the problems based on their features. The features that make the results different from each other include lower and higher order derivative of the function. The differential equations with impulse perturbations lie in a special significant position in the theory of differential equations. Among them, integral inequality methods are the important tools to investigate the qualitative characteristics of solutions of different kinds of equations such as differential equations, difference equations, partial differential equations, and impulsive differential equations; see ^{[18,19,20,21,22,23]} for more details.

The HH-type inequality (1.3) has been applied to various convex functions like s-geometrically convex functions ^[24], GA-convex functions ^[25], $MT$-convex function ^[10], $(\alpha, m)$-convex functions ^[26] and many other types can be found in ^[27]. Besides, the HH-type inequality (1.3) has been applied to a numerous type of convex functions in the sense of fractional calculus like $F$-convex functions ^[28], $\lambda_{\psi}$-convex functions ^[29], $MT$-convex functions ^[30], $(\alpha, m)$-convex functions ^[11], new class of convex functions ^[31] and many other types can be found in the literature. Meanwhile, it has been applied to other models of fractional calculus like standard Riemann-Liouville fractional operators ^[32,33], conformable fractional operators ^[34,35,36], generalized fractional operators ^[37], $\psi$-RL-fractional operators ^[38,39], Tempered fractional operators ^[40], and AB- and Prabhakar fractional operators ^[41].

In view of the above indices, we extend the work done in ^[42] to establish some modified HH-type inequalities for the 3-times differentiable quasi-convex functions.

2.   Main results

This section deals with our main results. Throughout this paper, we mean $g\in L[\beta_1, \beta_2]$ that the function $g$ is differential and continuous on $[\beta_1, \beta_2]$.

Lemma 2.1. Suppose that $g:J\subset \mathbb{R}\to \mathbb{R}$ is a differentiable function such that $\beta_1, \beta_2 \in J$ with $\beta_1 < \beta_2$. If $g'''\in L[\beta_1, \beta_2]$, then we have

Proof. By applying integration by parts three times to get

Making use of change of the variable $x = t\frac{\beta_1+\beta_2}{2}+(1-t)\beta_1$ for $t\in [0, 1]$ and multiplying by $\frac{(\beta_2-\beta_1)^3}{96}$ on both sides, we obtain

Analogously, we can deduce

Finally, by adding (2.3) and (2.4), we get the required identity (2.1).

Remark 2.1. Notice that f being quasi-convex is not equivalent to $|f|$ being quasi-convex. For instance, $g(x) = x^2-1$ is only quasi-convex (but not $|g(x)|$), whereas $g(x) = 1$ if $x\in[-1, 1]$ and $g(x) = -1$ otherwise, is not quasi-convex, but $|g(x)| = 1$ is quasi-convex.

Theorem 2.1. Suppose that $g:J\subseteq [0, +\infty)\to \mathbb{R}$ is a differentiable function such that $g'''\in L[\beta_1, \beta_2]$, where $\beta_1, \beta_2\in J$ with $\beta_1 < \beta_2$. If $|g'''|$ is quasi-convex function on $[\beta_1, \beta_2]$, then we have

where $K = \max\left\{\left|g'''\left(\frac{\beta_1+\beta_2}{2}\right)\right|, \left|g'''(\beta_1)\right|\right\}+ \max\left\{\left|g'''\left(\frac{\beta_1+\beta_2}{2}\right)\right|, \left|g'''(\beta_2)\right|\right\}.$

Proof. Making use of Lemma 2.1 and the quasi-convexity $|g'''|$, we have that

This rearranges to the desired result.

Example 2.1. To clarify the following expression occurs in Theorem 2.1

we consider the function $g(x) = \frac{x}{x^2+2}$ on the interval $[\beta_1, \beta_2] = [0, 1]$. Then, we have

Figure 1 demonstrates the intersections and relationships between the functions $g(x), y(x)$ and the point $p$ geometrically.

Corollary 2.1. Let the assumptions of Theorem 2.1 be valid and let

Then,

(i) if $|g'''|$ is increasing, then we have

(ii) if $|g'''|$ is decreasing, then we have

(iii) if $g'''\left(\frac{\beta_1+\beta_2}{2}\right) = 0$, then we have

(iv) if $g'''(\beta_1) = g'''(\beta_2) = 0$, then we have

Theorem 2.2. Suppose that $g:J\subseteq [0, +\infty)\to \mathbb{R}$ is a differentiable function such that $g'''\in L[\beta_1, \beta_2]$, where $\beta_1, \beta_2\in J$ with $\beta_1 < \beta_2$. If $|g'''|^q$ is quasi-convex function on $[\beta_1, \beta_2]$ and $q > 1$ with $\frac{1}{p}+\frac{1}{q} = 1$, then we have

where $K_q = \left(\max\left\{\left|g'''\left(\frac{\beta_1+\beta_2}{2}\right)\right|^q, |g'''(\beta_1)|^q\right\}\right)^{\frac{1}{q}}+\left(\max\left\{\left|g'''\left(\frac{\beta_1+\beta_2}{2}\right)\right|^q, |g'''(\beta_2)|^q\right\}\right)^{\frac{1}{q}}$.

Proof. Let $p > 1$. Then from Lemma 2.1 and using the Hölder inequality, we can deduce

The quasi-convexity of $|g'''|^q$ on $[\beta_1, \beta_2]$ implies that

and

Therefore, we obtain

where we used the identities

Thus, our proof is completely done.

Corollary 2.2. Let the assumptions of Theorem 2.2 be valid and let

Then,

(i) if $|g'''|$ is increasing, then we have

(ii) if $|g'''|$ is decreasing, then we have

(iii) if $g'''\left(\frac{\beta_1+\beta_2}{2}\right) = 0$, then we have

(iv) if $g'''(\beta_1) = g'''(\beta_2) = 0$, then we have

Theorem 2.3. Suppose that $g:J\subseteq [0, +\infty)\to \mathbb{R}$ is a differentiable function such that $g'''\in L[\beta_1, \beta_2],$ where $\beta_1, \beta_2\in J$ with $\beta_1 < \beta_2$. If $|g'''|^q$ is quasi-convex function on $[\beta_1, \beta_2]$ and $q\geq 1$ with $\frac{1}{p}+\frac{1}{q} = 1$, then we have

where $K_q$ is as before.

Proof. From Lemma 2.1, properties of modulus, and power mean inequality, we have

Then, by using the quasi-convexity of $|g'''|^q$ on $[\beta_1, \beta_2]$, we have

and

Therefore, we obtain

Hence, our proof is completely done.

Remark 2.2. If $q = \frac{p}{p-1} \ (p > 1)$, the constants of Theorem 2.2 are improved, since $\frac{1}{(3p+1)^{\frac{1}{p}}} < 1$.

The following corollary improves the inequalities (2.7)–(2.10).

Corollary 2.3. Let the assumptions of Theorem 2.3 be valid and let

Then,

(i) If $|g'''|$ is increasing, we obtain (2.7).

(ii) If $|g'''|$ is decreasing, we obtain (2.8).

(iii) If $g'''\left(\frac{\beta_1+\beta_2}{2}\right) = 0$, we obtain (2.9),

(iv) if $g'''(\beta_1) = g'''(\beta_2) = 0$, we obtain (2.10).

3.   Applications

3.1.   Applications for special means

Consider the special means of positive real numbers $\beta_1 > 0$ and $\beta_2 > 0$, define by:

● Arithmetic Mean:

● Logarithmic mean:

● Generalized log-mean:

Remark 3.1. Let $0 < \alpha\le 1$ and $x > 0$. Then, we consider

For each $x, y > 0$ and $t\in[0, 1]$, we see that $(tx+(1-t)y)^{\alpha}\le t^{\alpha}x^{\alpha}+(1-t)^{\alpha}y^{\alpha}$, then we see that $g'''(x)$ is $\alpha$-convex function on $(0, +\infty)$ and $g\left(\frac{\beta_1+\beta_2}{2}\right) = \frac{A^{\alpha+3}(\beta_1, \beta_2)}{(\alpha +1)(\alpha+2)(\alpha+3)}$. Furthermore, we have

Above we used the definition of $\alpha$-convexity ^[11]: A function $g:[0, r]\to\mathbb{R}, r > 0$ is said to be $\alpha$-convex, if the following holds:

Proposition 3.1. Let $0 < \alpha\leq 1$ and$\beta_1, \beta_2\in \mathbb{R}^+$ with $\beta_1 < \beta_2$, then we have

Proof. Since $x^{\alpha}$ is quasi-convex for each $x > 0$ and $\alpha\in(0, 1)$ because every non-decreasing continuous function is also quasi-convex, so the assertion follows from inequality (2.5) with $g(x) = \frac{x^{\alpha+3}}{(\alpha+1)(\alpha+2)(\alpha+3)}$.

Proposition 3.2. Let $\beta_1, \beta_2\in \mathbb{R}$ such that $\beta_1 < \beta_2$ and $[\beta_1, \beta_2]\subset (0, +\infty)$, then we have

Proof. The assertion follows from inequality (2.11) with $g(x) = \frac{1}{x}, \ x\in [\beta_1, \beta_2]$.

Proposition 3.3. Let $\beta_1, \beta_2\in \mathbb{R}$ with $0 < \beta_1 < \beta_2$ and $n\in \mathbb{N}$, $k\ge 5$, then for all $q\ge 1$, we have

Proof. Let $g(x) = x^k, \ x\in [\beta_1, \beta_2], \ k\in \mathbb{N}$ with $k\ge 5$, then we have

and it is easy to see that $g'''$ is an increasing and quasi-convex function. Then, by applying Corollary 2.2(ⅰ) above, we have

where

Then, by applying Corollary 2.1(9) to $g$ above, we get

and this completes the proof.

3.2.   Application for particular functions

Here, we consider two particular functions.

● First, we define $g:\mathbb{R}\to \mathbb{R}$, by $g(x) = e^x$.

Then, we have $g'''(x) = e^x$ and

By applying inequality (2.7) for above $||g'''||_{\infty}$, we can deduce

Particularly for $\beta_1 = 0$, it follows that

and for $\beta_2 = 1$, it follows that

● Now, we define $g:\mathbb{R^+}\to \mathbb{R}$, by $g(x) = \frac{1}{x}$.

Then, since $|g'''(x)| = \frac{6}{x^4}$ is quasi-convex in $[\beta_1, \beta_2]\subset \mathbb{R}^+$ and

By applying inequality (2.8) to the function $g$ above, we get

In view of (3.2) and Proposition 2, we can deduce

For further illustration on the inequalities (3.1) and (3.2), we present some plot examples. Figures 2 and 3 illustrate the inequalities (3.1) and (3.2), respectively.

Let

and

Furthermore, Figures 4 and 5 show $K(\beta_2)-k(\beta_2)$ and $D(\beta_1, \beta_2):= H(\beta_1, \beta_2)-h(\beta_1, \beta_2)$, receptively. From Figure 5, we can see that all values of $D(\beta_1, \beta_2)$ are positive which confirms the validity of (3.2).

4.   Conclusions

In this paper we have established new Hermite–Hadamard inequality mainly motivated by Alomari et al in ^[42] for quasi-convex functions with $g\in C^3([\beta_1, \beta_2])$ such that $g'''\in L([\beta_1, \beta_2])$ and we give some applications to some special means and for some particular functions. We hope that the ideas used in this paper may inspire interested readers to explore some new applications.

We believe that our results, this new understanding of Hermite-Hadamard integral inequalities for quasi-convex functions, will be vital information for the future studies of these models of integral inequality. One can obtain the similar results for other kind of convex functions

Acknowledgements

We would like to express our special thanks to the editor and referees. Also, the third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

Conflict of interest

The authors declare no conflict of interest.