Hermite-Hadamard and Jensen’s type inequalities for modified (p, h)-convex functions

1.   Introduction

In non-linear programming and optimization theory, convexity plays an important role. Three important areas of non-linear analysis are monotone operator theory, convex analysis and theory of non-expansive mapping. In early 1960 these theories get emerged. Theses areas have got the attention of many researcher and many connections have been identified between them over the past few years. The notion of convexity has been expanded and generalization in numerous ways utilizing novel and modern methods in recent years. Convexity also plays vital part in fields outside mathematics such as chemistry, biology, physics and other sciences. It is always interesting to generalize the definition of convexity from different aspects because the wide range of applications. Recently the object of numerous studies have been the convexity of functions and sets. For further studies on generalization of convexity one can see ^{[1,2,3,4,5,6,7,8,9]} and references therein.

In this paper, we introduce the concept of modified $(p, h)$-convex functions as generalization of convex functions. Some basic results under various conditions for the modified $(p, h)$-convex functions are investigated. We investigate the Jensen and Hermite-Hadamard type inequalities related to modified $(p, h)$-convex functions. For more on Hermite-Hadamard inequalities, we will refere to the reader ^{[10,11,12,13,14,15,16,17,18,19,20]}.

Let us see some basic definitions and generalizations of convex functions ^[21,22].

Definition 1.1. (Similarly ordered function) ^[17] Two functions $f$ and $g$ are called similarly ordered ($f$ is $g$-monotone) on $I\subset\mathbb{R}$, if

for all $r, s\in I$.

Definition 1.2. (Super (Sub) multiplicative function) ^[4] A function $h: J\rightarrow \mathbb{R}$ is said to be supermultiplicative if

for all $x, y \in J$, where $J\subset\mathbb{R}$. If inequality (1.1) is reversed, then $h$ is said to be a submultiplicative function.

Definition 1.3. ^[19] Let $f \in L_1[a, b]$, the Riemann-Liouville Integrals $J^\alpha_{a^+}f$ and $J^\alpha_{b^-}f$ of order $\alpha > 0$ are defined by

and

where

is the Gamma function.

Definition 1.4. (Convex function) Let $I\subseteq \mathbb{R}$ be an interval, then a function $f:I\rightarrow \mathbb{R}$ is called convex if the following inequality holds:

Definition 1.5. ($p$-convex set) ^[18] The interval $I$ is called $p$-convex set if

for $t \in [0, 1]$ and for all $r, s\in I$, whereas p = $\frac{n}{m}$ or p = 2k+1, m = 2t+1, n = 2u+1, and t, u, k$\in N$.

Definition 1.6. ($p$-convex function) ^[18] A function $f$ from $X$ to $\mathbb{R}$ is known as $p$-convex function whereas $I$ is a $p$-convex set, if

for $t\in [0, 1]$ and $\forall\, r, s\in I$.

Definition 1.7. (Modified $h$-convex function) ^[8] Let $f, h: J\subset\mathbb{R} \rightarrow \mathbb{R}$ be non-negative functions. A function $f:I\subset\mathbb{R} \rightarrow \mathbb{R}$ is called modified $h$-convex function if

for $t\in [0, 1]$ and $\forall\, r, s\in J$.

Definition 1.8. ($(p, h)$-convex function) ^[9] Assume $h: J\subset\mathbb {R} \rightarrow\mathbb {R}$ be a non-negative and non-zero function. A function $f:I\rightarrow\mathbb {R}$, where $I$ is $p-$convex set in $\mathbb {R}$ is called $(p, h)$-convex function, if $f$ is non-negative and

for $t\in(0, 1)$ and $\forall\, r, s\in I$, where $p > 0$.

Definition 1.9. (Modified $(p, h)$-convex function) Assume $h: J\subset\mathbb {R} \rightarrow\mathbb {R}$ be a non-negative and non-zero function. A function $f:I\rightarrow\mathbb {R}$, where $I$ is $p-$convex set in $\mathbb {R}$ is called modified $(p, h)$-convex function, if $f$ is non-negative and

for $t\in(0, 1)$ and $\forall\, r, s\in I$, where $p > 0$.

Likewise, if the inequality sign in (1.2) is inverted, then $f$ is known as a modified $(p, h)$-concave function.

Of course, if we put in (1.2)

1. $p = 1$ then we get modified $h$-convex function;

2. $p = 1$ and $h(t) = t$ then we get classical convex function.

The paper is organized as follows: In next section, we will derive some basic properties of this generalization. However, the third, fourth and fifth sections are devoted to develop Hermite-Hadamard inequality, Jensen type inequality and Fejér type inequalities for modified $(p, h)$-convex functions.

2.   Basic results

In this section, we will verify our basic properties.

Proposition 1. Assume $f_i:I\subset\mathbb{R}\rightarrow\mathbb{R}$ be modified $(p, h)$-convex function, suppose $\mu_i, \cdots, \mu_n$ be positive scalers. Consider a function $g$ from $\mathbb{R}$ to $\mathbb{R}$ so that

ADD then $g$ is modified $(p, h)$-convex function.

Proof. We know that $f_i:I\subset\mathbb{R}\rightarrow\mathbb{R}$ be modified $(p, h)$-convex functions. Then $t\in[0, 1]$ and $\forall\, r, s\in I$, we have

The proof is completed.

Proposition 2. Let $h: J\subset\mathbb {R} \rightarrow[0, 1]$. If $g:I\rightarrow\mathbb{R}$ is modified $(p, h)$-convex function, and $f:I\rightarrow\mathbb{R}$ is convex and increasing, then $f\circ g$ is also modified $(p, h)$-convex function.

Proof. Since $g$ is modified $(p, h)$-convex function on $I$, we obtained

Then by using the convexity of $f$, we obtain

which implies that $f\circ g$ is modified $(p, h)$-convex function.

Proposition 3. Let $h: J\subset\mathbb {R} \rightarrow[0, 1]$. Further let $\left\{f_j:I\rightarrow\mathbb{R}, \, j\in \mathbb{N}\right\}$ is non empty collection of modified $(p, h)$-convex functions such that for each $x\in I, \, \max\limits_{j\in J}f_j(x)$ exists in $\mathbb{R},$ then the function $f:I\rightarrow\mathbb{R}$ defined by $f(x) = \max\limits_{j\in J}f_j(x)$ for each $x\in I$ is modified $(p, h)$-convex.

Proof. For any $x, y \in I$ and $t\in [0, 1]$, we have

which is required.

Proposition 4. Let $h: J\subset\mathbb {R} \rightarrow[0, 1]$. Further let $g$ and $f$ are two modified $(p, h)$-convex functions. Then the product of $f$ and $g$ will be a modified $(p, h)$-convex function if $g$ and $f$ are similarly ordered.

Proof. We know that $g$ and $f$ are modified $(p, h)$-convex function. Then

That's the required result.

3.   Hermite-Hadamard type inequalities

Theorem 3.1. Assume $f$ from $I$ to $\mathbb{R}$ be modified $(p, h)$-convex function on the interval $[a_1, a_2]$ with $a_1 < a_2$ then we have

Proof. Let $u^p = ta_1^p+(1-t)a_2^p$ and $v^p = (1-t)a_1^p+ta_2^p$, then we get

Integrating inequality over $t\in[0, 1]$, we get

Now, we know that

Thus,

Combining (3.2) and (3.3), we obtain the required result.

Remark 1. If we put $p = 1$ in (3.1) then we gain Hermite-Hadamard type inequality for modified $h$-convexity, (see ^[8]).

Remark 2. If we put $p = 1$ and $h(t) = t$ in (3.1) then we attain classical Hermite-Hadamard type inequality.

Theorem 3.2. Assume $f$ be modified $(p, h)$-convex function and $f\in L_1[a_1, a_2]$, with $a_1 < a_2$. Then

and

Proof. We know that $f$ is modified $(p, h)$-convex function, so we have

Let $r^p = (ta_1^p+(1-t)a_2^p)$, and $s^p = ((1-t)a_1^p+t a_2^p)$, then

Multiplying above inequality by $t^{\alpha-1}$, then integrating over $t\in[0, 1]$, we get

which is (3.4).

We know that $f$ is a modified $(p, h)$-convex function, then

Multiplying above inequality by $t^{\alpha-1}$ and then integrating over $t\in[0, 1]$, we get

from which we have

The result is completed.

Remark 3. If $p = 1$ in (3.4) and (3.5) then we will get the result for modified $h$-convex function, (see ^[8]).

4.   Jensen type inequality

The following expression is useful to prove Jensen type inequality for modified $(p, h)$-convex functions.

Assume $f$ from $I$ to $\mathbb{R}$ be an modified $(p, h)$-convex function. For $r_1, r_2 \in I$ and $\alpha_1 + \alpha_2 = 1$, we have $f(\alpha_1r_1^p + \alpha_2r_2^p)^\frac{1}{p} \leq h(\alpha_{1})f(r_1) + (1-h(\alpha_{1}))f(r_2).$

Also when $m > 2$ for $r_1, r_2, ..., r_m \in I$, $\sum_{i = 1}^{m} {\alpha_i} = 1$ and $T_i = \sum_{j = 1}^{i}{\alpha_j}$, we have

Theorem 4.1. Assume $f$ from $I$ to $\mathbb{R}$ be an modified $(p, h)$-convex function and $h$ be non-negative super-multiplicative function. If $T_i = \sum_{j = 1}^{i} \alpha_j$ for $i = 1, ..., m$, so that $T_m = 1$ where $m\in \mathbb{N},$ then

Proof. By using (4.1) it follows that:

using the fact that $h$ is supermultiplicative function, we have

from (4.1), we get

using the fact that $h$ is supermultiplicative function,

The result is completed.

Remark 4. If we choose $p = 1$ and $h(t) = t$ in (4.2) then we have classical Jensen type inequality.

5.   Fejér type inequality

Theorem 5.1. Assume that $g$ and $f$ are functions such that $f$ is modified $(p, h_1)$-convex function and $g$ is modified $(p, h_2)$-convex function, $fg \in L_1([v_1, v_2])$ and $h_1 h_2 \in L_1([0, 1])$ along $v_1, v_2\in I$ and $v_1 < v_2$. Then we get

where

Proof. We know that $f$ be modified $(p, h_1)$-convex function and $g$ be modified $(p, h_2)$-convex function, we have

for each $\, t\in[0, 1]$. Because $f$ and $g$ are non-negative, we attain the inequality

integrating above inequality over (0, 1), we attain the inequality

By setting $r = [tv_1^p+(1-t)v_2^p]^\frac{1}{p}$, we get

Remark 5. If we take $p = 1$ and $h(t) = t$ in (5.1) then we will get the result for convex function.

6.   Conclusions

Convexity play an important rule in applied sciences and mathematics. In this paper, we introduced modified $(p, h)$-convex functions which unify $p$-convexity with modified $h$-convexity. We investigated the fundamental properties of modified $(p, h)$-convexity and gave the Hermite-Hadamard, Fejér and Jensen's type inequalities.

Acknowledgment

1. Research Project on Basic Scientific Research Operating Costs of Higher Education Institutions in Heilongjiang Province (1353MSYYB017), Mudanjiang City 2018 Guidance Science and Technology Plan Project Technical Attack Project of Mudanjiang City (G2018q2465), Nature Science Fund of Daqing Normal University (19ZR04).

2. The research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202, 61673169, 11701176, 11626101, 11601485).

Conflict of interest

The authors declare that no competing interests exist.