Abstract

In this study, we present a new definition of convexity. This definition is the class of strongly multiplicatively P-functions. Some new Hermite-Hadamard type inequalities are derived for strongly multiplicatively -functions. Some applications to special means of real numbers are given. Ideas of this paper may stimulate further research.

1. Introduction

In this section, we firstly give several definitions and some known results.

Definition 1: A function is said to be convex if the inequality

is valid for all and If this inequality reverses, then the function is said to be concave on interval

This definition is well known in the literature. Convexity theory has appeared as a powerful technique to study a wide class of unrelated problems in pure and applied sciences.

One of the most important integral inequalities for convex functions is the Hermite-Hadamard inequality. The classical Hermite–Hadamard inequality provides estimates of the mean value of a continuous convex function The following double inequality is well known as the Hadamard inequality in the literature.

Definition 2: Let be a convex function, then the inequality

is known as the Hermite-Hadamard inequality.

Some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors ^{1, 2, 3, 4, 5, 6} and the Authors obtained a new refinement of the Hermite-Hadamard inequality for convex functions.

Definition 3: A nonnegative function is said to be -function if the inequality

holds for all and

We will denote by the set of -functions on the interval Note that contain all nonnegative convex and quasi-convex functions.

In ⁷, Dragomir et al. proved the following inequality of Hadamard type for class of -functions.

Theorem 1: Let with and Then

Definition 4: ⁸ Let be an interval and be a positive number. A function is called strongly convex with modulus if

for all and

In ⁹, Kadakal gave the definition of multiplicatively -function (or log--function) and related Hermite-Hadamard inequality. It should be noted that the concept of log--convex, which we consider in our study and given below, was first defined by Noor et al in 2013 ¹⁰. Then, the algebraic properties of this definition with the name of multiplicatively -function are examined in detail by us.

Definition 5: ^{9, 10} Let be an interval in The function is said to be multiplicatively -function (or -function), if the inequality

holds for all and

Denote by the class of all multiplicatively -functions on Clearly, is multiplicatively -function if and only if is -function. The range of the multiplicatively -functions is greater than or equal to 1.

Theorem 2: Let the function be a multiplicatively -function and with If then the following inequalities hold:

i)

ii)

Dragomir and Agarwal in ¹¹ used the following lemma to prove Theorems.

Lemma 1: The following equation holds true:

In ¹², U. S. Kırmacı used the following lemma to prove Theorems.

Lemma 2: Let be a differentiable mapping on ( is the interior of ) with If then we have the following equation holds true:

The main purpose of this paper is to establish new estimations and refinements of the Hermite–Hadamard inequality for functions whose derivatives in absolute value are strongly multiplicatively -function.

2. Strongly Multiplicatively P-functions and Their Some Properties

In this section, we begin by setting some algebraic properties for strongly multiplicatively -functions.

Definition 6: Let be an interval in The function is said to be strongly multiplicatively -function with modulus if the inequality

holds for all and

We will denote by the class of all strongly multiplicatively -functions on interval

Remark 1: The range of the strongly multiplicatively -functions is greater than or equal to 1.

Proof: Using the definition of the strongly multiplicatively -function, for

Here, so we obtain Similarly, for

Since we get

3. Hermite-Hadamard Type Inequalities for Strongly Multiplicatively P-functions

The goal of this paper is to develop concept of the strongly multiplicatively P-functions and to establish some inequalities of Hermite-Hadamard type for these classes of functions.

Theorem 3: Let the function be a strongly multiplicatively -function and with If then the following inequalities hold:

i)

Proof: i) Since the function is a strongly multiplicatively -function, we write the following inequality:

By integrating this inequality on and changing the variable as then

Moreover, a simple calculation give us that

So, we get the desired result.

ii) Similarly, as is a strongly multiplicatively -function, we write the following:

Here, by integrating this inequality on and changing the variable as then, we have

Since,

we obtain

This completes the proof of theorem.

Remark 2: Above Theorem (i) and (ii) can be written together as follows:

Proof: By integrating the following inequality on the desired result can be obtained:

where

Theorem 4: Let be a differentiable function on such that the function is strongly multiplicatively -function. Suppose that with and Then the following inequality holds:

(3.1)

Proof: Using Lemma 1, since is strongly multiplicatively -function, we obtain

where

This completes the proof of theorem.

Theorem 5: Let be a differentiable function on Assume is such that the function is strongly multiplicatively -function. Suppose that with and Then the following inequality holds:

where

Proof: Let By assumption, Hölder’s integral inequality, Lemma 1 and the inequality

we have

where

This completes the proof of theorem.

A more general inequality using Lemma 1 is as follows.

Theorem 6: Let be a differentiable function on Assume is such that the function is strongly multiplicatively -function. Suppose that with and Then the following inequality holds:

(3.2)

Proof: Let Since the function is a strongly multiplicatively -function, from Lemma 1 and the power-mean integral inequality, we have

This completes the proof.

Corollary 1: If we take in inequality (3.2), we obtain the following inequality:

This inequality coincides with the inequality (3.1).

Theorem 7: Let be a differentiable function on such that the function is strongly multiplicatively -function. Suppose that t and Then the following inequality holds:

(3.3)

Proof: Using Lemma 2, since is strongly multiplicatively -function, we obtain

where

This completes the proof of theorem.

Theorem 8: Let be a differentiable function on Assume is such that the function is strongly multiplicatively -function. Suppose that with and Then the following inequality holds:

where

Proof: Since the function is a strongly multiplicatively -function, from Lemma 2 and the Hölder’s integral inequality, we have

where

This completes the proof of theorem.

Theorem 9: Let be a differentiable function on Assume is such that the function is multiplicatively -function. Suppose that with and Then the following inequality holds:

(3.4)

where

Proof: Since the function is a multiplicatively -function, from Lemma 2 and the power-mean integral inequality, we obtain

where

Corollary 2: If we take in inequality (3.4), we obtain the following inequality:

This inequality coincides with the inequality (3.3).

4. Conclusion

We derived some new Hermite-Hadamard type inequalities for strongly multiplicatively P-functions. Similar method can be applied to the different type of convex functions.

References