Abstract

In this paper, we establish integral inequalities of Hermite-Hadamard type for multiplicatively h-preinvex functions. We also obtain some new inequalities involving multiplicative integrals by using some properties of preinvex and multiplicatively h-preinvex functions.

1. Introduction

Many inequalities have been established for convex functions but the most famous is the Hermite-Hadamard inequality, due to its rich geometrical significance and applications, which is stated in ^{1, 2, 3} as:

Let be a convex function on the interval of real numbers with then

(1)

Both the inequalities hold in the reversed direction if is concave. For some results which generalize, extend and improve the inequality (1), we refer the interested reader ^{4, 5, 6, 7, 8}.

A significant generalization of convex functions is that of preinvex functions. In recent years, lots of efforts have been made by many researchers to generalize Hermite-Hadamard inequality for preinvex functions ^{9, 10, 11, 12, 13, 14, 15}. These studies include among others the work of Hanson ¹⁶, Ben-Israel and Mond ¹⁷, Pini ¹⁸, Weir and Mond ¹⁹, Noor ²⁰ and Yang and Li ²¹ have studied the basic properties of the preinvex function and their role in optimization, variational inequalities and equilibrium problems. Hanson ¹⁶ introduced a significant class of generalized convex functions, which is called invex functions. Ben-Israel and Mond ¹⁹ introduced the notions of invex sets and preinvex functions. Yang and Li ²¹ studied the basic properties of the preinvex function and their role in optimization, variational inequalities and equilibrium problems. Let us recall some definitions and known results concerning invexity and preinvexity.

Definition 1.1 ²¹ A set is said to be invex if there exist a function such that

The invex set is also called a -connected set.

Definition 1.2 ¹⁹ Let be a function on the invex set Then, is said to be preinvex with respect to , if

It is to be noted that every convex function is preinvex with respect to the map but the converse is not true, see for example ^{19, 22}.

In ²³ Noor has obtained the following Hermite-Hadamard inequalities for the preinvex functions.

Theorem 1.1 Let be a preinvex function on the interval of real numbers and with Then the following inequality holds:

Definition 1.3 ²⁴ Let be an interval in and let be an invex set with respect to η. A nonnegative function is called h-preinvex with respect to η if

Definition 1.4 ²⁵ A nonnegative function is said to be multiplicatively (or logarithmically) -preinvex with respect to if

From the above definition, we have

Recall that the notion of multiplicative integral is denoted by while the ordinary integral is denoted by This comes from the fact that the sum of the terms of product is used in the definition of a classical Riemann integral of on the product of terms raised to certain powers is used in the definition of multiplicative integral of on

There is the following relation between Riemann integral and multiplicative integral ²⁶.

Proposition 1.1 If is Riemann integrable on then is multiplicative integrable on and

In ²⁶, Bashirov et al. show that multiplicative integral has the following results:

Proposition 1.2 If is positive and Riemann integrable on then is multiplicative integrable on and

1.

2.

3.

4.

5. and

2. Main Results

In this section we establish some Hermite-Hadamard type inequalities for multiplicatively -preinvex functions. We also obtain integral inequalities of Hermite-Hadamard type for product and quotient of multiplicatively -preinvex and preinvex positive functions.

Theorem 2.1 be an open invex subset with respect to and with If is a positive and multiplicatively -preinvex function on the interval such that and satisfies Condition C, then

(2)

Proof Since is a multiplicatively -preinvex function, we have for every with

Now, let From Condition C, we have

Taking logarithms of both sides of the above inequality leads to

Integrating the above inequality with respect to on

Thus,

Hence, we have

(3)

which completes the proof of the first inequality in (2).

Now consider the second inequality in (2).

Hence, we get the inequality

(4)

Combining (3) and (4) gives the desired result.

Corollary 2.1 Let an open invex subset with respect to and with If and are positive and multiplicatively -preinvex functionson such that and satisfies Condition C, then

Since and are multiplicatively -preinvex functions, is a multiplicatively -preinvex function. Thus, if we apply Theorem 2.1 to the function then we obtain the required result.

Corollary 2.2 Let an open invex subset with respect to and with If and are positive and multiplicatively -preinvex functions on such that and satisfies Condition C, then

Proof Since and are multiplicatively -preinvex functions, is a multiplicatively -preinvex function. Thus, if we apply Theorem 2.1 to the function then we obtain the desired result.

Theorem 2.2 Let an open invex subset with respect to and with Let and be preinvex and multiplicatively h-preinvex positive functions, respectively, on the interval Then, we have

Proof Note that,

Thus, we have

which completes the proof.

Theorem 2.3 Let an open invex subset with respect to and with Let and be multiplicatively -preinvex and preinvex positive functions, respectively, on the interval Then, we have

Proof Note that

Hence,

which is the desired result.

Theorem 2.4 Let an open invex subset with respect to and with Let and be preinvex and multiplicatively -preinvex positive functions, respectively, on the interval Then, we have

Proof Note that

Consequently,

This completes the proof.

Remark 2.1 Now we point out some special cases which are included in our main results.

1. If , then our results reduce to the results for multiplicatively preinvex functions given in ¹⁵.

2. If and then our results reduce to the results for multiplicatively convex functions given in ⁴.

3. If and then our results reduce to the results for multiplicatively s-convex functions given in ⁶.

4. If and then our results reduce to the results for multiplicatively P-functions given in ⁷.

Availability of Data and Materials

Not applicable.

Competing Interests

The author declares that no competing interests.

Funding

There is no funding for this research article.

Authors’ Contributions

All authors read and approved the final manuscript.

Acknowledgements

Author is thankful to editor and anonymous referees for their valuable comments and suggestions.

References