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Hermite-Hadamard Type Inequalities for Multiplicatively h-Preinvex Functions

Serap Özcan
Turkish Journal of Analysis and Number Theory. 2021, 9(3), 65-70. DOI: 10.12691/tjant-9-3-5
Received November 11, 2021; Revised December 14, 2021; Accepted December 24, 2021

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

1.1. Multiplicative Calculus

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

[1]  Dragomir, S. S. and Pearce, C. E. M.: Selected topics on Hermite-Hadamard inequalities and applications. RGMIA Monographs, Victoria University, 2000.
In article      
 
[2]  Hadamard, J.: Étude sur les propriétés des fonctions entières en particulier d'une fonction considérée par Riemann. J. Math. Pure Appl. 58, 171-215 (1893).
In article      
 
[3]  Pecaric, J. E., Proschan, F. and Tong, Y. L.: Convex functions, partial orderings and statistical applications. Academic Press, Boston, 1992.
In article      
 
[4]  Ali, M. A., Abbas, M., Zhang, Z., Sial, I. B. and Arif, R.: On integral inequalities for product and quotient of two multiplicatively convex functions. Asian Research J. Math. 12(3), 1-11 (2019).
In article      View Article
 
[5]  Özcan, S.,: Hermite-Hadamard type inequalities for multiplicatively h-convex functions. Konuralp J. Math. 8(1), 158-164 (2020).
In article      View Article
 
[6]  Özcan, S.: Hermite-Hadamard type inequalities for multiplicatively s-convex functions. Cumhuriyet Sci. J. 41(1), 245-259 (2020).
In article      View Article
 
[7]  Özcan, S.: Hermite-Hadamard type inequalities for multiplicatively P-functions. Gumushane Univ. J. Sci. Tech. Inst. 10(2), 486-491 (2020).
In article      View Article
 
[8]  Toplu, T., Kadakal, M. and İşcan, İ.: On n-polynomial convexity. AIMS Math. 5(2), 1304-1318 (2020).
In article      View Article
 
[9]  Antczak, T.: Mean value in invexity and analysis, Nonlinear Analysis 60, 1471-1484 (2005).
In article      View Article
 
[10]  İşcan, İ., Kadakal, M. and Kadakal, H.: On two times differentiable preinvex and prequasiinvex functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68(1), 950-963 (2019).
In article      View Article
 
[11]  Kadakal, H., Kadakal, M. and İşcan, İ.: New type integral inequalities for three times differentiable preinvex and prequasiinvex functions. Open J. Math. Anal. 2(1), 33-46 (2018).
In article      View Article
 
[12]  Kadakal, H.: Differentiable preinvex and prequasiinvex functions. Universal J. Math. Appl. 3(2), 69-77 (2020).
In article      View Article
 
[13]  Latif, M. A. and Shoaib, M.: Hermite-Hadamard type integral inequalities for differentiable m-preinvex and (, m)-preinvex functions. J. Egyptian Math. Soc. 23, 236-241 (2015).
In article      View Article
 
[14]  Özcan, S.: On refinements of some integral inequalities for differentiable prequasiinvex functions. Filomat 33(14), 4377-4385 (2019).
In article      View Article
 
[15]  Özcan, S.: Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions. AIMS Math. 5(2), 1505-1518 (2020).
In article      View Article
 
[16]  Hanson, M. A.: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 1, 545-550 (1981).
In article      View Article
 
[17]  Ben-Israel, A. and Mond, B.: What is invexity. J. Australian Math. Soc. Ser. B 28(1), 1-9 (1986).
In article      View Article
 
[18]  Pini, R.: Invexity and Generalized convexity. Optimization 22, 513-523 (1991).
In article      View Article
 
[19]  Weir, T. and Mond, B.: Preinvex functions in multiple objective optimization. J. Math. Anal. Appl. 136, 29-38 (1998).
In article      View Article
 
[20]  Noor, M. A.: Variational like inequalities. Optimization 30, 323-330 (1994).
In article      View Article
 
[21]  Yang, X. M. and Li. D.: On properties of preinvex functions. J. Math. Anal. Appl. 256, 229-241 (2001).
In article      View Article
 
[22]  Yang, X. M., Yang, X. Q. and Teo, K. L.: Generalized invexity and generalized invariant monotonicity. J. Optimization Theory and Appl. 117, 607-625 (2003).
In article      View Article
 
[23]  Noor, M. A.: Hermite-Hadamard integral inequalities for log-preinvex functions. J. Math. Anal. Approx. Theory 2, 126-131 (2007).
In article      
 
[24]  Noor, M. A., Noor, K. I., Awan, M. U. and Qi, F.: Integral inequalities of Hermite-Hadamard type for logarithmically h-preinvex functions. Cogent Math. Stat. 28(7), 1463-1474 (2014).
In article      View Article
 
[25]  Noor, M. A., Noor, K. I., Awan, M. U. and Li, J.: On Hermite-Hadamard inequalities for h-preinvex functions. Filomat 2, Article ID 10335856 (2015).
In article      View Article
 
[26]  Bashirov, A. E., Kurpınar, E. M. and Özyapıcı, A.: Multiplicative calculus and applications. J. Math. Anal. and Appl. 337(1), 36-48 (2008).
In article      View Article
 
[27]  Ali, M.A., Abbas, M., Zafer, A.A.: On some Hermite-Hadamard integral inequalities in multiplicative calculus. J. Ineq. Special Func. 10(1), 111-122 (2019).
In article      
 

Published with license by Science and Education Publishing, Copyright © 2021 Serap Özcan

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Normal Style
Serap Özcan. Hermite-Hadamard Type Inequalities for Multiplicatively h-Preinvex Functions. Turkish Journal of Analysis and Number Theory. Vol. 9, No. 3, 2021, pp 65-70. http://pubs.sciepub.com/tjant/9/3/5
MLA Style
Özcan, Serap. "Hermite-Hadamard Type Inequalities for Multiplicatively h-Preinvex Functions." Turkish Journal of Analysis and Number Theory 9.3 (2021): 65-70.
APA Style
Özcan, S. (2021). Hermite-Hadamard Type Inequalities for Multiplicatively h-Preinvex Functions. Turkish Journal of Analysis and Number Theory, 9(3), 65-70.
Chicago Style
Özcan, Serap. "Hermite-Hadamard Type Inequalities for Multiplicatively h-Preinvex Functions." Turkish Journal of Analysis and Number Theory 9, no. 3 (2021): 65-70.
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[1]  Dragomir, S. S. and Pearce, C. E. M.: Selected topics on Hermite-Hadamard inequalities and applications. RGMIA Monographs, Victoria University, 2000.
In article      
 
[2]  Hadamard, J.: Étude sur les propriétés des fonctions entières en particulier d'une fonction considérée par Riemann. J. Math. Pure Appl. 58, 171-215 (1893).
In article      
 
[3]  Pecaric, J. E., Proschan, F. and Tong, Y. L.: Convex functions, partial orderings and statistical applications. Academic Press, Boston, 1992.
In article      
 
[4]  Ali, M. A., Abbas, M., Zhang, Z., Sial, I. B. and Arif, R.: On integral inequalities for product and quotient of two multiplicatively convex functions. Asian Research J. Math. 12(3), 1-11 (2019).
In article      View Article
 
[5]  Özcan, S.,: Hermite-Hadamard type inequalities for multiplicatively h-convex functions. Konuralp J. Math. 8(1), 158-164 (2020).
In article      View Article
 
[6]  Özcan, S.: Hermite-Hadamard type inequalities for multiplicatively s-convex functions. Cumhuriyet Sci. J. 41(1), 245-259 (2020).
In article      View Article
 
[7]  Özcan, S.: Hermite-Hadamard type inequalities for multiplicatively P-functions. Gumushane Univ. J. Sci. Tech. Inst. 10(2), 486-491 (2020).
In article      View Article
 
[8]  Toplu, T., Kadakal, M. and İşcan, İ.: On n-polynomial convexity. AIMS Math. 5(2), 1304-1318 (2020).
In article      View Article
 
[9]  Antczak, T.: Mean value in invexity and analysis, Nonlinear Analysis 60, 1471-1484 (2005).
In article      View Article
 
[10]  İşcan, İ., Kadakal, M. and Kadakal, H.: On two times differentiable preinvex and prequasiinvex functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68(1), 950-963 (2019).
In article      View Article
 
[11]  Kadakal, H., Kadakal, M. and İşcan, İ.: New type integral inequalities for three times differentiable preinvex and prequasiinvex functions. Open J. Math. Anal. 2(1), 33-46 (2018).
In article      View Article
 
[12]  Kadakal, H.: Differentiable preinvex and prequasiinvex functions. Universal J. Math. Appl. 3(2), 69-77 (2020).
In article      View Article
 
[13]  Latif, M. A. and Shoaib, M.: Hermite-Hadamard type integral inequalities for differentiable m-preinvex and (, m)-preinvex functions. J. Egyptian Math. Soc. 23, 236-241 (2015).
In article      View Article
 
[14]  Özcan, S.: On refinements of some integral inequalities for differentiable prequasiinvex functions. Filomat 33(14), 4377-4385 (2019).
In article      View Article
 
[15]  Özcan, S.: Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions. AIMS Math. 5(2), 1505-1518 (2020).
In article      View Article
 
[16]  Hanson, M. A.: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 1, 545-550 (1981).
In article      View Article
 
[17]  Ben-Israel, A. and Mond, B.: What is invexity. J. Australian Math. Soc. Ser. B 28(1), 1-9 (1986).
In article      View Article
 
[18]  Pini, R.: Invexity and Generalized convexity. Optimization 22, 513-523 (1991).
In article      View Article
 
[19]  Weir, T. and Mond, B.: Preinvex functions in multiple objective optimization. J. Math. Anal. Appl. 136, 29-38 (1998).
In article      View Article
 
[20]  Noor, M. A.: Variational like inequalities. Optimization 30, 323-330 (1994).
In article      View Article
 
[21]  Yang, X. M. and Li. D.: On properties of preinvex functions. J. Math. Anal. Appl. 256, 229-241 (2001).
In article      View Article
 
[22]  Yang, X. M., Yang, X. Q. and Teo, K. L.: Generalized invexity and generalized invariant monotonicity. J. Optimization Theory and Appl. 117, 607-625 (2003).
In article      View Article
 
[23]  Noor, M. A.: Hermite-Hadamard integral inequalities for log-preinvex functions. J. Math. Anal. Approx. Theory 2, 126-131 (2007).
In article      
 
[24]  Noor, M. A., Noor, K. I., Awan, M. U. and Qi, F.: Integral inequalities of Hermite-Hadamard type for logarithmically h-preinvex functions. Cogent Math. Stat. 28(7), 1463-1474 (2014).
In article      View Article
 
[25]  Noor, M. A., Noor, K. I., Awan, M. U. and Li, J.: On Hermite-Hadamard inequalities for h-preinvex functions. Filomat 2, Article ID 10335856 (2015).
In article      View Article
 
[26]  Bashirov, A. E., Kurpınar, E. M. and Özyapıcı, A.: Multiplicative calculus and applications. J. Math. Anal. and Appl. 337(1), 36-48 (2008).
In article      View Article
 
[27]  Ali, M.A., Abbas, M., Zafer, A.A.: On some Hermite-Hadamard integral inequalities in multiplicative calculus. J. Ineq. Special Func. 10(1), 111-122 (2019).
In article