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Research Article

Open Access Peer-reviewed

Serap Özcan^{ }

Received November 11, 2021; Revised December 14, 2021; Accepted December 24, 2021

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.

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.

**De****fi****nition 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*.

**De****fi****nition 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: *

**De****fi****nition 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

**De****fi****nition 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

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}.

Not applicable.

The author declares that no competing interests.

There is no funding for this research article.

All authors read and approved the final manuscript.

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

[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

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

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

Özcan, Serap. "Hermite-Hadamard Type Inequalities for Multiplicatively *h*-Preinvex Functions." *Turkish Journal of Analysis and Number Theory* 9.3 (2021): 65-70.

Özcan, S. (2021). Hermite-Hadamard Type Inequalities for Multiplicatively *h*-Preinvex Functions. *Turkish Journal of Analysis and Number Theory*, *9*(3), 65-70.

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 | |||