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Research Article

Open Access Peer-reviewed

Mahir Kadakal^{ }

Received March 19, 2019; Revised April 22, 2019; Accepted May 05, 2019

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.

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.

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

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* **-f*unction*. *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).

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.

[1] | Dragomir, S.S. and Pearce, C.E.M. Selected Topics on Hermite-Hadamard Inequalities and Its Applications, RGMIA Monograph, 2002. | ||

In article | |||

[2] | Hadamard, J. Etude sur les proprietes des fonctions entieres en particulier d’une fonction consideree par Riemann, J. Math. Pures Appl. 58, 171-215, 1893. | ||

In article | |||

[3] | Kadakal, H., New Inequalities for Strongly r-Convex Functions, Journal of Function Spaces, Volume 2019, Article ID 1219237, 10 pages. | ||

In article | View Article | ||

[4] | Kadakal, M., Kadakal, H. and İşcan, İ., Some New Integral Inequalities for n-Times Differentiable Strongly Convex Functions, Karaelmas Science and Engineering Journal, 8(1): 147-150, 2018. | ||

In article | View Article | ||

[5] | Pečarić, J.E. Proschan, F. and Tong, Y.L. Convex Functions, Partial Orderings, and Statistical Applications, Academic Press Inc., 1992, p. 137. | ||

In article | |||

[6] | Zabandan, G. A new refinement of the Hermite-Hadamard inequality for convex functions, J. Inequal. Pure Appl. Math. 10(2), Article ID 45, 2009. | ||

In article | |||

[7] | Dragomir, S.S. Pečarić, J. and Persson, L.E. Some inequalities of Hadamard Type, Soochow Journal of Mathematics, Vol.21, No:3, pp. 335-341, July 1995. | ||

In article | |||

[8] | Polyak, B.T., Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl. 7, 72-75, 1966. | ||

In article | |||

[9] | Kadakal, H., Multiplicatively P-functions and some new inequalities, New Trends in Mathematical Sciences, NTMSCI 6, No. 4, 111-118, 2018. | ||

In article | View Article | ||

[10] | Noor, M.A., Qi, F. and Awan, M.U., Some Hermite-Hadamard type inequalities for log-h-convex functions, Analysis 33, 1-9, 2013. | ||

In article | View Article | ||

[11] | Dragomir, S. S. and Agarwal, R. P. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11, 1998, 91-95. | ||

In article | View Article | ||

[12] | Kirmaci U. S. Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Applied Mathematics and Computation 147, 2004. 137-146. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2019 Mahir Kadakal

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/

Mahir Kadakal. Strongly Multiplicatively -function and Some New Inequalities. *Turkish Journal of Analysis and Number Theory*. Vol. 7, No. 3, 2019, pp 59-64. http://pubs.sciepub.com/tjant/7/3/1

Kadakal, Mahir. "Strongly Multiplicatively -function and Some New Inequalities." *Turkish Journal of Analysis and Number Theory* 7.3 (2019): 59-64.

Kadakal, M. (2019). Strongly Multiplicatively -function and Some New Inequalities. *Turkish Journal of Analysis and Number Theory*, *7*(3), 59-64.

Kadakal, Mahir. "Strongly Multiplicatively -function and Some New Inequalities." *Turkish Journal of Analysis and Number Theory* 7, no. 3 (2019): 59-64.

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[1] | Dragomir, S.S. and Pearce, C.E.M. Selected Topics on Hermite-Hadamard Inequalities and Its Applications, RGMIA Monograph, 2002. | ||

In article | |||

[2] | Hadamard, J. Etude sur les proprietes des fonctions entieres en particulier d’une fonction consideree par Riemann, J. Math. Pures Appl. 58, 171-215, 1893. | ||

In article | |||

[3] | Kadakal, H., New Inequalities for Strongly r-Convex Functions, Journal of Function Spaces, Volume 2019, Article ID 1219237, 10 pages. | ||

In article | View Article | ||

[4] | Kadakal, M., Kadakal, H. and İşcan, İ., Some New Integral Inequalities for n-Times Differentiable Strongly Convex Functions, Karaelmas Science and Engineering Journal, 8(1): 147-150, 2018. | ||

In article | View Article | ||

[5] | Pečarić, J.E. Proschan, F. and Tong, Y.L. Convex Functions, Partial Orderings, and Statistical Applications, Academic Press Inc., 1992, p. 137. | ||

In article | |||

[6] | Zabandan, G. A new refinement of the Hermite-Hadamard inequality for convex functions, J. Inequal. Pure Appl. Math. 10(2), Article ID 45, 2009. | ||

In article | |||

[7] | Dragomir, S.S. Pečarić, J. and Persson, L.E. Some inequalities of Hadamard Type, Soochow Journal of Mathematics, Vol.21, No:3, pp. 335-341, July 1995. | ||

In article | |||

[8] | Polyak, B.T., Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl. 7, 72-75, 1966. | ||

In article | |||

[9] | Kadakal, H., Multiplicatively P-functions and some new inequalities, New Trends in Mathematical Sciences, NTMSCI 6, No. 4, 111-118, 2018. | ||

In article | View Article | ||

[10] | Noor, M.A., Qi, F. and Awan, M.U., Some Hermite-Hadamard type inequalities for log-h-convex functions, Analysis 33, 1-9, 2013. | ||

In article | View Article | ||

[11] | Dragomir, S. S. and Agarwal, R. P. Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11, 1998, 91-95. | ||

In article | View Article | ||

[12] | Kirmaci U. S. Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Applied Mathematics and Computation 147, 2004. 137-146. | ||

In article | View Article | ||