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Research Article

Open Access Peer-reviewed

Huriye Kadakal^{ }

Received October 11, 2018; Revised December 24, 2018; Accepted February 25, 2019

In this paper, we introduce a new class of extended (m,r)-convex function and we establish the Hermite-Hadamard inequality for (m,r)-convex functions. Some special cases are discussed. Results represent significant refinement and improvement of the previous results. The definition of (m,r)-convex function is given for the first time in the literature and moreover, the results obtained in special cases coincide with the well-known results in the literature.

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

is valid for all and If this inequality reverses, then is said to be concave on interval This definition is well known in the literature. Denote by the set of the convex functions on the interval

**Definition 2****:**** *** *be a convex function defined on the interval* ** *of real numbers and* ** *with* ** *The following inequality

(1.1) |

holds. This double inequality is known in the literature as Hermite-Hadamard integral inequality for convex functions.

The inequality (1.1) is well known as the Hermite Hadamard integral inequality. Readers can find more information in ^{ 1}. Some refinements of the Hermite-Hadamard integral inequality on convex functions have been extensively investigated by a number of authors (see ^{ 2, 3, 4}).

**Definition 3****:**** **^{ 5} The function* ** *is said to be m-convex, where* **, *if for every* ** *and* ** *we have

Denote by the set of the m-convex functions on for which

In ^{ 5}, the author gave the following theorem about the inequalities of Hermite-Hadamard type for m-convex functions.

**Theorem 1****:**** **Let* ** *be a* **m**-*convex functions* *with* ** *If* ** *and* ** *then* *one has the inequality:

**Definition 4****:**** **^{ 6} A positive function* ** is *called* **-*convex on interval* ** *if for each* ** *and* *

If the equality is reversed, then the function is said to be -concave.

The definition of -convexity naturally complements the concept of -concavity, in which the inequality is reversed ^{ 7} and which plays an important role in statistics. It is obvious 0-convex functions are simply -convex functions, -convex functions are ordinary convex functions and -convex functions are arithmetically harmonically convex. If is r-convex in the interval then is a convex function and If is -concave in the interval then is a concave function We note that if and are convex and is increasing, then is convex; moreover, since it follows that a -convex function is convex.

Some refinements of the Hadamard integral inequality for -convex functions could be found in ^{ 8, 9}. In ^{ 10}, Bessenyei studied Hermite-Hadamard-type inequalities for generalized -convex functions. In ^{ 8}, the authors showed that if * *is -convex in and then

In ^{ 11}, the authors show that is -convex in the interval and then

and they prove the following inequality for -convex functions:

The main aim of this paper is to establish new inequalities of Hermite-Hadamard type for the class of functions whose derivatives in absolutely value at certain powers are (m,r)-convex.

**Defi****nition 5: **A positive function* ** *is called* *(m,r)*-*convex on interval* ** *if for each* ** *and* *

(2.1) |

If the equality is reversed, then the function is said to be -concave.

**Theorem 2****:**** **Let* ** *be* *a* *(m,r)*-*convex and* ** *and* ** *If* ** *and* * then the following inequalities hold:

(2.2) |

**Proof:** Suppose that

i. Using the definition of (m,r)-convexity and changing variable as we get

For we obtain,

ii. Let For we obtain,

For we have

This completes the proof of theorem.

**Corollary 1****:**** **Suppose that all the assumptions of Theorem 2 are satisfied. In the inequality* *2.2*, *If we choose* ** *we obtain the inequality in ^{ 11}.

In ^{ 12}, İşcan obtained main results using the following lemma. We will use the same lemma to obtain the main results for (m,r)-convex functions.

**Lemma 1****:**** *** *Let* ** *be differentiable mapping on* ** ** *with* ** *If* ** *and* ** ** *then following equality holds:

(2.3) |

**Theorem 3****:**** **Let be a differentiable mapping on such that where with If is (m,r)-convex on for some fixed with and then the following inequalities holds for

(2.4) |

**Proof:** Let From Lemma 1 and the definition of (m,r)-convexity of that is,

we have

Let From Lemma 1, the Power-mean integral inequality and the (m,r)-convexity of we can write

(2.5) |

Let calculate the integral in (2.5) respectively: Firstly, sample calculation give us

(2.6) |

Using the partial integration, we get

(2.7) |

Substituting (2.6) and (2.7) inequalities in (2.5), we obtain

**Corollary 2****:**** **Under the conditions of Theorem 3,* *

(i) In the inequality (2.4), for* ** *we get the following inequality:

(ii) In the inequality (2.4), for* ** ** *and * *we get the following:

where* ** *is the arithmetic mean.

(iii) In the inequality (2.4), for* ** ** ** *and* ** *we get:

where* ** *is the arithmetic mean. This inequality coincides with in ^{ 12} for* *

**Theorem 4****:**** *** *Let* ** *be a differentiable mapping on* ** *such that* ** *where* ** *with* ** If ** *is (m,r)*-*convex on the interval* ** *for some fixed* ** ** *with* ** *and and* * then the following inequalities holds for

(2.8) |

where

**Proof:** Using Lemma 1, the Hölder inequality and (m,r)-convexity of we get

where we use the fact that

**Corollary 3****:**** **Under the conditions of Theorem 4,

(i) In the inequality (2.8), for* ** *we get the following inequality:

(ii) In the inequality (2.8), for* ** *and* ** *we get the following inequality:

where * *is the arithmetic mean.* *

(iii) In the inequality (2.8), for* ** ** *and* ** *we get the following inequality:

where* ** *is the arithmetic mean.

**Theorem 5****:**** ** Let be a differentiable mapping on such that where with If is (m,r)-convex on the interval for some fixed with and then the following inequalities holds for

(2.9) |

**Proof:** From Lemma 1 and using the Power-mean integral inequality and (m,r)-convexity of we get

**Corollary 4****:**** **Under the conditions of the Theorem 5,

(i) In the inequality (2.9), if we choose* ** *we obtain the following inequality:

(ii) In the inequality (2.9), if we choose* * and* ** *we obtain the following inequality:

(iii) In the inequality (2.9), if we choose* ** ** *and* ** *we obtain the following inequality:

**Corollary 5****:**** **Suppose that all the assumptions of Theorem 5 are satisfied. In the inequality (2.9), if we choose* ** ** ** *and* ** *we get the following inequality:

where is the arithmetic mean.

[1] | Hadamard, J., Étude sur les proprietes des fonctions entieres en particulier d’une fonction considérée par Riemann. J. Math. Pures Appl. 58, 171-215 (1893). | ||

In article | |||

[2] | Dragomir, S.S., Refinements of the Hermite-Hadamard integral inequality for log-convex functions. Aust. Math. Soc. Gaz. 28(3), 129-134 (2001). | ||

In article | |||

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

In article | |||

[4] | Mihaly, B., Hermite-Hadamard-type inequalities for generalized convex functions. J. Inequal. Pure Appl. Math. 9(3), Article ID 63 (2008) (PhD thesis). | ||

In article | |||

[5] | Dragomir, S.S. and Toader, G.H., Some inequalities for m-convex functions, Studia Univ. Babeş-Bolyai, Math., 38(1993), 21-28. | ||

In article | |||

[6] | Pearce, C.E.M, Pečaric, J. and Šimić, V., Stolarsky means and Hadamard’s inequality. J. Math. Anal. Appl. 220, 99-109 (1998). | ||

In article | View Article | ||

[7] | Uhrin, B., Some remarks about the convolution of unimodal functions, Ann. Probab. 12 1984, 640-645. | ||

In article | View Article | ||

[8] | Ngoc, N.P.N, Vinh, N.V. and Hien, P.T.T., Integral inequalities of Hadamard type for r-convex functions. Int. Math. Forum 4(35), 1723-1728 (2009). | ||

In article | |||

[9] | Yang, G.S., Refinementes of Hadamard inequality for r-convex functions. Indian J. Pure Appl. Math. 32(10), 1571-1579 (2001). | ||

In article | |||

[10] | Bessenyei, M., Hermite-Hadamard-type inequalities for generalized 3-convex functions, Publ. Math. (Debr.) 65(1-2), 223-232 (2004). | ||

In article | View Article | ||

[11] | Zabandan, G., Bodaghi, A., and Kılıçman, A., The Hermite-Hadamard inequality for r-convex functions, Journal of Inequalities and Applications 2012, 2012:215. | ||

In article | View Article | ||

[12] | İşcan, İ., Hermite-Hadamard type inequalities for functions whose derivatives are (α,m)-convex, International Journal of Engineering and Applied Sciences, (EAAS), 2 (3) (2013), 69-78. | ||

In article | |||

Published with license by Science and Education Publishing, Copyright © 2019 Huriye 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/

Huriye Kadakal. (m,r)-Convex Functions. *Turkish Journal of Analysis and Number Theory*. Vol. 7, No. 1, 2019, pp 23-32. http://pubs.sciepub.com/tjant/7/1/5

Kadakal, Huriye. "(m,r)-Convex Functions." *Turkish Journal of Analysis and Number Theory* 7.1 (2019): 23-32.

Kadakal, H. (2019). (m,r)-Convex Functions. *Turkish Journal of Analysis and Number Theory*, *7*(1), 23-32.

Kadakal, Huriye. "(m,r)-Convex Functions." *Turkish Journal of Analysis and Number Theory* 7, no. 1 (2019): 23-32.

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[1] | Hadamard, J., Étude sur les proprietes des fonctions entieres en particulier d’une fonction considérée par Riemann. J. Math. Pures Appl. 58, 171-215 (1893). | ||

In article | |||

[2] | Dragomir, S.S., Refinements of the Hermite-Hadamard integral inequality for log-convex functions. Aust. Math. Soc. Gaz. 28(3), 129-134 (2001). | ||

In article | |||

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

In article | |||

[4] | Mihaly, B., Hermite-Hadamard-type inequalities for generalized convex functions. J. Inequal. Pure Appl. Math. 9(3), Article ID 63 (2008) (PhD thesis). | ||

In article | |||

[5] | Dragomir, S.S. and Toader, G.H., Some inequalities for m-convex functions, Studia Univ. Babeş-Bolyai, Math., 38(1993), 21-28. | ||

In article | |||

[6] | Pearce, C.E.M, Pečaric, J. and Šimić, V., Stolarsky means and Hadamard’s inequality. J. Math. Anal. Appl. 220, 99-109 (1998). | ||

In article | View Article | ||

[7] | Uhrin, B., Some remarks about the convolution of unimodal functions, Ann. Probab. 12 1984, 640-645. | ||

In article | View Article | ||

[8] | Ngoc, N.P.N, Vinh, N.V. and Hien, P.T.T., Integral inequalities of Hadamard type for r-convex functions. Int. Math. Forum 4(35), 1723-1728 (2009). | ||

In article | |||

[9] | Yang, G.S., Refinementes of Hadamard inequality for r-convex functions. Indian J. Pure Appl. Math. 32(10), 1571-1579 (2001). | ||

In article | |||

[10] | Bessenyei, M., Hermite-Hadamard-type inequalities for generalized 3-convex functions, Publ. Math. (Debr.) 65(1-2), 223-232 (2004). | ||

In article | View Article | ||

[11] | Zabandan, G., Bodaghi, A., and Kılıçman, A., The Hermite-Hadamard inequality for r-convex functions, Journal of Inequalities and Applications 2012, 2012:215. | ||

In article | View Article | ||

[12] | İşcan, İ., Hermite-Hadamard type inequalities for functions whose derivatives are (α,m)-convex, International Journal of Engineering and Applied Sciences, (EAAS), 2 (3) (2013), 69-78. | ||

In article | |||