**Turkish Journal of Analysis and Number Theory**

## Integral Inequalities for Mappings Whose Derivatives Are s-Convex in the Second Sense and Applications to Special Means for Positive Real Numbers

**MEVLÜT TUNÇ**^{1,}, **SEVIL BALGEÇTİ**^{1}

^{1}Mustafa Kemal University, Faculty of Science and Arts, Department of Mathematics, Hatay, Turkey

### Abstract

In this paper, the authors establish a new type integral inequalities for differentiable s-convex functions in the second sense. By the well-known Hölder inequality and power mean inequality, they obtain some integral inequalities related to the s-convex functions and apply these inequalities to special means for positive real numbers.

**Keywords:** s-convexity, Hermite-Hadamard Inequality, Bullen’s inequality, Special Means.

Received May 07, 2015; Revised April 21, 2016; Accepted April 29, 2016

**Copyright**© 2016 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- MEVLÜT TUNÇ, SEVIL BALGEÇTİ. Integral Inequalities for Mappings Whose Derivatives Are s-Convex in the Second Sense and Applications to Special Means for Positive Real Numbers.
*Turkish Journal of Analysis and Number Theory*. Vol. 4, No. 2, 2016, pp 48-53. http://pubs.sciepub.com/tjant/4/2/5

- TUNÇ, MEVLÜT, and SEVIL BALGEÇTİ. "Integral Inequalities for Mappings Whose Derivatives Are s-Convex in the Second Sense and Applications to Special Means for Positive Real Numbers."
*Turkish Journal of Analysis and Number Theory*4.2 (2016): 48-53.

- TUNÇ, M. , & BALGEÇTİ, S. (2016). Integral Inequalities for Mappings Whose Derivatives Are s-Convex in the Second Sense and Applications to Special Means for Positive Real Numbers.
*Turkish Journal of Analysis and Number Theory*,*4*(2), 48-53.

- TUNÇ, MEVLÜT, and SEVIL BALGEÇTİ. "Integral Inequalities for Mappings Whose Derivatives Are s-Convex in the Second Sense and Applications to Special Means for Positive Real Numbers."
*Turkish Journal of Analysis and Number Theory*4, no. 2 (2016): 48-53.

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### 1. Introduction

**1.1. Definitions**

**De****fi****nition 1.** ^{[10]} *A function** ** **is said to be convex on** ** **if inequality*

(1.1) |

*holds for all** ** **and** ** **We say that** ** **is concave if** ** **is convex*.

**De****fi****nition 2.** ^{[8]} *Let** ** **A function** ** **is said to be s-convex in the second sense if*

(1.2) |

*for all** ** **and** ** **This class of s-convex functions is usually denoted by*

Certainly, s-convexity means just ordinary convexity when *s* = 1.

**1.2. Theorems**

**Theorem 1**. **The Hermite-Hadamard inequality***: Let** ** **be a convex function and** ** **with** ** **The following double inequality*:

(1.3) |

*is known in the literature as Hadamard**’**s inequality (or Hermite-Hadamard inequality) for convex functions. If** ** **is a positive concave function, then the inequality is** **revers**ed*.

**Theorem 2.** ^{[6]} *Suppose that** ** **is an **s**-**convex function in the second sense, where** ** **and let** **, **. **If** ** then the following inequalities hold:*

(1.4) |

*The constant** ** **is the best possible in the second inequality in (1.4). The above inequalities are sharp. If** ** **is an s-concave function in the second sense, then the inequality is reversed*.

**Theorem 3.** *Let** ** **be a convex function on the interval** ** **of real numbers and** ** **with** ** **The inequality*

*is known as Bullen’s inequality for convex functions *[^{[5]}, p.39].

In ^{[4]}, Dragomir and Agarwal obtained inequalities for differentiable convex mappings which are connected to Hadamard’s inequality, as follow:

**Theorem 4.** *Let** ** **be a di**ff**erentiable mapping on** **,** where** **, with** ** **If** ** **is convex on [a; b], then the following inequality holds:*

(1.5) |

In ^{[11]}, Pearce and Pečarić obtained inequalities for differentiable convex mappings which are connected with Hadamard’s inequality, as follow:

**Theorem 5.** *Let** ** **be di**ff**erentiable mapping on** **, where** **, with** ** If** ** **is convex on** ** **for some** **, then the following inequality holds:*

(1.6) |

*If** ** **is concave on** ** **for some** ** **then*

(1.7) |

In ^{[1]}, Alomari, Darus and Kırmacı obtained inequalities for differentiable* s*-convex and concave mappings which are connected with Hadamard’s inequality, as follow:

**Theorem 6.** *Let** ** **be di**ff**erentiable mapping on** ** **such that** **, **where** ** **with** ** **If** ** ** **is **concave on** ** **for some fixed** ** **then the following inequality holds:*

(1.8) |

In ^{[12]}, Tunç and Balgeçti obtained inequalities for differentiable convex functions which are connected with a new type integral inequality, as follow:

**Lemma 1.** *Le** ** **be a di**ff**erentiable function on** ** **If** ** **then*

(1.9) |

*for each** ** **and** *

**Theorem 7.** ^{[12]} *Let** ** be a di**ff**erentiable function on** ** **If** ** **is convex on** **, then*

(1.10) |

*for each*

**Theorem 8.** ^{[12]} *Let** ** **R be a di**ff**erentiable function on** ** **If** ** **is convex on** ** **and** ** **with** ** **then*

(1.11) |

**Theorem 9.** ^{[12]} *Let** ** **be a di**ff**erentiable function on** ** **If** ** **is convex** **on** ** **and** **, then*

(1.12) |

For recent results and generalizations concerning Hadamard’s inequality and concepts of convexity and *s*-convexity see [1-12]^{[1]} and the references therein.

Throughout this paper we will use the following notations and conventions. Let and with and and

be the arithmetic mean, geometric mean, generalized logarithmic mean for respectively.

### 2. Inequalities for s-convex Functions and Applications

**Theorem 10.** Let* ** **be a di**ff**erentiable function on** ** **If** ** **is s-convex on** ** for some **fi**xed** ** **then*

(2.1) |

*for each*

*Proof.* Using Lemma 1 and from properties of modulus, and since is *s*-convex on J, then we obtain

**Proposition 1**.* Let** ** ** **and** ** **then*

(2.2) |

*Proof.* The proof follows from (2.1) applied to the *s*-convex function and

**Proposition 2.** *Let** ** ** ** **then*

(2.3) |

*Proof*. The proof follows from (2.1) applied to the *s*-convex function and with

**Remark 1.** *In (2.1), (2.2), if we take** ** **then (2.1), (2.2) reduces to (1.10), *[^{[12]}, Proposition 2]*, respectively.*

**Theorem 11.** *Let** ** **be a differentiable function on** ** **If** ** **is s-convex on** ** **for some fixed** ** **and** ** **with** ** **then*

(2.4) |

*for each** *

*Proof.* Using Lemma 1 and from properties of modulus, and since is s-convex on J, then we obtain

(2.5) |

Since is *s*-convex, by the Hölder inequality, we have

(2.6) |

and

(2.7) |

It can be easily seen that

(2.8) |

If expressions (2.6)-(2.8), we obtain

The proof is completed.

**Proposition 3**. *Let** ** ** **and** **, then*

(2.9) |

*Proof.* The proof follows from (2.4) applied to the *s*-convex function and

**Proposition 4.** *Let** ** ** **and** **, then*

(2.10) |

*Proof.* The proof follows from (2.4) applied to the s-convex function and

**Remark 2.** *In (2.4), (2.9), if we take** ** **then (2.4), (2.9) reduces to (1.11), *[^{[12]}, Proposition 5]*, respectively.*

**Theorem 12.** *Let** ** **be a di**ff**erentiable function on** **. **If** ** **is s-convex on** ** **for some **fi**xed** ** **and ** **then*

(2.11) |

*Proof.* From Lemma 1 and using the well-known power mean inequality and since is *s*-convex on , we can write

The proof is completed.

**Proposition 5**. *Let** ** ** **and** **, then*

(2.12) |

*Proof*. The proof follows from (2.11) applied to the s-convex function and

**Proposition 6.** Let and , then

(2.13) |

*Proof*. The proof follows from (2.11) applied to the s-convex function and

**Remark 3**. *In (2.11), (2.12), if we take** **, then (2.11), (2.12) reduces to (1.12), *[^{[12]}, Proposition 8]* respectively*.

### References

[1] | M. Alomari, M. Darus, U.S. Kırmacı, Some Inequalities of Hermite-Hadamard type for s-convex Functions, Acta Math. Sci. 31B(4):1643-1652 (2011). | ||

In article | View Article | ||

[2] | P. Burai, A. Házy, and T. Juhász, Bernstein-Doetsch type results for s-convex functions, Publ. Math. Debrecen 75 (2009), no. 1-2, 23-31. | ||

In article | |||

[3] | P. Burai, A. Házy, and T. Juhász, On approximately Breckner s-convex functions, Control Cybernet. 40 (2011), no. 1, 91-99. | ||

In article | |||

[4] | S.S. Dragomir, R.P. Agarwal, 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 | ||

[5] | S. S. Dragomir and C. E. M. Pearce: Selected topics on Hermite-Hadamard inequalities and applications, RGMIA monographs, Victoria University, 2000. [Online:http://www.sta¤.vu.edu.au/RGMIA/monographs/hermite-hadamard.html]. | ||

In article | |||

[6] | S.S. Dragomir, S. Fitzpatrick, The Hadamard’s inequality for s-convex functions in the second sense, Demonstration Math., 32 (4) (1999), 687-696. | ||

In article | |||

[7] | J. Hadamard, Étude sur les propriétés des fonctions entières en particulier d.une function considérée par Riemann, J. Math. Pures Appl. 58 (1893) 171-215. | ||

In article | |||

[8] | H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 48, 100-111, (1994). | ||

In article | View Article | ||

[9] | D.S. Mitrinović, I.B. Lacković, Hermite and convexity, Aequationes Math. 28 (1985) 229-232. | ||

In article | View Article | ||

[10] | D. S. Mitrinović, J. Peµcarić, and A.M. Fink, Classical and new inequalities in analysis, KluwerAcademic, Dordrecht, 1993. | ||

In article | View Article | ||

[11] | C.E.M. Pearce, J. Peµcarić, Inequalities for differentiable mappings with application to special means and quadrature formula, Appl. Math. Lett. 13 (2000) 51-55. | ||

In article | View Article | ||

[12] | M Tunç, S Balgeçti, Some inequalities for differentiable convex functions with applications, http://arxiv.org/pdf/1406.7217.pdf, submitted. | ||

In article | |||