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

MEVLÜT TUNÇ*, SEVIL BALGEÇTİ

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.

Citation: 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, 4 (2). 48-53. January 2016. doi:10.12691/tjant-4-2-5

1. Introduction

1.1. Definitions

Definition 1. [10] A function is said to be convex on if inequality

[picture] | (1.1)

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

Definition 2. [8] Let A function is said to be s-convex in the second sense if

[picture] | (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:

[picture] | (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 reversed.

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

[picture] | (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 differentiable mapping on , where , with If is convex on [a; b], then the following inequality holds:

[picture] | (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 differentiable mapping on , where , with If is convex on for some , then the following inequality holds:

[picture] | (1.6)

If is concave on for some then

[picture] | (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 differentiable mapping on such that , where with If is concave on for some fixed then the following inequality holds:

[picture] | (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 differentiable function on If then

[picture] | (1.9)

for each and

Theorem 7. [12] Let be a differentiable function on If is convex on , then

[picture] | (1.10)

for each

Theorem 8. [12] Let R be a differentiable function on If is convex on and with then

[picture] | (1.11)

Theorem 9. [12] Let be a differentiable function on If is convex on and , then

[picture] | (1.12)

For recent results and generalizations concerning Hadamard’s inequality and concepts of convexity and s-convexity see [1-12] 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 differentiable function on If is s-convex on for some fixed then

[picture] | (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

[picture] | (2.2)

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

Proposition 2. Let then

[picture] | (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

[picture] | (2.4)

for each

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

[picture] | (2.5)

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

[picture] | (2.6)

and

[picture] | (2.7)

It can be easily seen that

[picture] | (2.8)

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

The proof is completed.

Proposition 3. Let and , then

[picture] | (2.9)

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

Proposition 4. Let and , then

[picture] | (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 differentiable function on . If is s-convex on for some fixed and then

[picture] | (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

[picture] | (2.12)

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

Proposition 6. Let and , then

[picture] | (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).
[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.
[3]   | P. Burai, A. Házy, and T. Juhász, On approximately Breckner s-convex functions, Control Cybernet. 40 (2011), no. 1, 91-99.
[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.
[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].
[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.
[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.
[8]   | H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math., 48, 100-111, (1994).
[9]   | D.S. Mitrinović, I.B. Lacković, Hermite and convexity, Aequationes Math. 28 (1985) 229-232.
[10]   | D. S. Mitrinović, J. Peµcarić, and A.M. Fink, Classical and new inequalities in analysis, KluwerAcademic, Dordrecht, 1993.
[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.
[12]   | M Tunç, S Balgeçti, Some inequalities for differentiable convex functions with applications, http://arxiv.org/pdf/1406.7217.pdf, submitted.