Cite This Article: MEVLÜT TUNÇ, IBRAHİM KARABAYIR, and EBRU YÜKSEL, “On Some Inequalities for Functions Whose Second Derivatives Absolute Values Are S-Geometrically Convex.” Turkish Journal of Analysis and Number Theory, vol. 2, no. 4 (2014): 113-118. doi: 10.12691/tjant-2-4-2.

1. Introduction

Let be a convex mapping defined on the interval I of real numbers and , with . The following double inequalities:

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

In recent years many authors established several inequalities connected to this fact. For recent results, refinements, counterparts, generalizations and new Hermite-Hadamard-type inequalities see [5-9]^[5].

In this section we will present definitions and some results used in this paper.

Definition 1. Let I be an interval in : Then is said to be convex if

(1.1)

for all x,y∈I and t∈ [0,1].

Definition 2. ^[5] Let . A function is said to be s-convex in the second sense if

(1.2)

for all and .

It can be easily checked for s = 1, s-convexity reduces to the ordinary convexity of functions defined on .

Recently, in ^[12], the concept of geometrically and s-geometrically convex functions was introduced as follows.

Definition 3. ^[12] A function is said to be a geometrically convex function if

(1.3)

for all and .

Definition 4. ^[12] A function is said to be a s-geometrically convex function if

(1.4)

for some , where and .

If s = 1, the s-geometrically convex function becomes a geometrically convex function on.

Example 1. ^[12] Let ; , , , and then the function

(1.5)

is monotonically decreasing on (0,1]. For , we have

(1.6)

Hence, is s-geometrically convex on (0,1] for .

2. Hadamardfis Type Inequalities

In order to prove our main theorems, we need the following lemma ^{[1, 3]}.

Lemma 1. ^{[1, 3]} Let be twice differentiable mapping on , with and is integrable on [a,b], then the following equality holds:

(2.1)

A simple proof of this equality can be also done integrating by parts twice in the right hand side. The details are left to the interested reader.

The next theorems gives a new result of the upper Hermite-Hadamard inequality for s-geometrically and geometrically convex functions.

Theorem 1. Let be twice differentiable mapping on , with and is integrable on [a,b] and . If is s-geometrically convex and monotonically decreasing on [a,b]; and then the following inequality holds:

(2.2)

where

(2.3)

Proof. Since is a s-geometrically convex and monotonically decreasing on [a,b], from Lemma 1, we get

If

(2.4)

When , by(2.4), we get

which completes the proof.

Theorem 2. Let f be differentiable on , a,b∈I with and and . If is s-geometrically convex and monotonically decreasing on [a,b] for and ; then

(2.5)

where

Proof. Since is a s-geometrically convex and monotonically decreasing on [a,b], from Lemma 1 and the well known Hölder inequality, we have

(2.6)

If , by (2.4), we obtain

(2.7)

and then from (2.6)-(2.7), (2.8) holds.

(2.8)

where . We have to note that, the Beta and Gamma Functions (see ^[1]), are described respectively, as follows.

and

are used to evaluate the integral

Using the proprieties of Beta function, that is, and , we can achieve that

where , which completes the proof.

Corollary 1. Let be differentiable on , with and . If is s-geometrically convex and monotonically decreasing on [a,b] for and , then

i) When , one has

where

ii) If we take in (2.5), we have for geometrically convex, one has

Theorem 3. Let be twice differentiable on , with and and . If is s-geometrically convex and monotonically decreasing on [a,b], for and , then

(2.9)

where α (u,v) is same with above (2.3).

Proof. Since is a s-geometrically convex and monotonically decreasing on [a,b], from Lemma 1 and the well known power mean integral inequality, we have

When , by (2.4), we get

which completes the proof.

Theorem 4. Let be twice differentiable on , with and and . If is s-geometrically convex and monotonically decreasing on [a,b] for with and , then

where α(u,v) is same with above (2.3) and

Proof. Since is a s-geometrically convex and monotonically decreasing on [a,b], from Lemma 1, we have

(2.10)

for all . Using the well known inequality , on the right side of (2.10), we get

When , by (2.4), we get

and then, we have

We have to note that, using the Beta and Gamma Functions and evaluating the integral, we get

And, using the proprieties of Beta function, that is, and , we achieve

Where , which completes the proof.

3. Applications to Special Means for Positive Numbers

Let

be the arithmetic, logarithmic, generalized logarithmic means for a, b > 0 respectively.

Proposition 1. Let . Then, we have

Proof. The assertion follows from Theorem 1 applied to s-geometrically convex mapping .

Proposition 2. Let , and . Then, we have

Proof. The assertion follows from Theorem 2 applied to s-geometrically convex mapping .

Proposition 3. Let , and . Then, we have

In here, and we can write this; if and , and if and .

Proof. The assertion follows from Theorem 3 applied to s-geometrically convex mapping .

Proposition 4. Let Let ; and . Then, we have

In here, and we can write this; if and d , and if and .

Proof. The assertion follows from Theorem 4 applied to s-geometrically convex mapping .

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