Cite This Article: MEVLÜT TUNÇ, IBRAHİM KARABAYIR, and EBRU YÜKSEL, “On Some Inequalities for Functions Whose Second Derivatives Absolute Values Are SGeometrically Convex.” Turkish Journal of Analysis and Number Theory, vol. 2, no. 4 (2014): 113118. doi: 10.12691/tjant242.
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 HermiteHadamard inequality for convex functions.
In recent years many authors established several inequalities connected to this fact. For recent results, refinements, counterparts, generalizations and new HermiteHadamardtype inequalities see [59]^{[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 sconvex in the second sense if
 (1.2) 
for all and .
It can be easily checked for s = 1, sconvexity reduces to the ordinary convexity of functions defined on .
Recently, in ^{[12]}, the concept of geometrically and sgeometrically 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 sgeometrically convex function if
 (1.4) 
for some , where and .
If s = 1, the sgeometrically 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 sgeometrically 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 HermiteHadamard inequality for sgeometrically and geometrically convex functions.
Theorem 1. Let be twice differentiable mapping on , with and is integrable on [a,b] and . If is sgeometrically convex and monotonically decreasing on [a,b]; and then the following inequality holds:
 (2.2) 
where
 (2.3) 
Proof. Since is a sgeometrically 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 sgeometrically convex and monotonically decreasing on [a,b] for and ; then
 (2.5) 
where
Proof. Since is a sgeometrically 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 sgeometrically 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 sgeometrically 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 sgeometrically 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 sgeometrically 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 sgeometrically 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 sgeometrically convex mapping .
Proposition 2. Let , and . Then, we have
Proof. The assertion follows from Theorem 2 applied to sgeometrically 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 sgeometrically 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 sgeometrically convex mapping .
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