Abstract

In this work, by using an integral identity together with both the Hölder integral inequality and the power-mean integral inequality we establish several new inequalities for three-times differentiable arithmetic-harmonically-convex function. Then, using this inequalities, we obtain some new inequalities connected with means.

1. Preliminaries and Fundamentals

It is well known that theory of convex sets and convex functions play an important role in mathematics and the other pure and applied sciences. Convexity theory has appeared as a powerful technique to study a wide class of unrelated problems in pure and applied sciences.

Definition 1.1 A function is said to be convex if the inequality

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

Theorem 1.2 Let be a convex function defined on the interval of real numbers and with . The following inequality

(1.1)

holds.

The Hermite-Hadamard integral inequality is very important for convex functions. See ^{1, 2, 3, 4, 5, 6}, for the results of the generalization, improvement and extention of the famous integral inequality (1.1).

Definition 1.3 ^{7, 8} A function is said to be arithmetic-harmonically (AH) convex function if for all and the equality

(1.2)

holds. If the inequality (1.2) is reversed then the function is said to be arithmetic-harmonically (AH) concave function.

Theorem 1.4 (Hölder Inequality for Integrals) Let and . If and are real functions defined on and if , are integrable functions on then

with equality holding if and only if almost everywhere, where and are constants.

In order to establish some inequalities of Hermite-Hadamard type integral inequalities for-AH convex functions, we will use the following lemma ⁵.

Lemma 1.5 Let be three-times differentiable mapping on and where with , we have the identity

(1.3)

In this study, using both Hölder integral inequality, power-mean integral inequality and the identity (1.3) in order to provide inequality for functions whose third derivatives in absolute value at certain power are arithmetic-harmonically-convex functions.

For shortness, throught this paper, we will use the following notations for special means of two nonnegative numbers with :

1. The arithmetic mean

2. The geometric mean

3. The logarithmic mean

4. The -logaritmic mean

These means are often used in numerical approximation and in other areas. However, the following simple relationships are known in the literature:

It is also known that is monotonically increasing over denoting and In addition, we will use the following notation for shortness:

2. Main Results

Theorem 2.1 Let be a three times-differentiable mapping on , and with . If is an arithmetic-harmonically convex function on the interval , then the following inequality holds:

i) If then,

(2.1)

ii) If then,

where

Proof. i) Let If is an arithmetic-harmonically convex function on the interval , using Lemma 1.5 and

we get

(2.2)

We can write the following inequality according to and :

Therefore, we obtain the desired inequality.

ii) Let Then, substituting in the inequality (2.2), we obtain

(2.3)

This completes the proof of theorem.

Theorem 2.2 Let be a there-times differentiable mapping on , and with . If is an arithmetic-harmonically convex function on the interval , then the following inequality holds:

i) If then,

(2.4)

ii) If then,

where

and

Proof. i) Let . If for is an arithmetic-harmonically convex function on the interval , then by applying the well known Hölder integral inequality to the right-hand side of the Lemma 1.5 and using the following identity

we have

(2.5)

From here, we can write the following inequality

where

ii) Let . In this case, substituting in the inequality (2.5) we get the following inequality:

(2.6)

This completes the proof of the Theorem.

Theorem 2.3 Let be a there-times differentiable mapping on , and with . If is an arithmetic-harmonically convex function on the interval , then the following inequality holds:

i) If then,

(2.7)

ii) If then,

where

Proof. i) Let . If for is an arithmetic-harmonically convex function on the interval , then by applying well known power-mean integral inequality to the right-hand side of the Lemma 1.5 and using

we have

(2.8)

where

ii) Let . By using the inequality (2.8), we have

(2.9)

This completes the proof of the Theorem.

Corollary 2.4 If we take in the inequality (2.7), we get the following inequality:

(2.10)

3. Applications for Special Means

If then the function is an arithmetic harmonically-convex function ⁷. Using this function we obtain following propositions related to means:

Proposition 3.1 Let and . Then we have the following inequalities:

Proof. is convex function for . Therefore, the assertion follows from the inequality (2.1) in the Theorem 2.1, for

Proposition 3.2 Let with , and . Then, we have the following inequality:

Proof. The assertion follows from the inequality (2.4) in the Theorem 2.2. Let

Then is an arithmetic harmonically-convex on and the result follows directly from Theorem 2.2.

Proposition 3.3 Let with , and . Then, we have the following inequality:

(3.1)

Proof. The assertion follows from the inequality (2.6) in the Theorem 2.3. Let

Then is an arithmetic harmonically-convex on and the result follows directly from Theorem 2.3.

Corollary 3.4 If we take in the inequality (3.1), we get the following inequality:

4. Conclusion

We established several new inequalities for three-times differentiable arithmetic-harmonically-convex function and obtained some new inequalities connected with means. Similar method can be applied the different type of convexity.

References