1. Introduction

In the studies of inequality theory, the efficiency of obtained results can be observed via applicability in mathematics or in other applied sciences, the simplicity and esthetic. Lots of generalization or improvement of the previous results can be found in the literature. Unchallenged the fact that some of the obtained results have enormous importance because of the utilization in different fields. Besides several definitions of real functions, the concept of convex functions comes to mind with powerful relationship to inequalities. The notion of convexity can be given as follows:

A function is an interval, is said to be a convex function on if

holds for all and

Several researchers have studied on convex functions and they contributed numerous inequalities. For example, the classical Hermite-Hadamard inequality that gives us an estimate of the mean value of a convex function is quite useful. This famous integral inequality can be traced back to the papers presented by Hermite (see ^[15]) and Hadamard (see ^[14]). Furthermore, in 1906, Fejér, while studying trigonometric polynomials, obtained inequalities which generalize Hermite-Hadamard integral inequality (see ^[17]). Another striking inequality is Ostrowski’s inequality for differentiable functions that were proven by Ostrowski in 1938 (see ^[16]).

In ^[7], Niculescu mentioned log-convex functions as follows:

A positive function defined on an interval (or, more generally, on a convex subset of some vector space) is called log-convex if is a convex function of or equivalently, if for any two points and in its domain and any in we have

A function f is called log-concave if the inequality above works in the reverse way (that is, when is log-convex). The arithmetic mean-geometric mean inequality easily yields that every log-convex function is also convex.

Many different extensions, generalizations and improvements related to log-convex functions can be found in [2-13]^[2].

The main purpose of this paper is to prove some new integral inequalities for functions whose derivatives are logarithmically convex functions by using some identities which are obtained by Tseng et al. in ^[1]. We will give some applications of our results in numerical integration via quadrature formulas.

2. Main Results

In sequel of paper, we will denote and In order to prove our main results, we need the following lemmas which have been established by Tseng et al. in ^[1].

Lemma 1. [^[1], Lemma 1] Let be a differentiable mapping on with and If then, for all then

Lemma 2. [^[1], Theorem 2] Under the assumptions of Lemma 1, we have

where

Define

Then

Theorem 1. Let is a differentiable mapping on If is logarithmically convex for and is a continuous mapping on Then, for all ,

Proof. From Lemma 1 and by using the power mean inequality, we can write

Since is logarithmically convex, we obtain

By taking into account that

the proof is completed.

Proposition 1. Under the assumptions of Theorem 1, if we use the function we obtain

for all

Corollary 1. Under the assumptions of Theorem 1, if we choose then

Corollary 2. Let be symmetric to and in Theorem 1. Then

(2.1)

Remark 1. The above inequality is the "weighted trapezoid" inequality for the logarithmically convex functions.

Corollary 3. If we choose in the inequality (2.1), then

Theorem 2. Under the assumptions of Theorem 1 the inequality:

(2.2)

holds for

Proof. From Lemma 2 and by using the power mean inequality, we have

Since is logarithmically convex, we obtain

(2.3)

By making use of the neccessary process, one can see that

By substituting these equalities in (2.3), the proof is completed.

Proposition 2. Under the assumptions of Theorem 1, if we use the function we obtain

for all

Remark 2. The inequality (2.2) is the "weighted Ostrowski" inequality for the logarithmically convex functions.

Corollary 4. If we choose in the inequality (2.2), then:

Remark 3. The above inequality is the "weighted midpoint" inequality for the logarithmically convex functions.

3. Applications in Numerical Integration

In this section, we will use the following notation:

Let be a continuous mapping be an integrable mapping and let be a division of such that and

(1) The trapezoidal formula

(2) The weighted trapezoidal formula

(3) The midpoint formula

(4) The weighted midpoint formula

(5) The approximation error of by

(6) The approximation error of by

(7) The approximation error of by

(8) The approximation error of by

Proposition 3. Let is a differentiable mapping on If is logarithmically convex for and let be a continuous mapping on then

for division d of

Proof. By applying Corollary 1 on we have

By summing over from to , we acquire

The required proof is complete.

Proposition 4. Under the assumptions of Corollary 3, we have

Proof. By a similar argument to the proof of Proposition 1 and by using Corollary 3, we get the desired result.

Proposition 5. Under the assumptions of Corollary 4, we have

Proof. The proof is immediately follows from Corollary 4. We omit the details.

Proposition 6. If we choose in Corollary 4, we have

Proof. By using the midpoint formula and Corollary 4, the proof is obvious.

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