Improvements of Logarithmic and Identric Mean Inequalities for Scalars and Operators

Abstract

In this article, we provide refined inequalities for a convex Riemann’s integrable function using refinements of the classical Hermite-Hadamard inequality. The obtained results are applied on special functions to establish new improvements of inequalities on the weighted logarithmic mean and weighted identric mean. Moreover, corresponding operator inequalities are introduced based on the scalar inequalities and the monotonicity property for operators.

1. Introduction

The theory of means has grown to occupy a prominent position in mathematics with hundreds of papers on the subject appearing every year. It has applications in so many diverse field, these include electrostatics, heat conduction, chemistry, and even medicine.

The inequalities on means have attracted the attention of many mathematicians because of its implications. Pal et al. [1], introduced the weighted logarithmic mean and the weighted identric mean and presented the inequalities among weighted means as where the weighted geometric mean , the weighted arithmetic mean , the weighted logarithmic mean and the weighted identric mean for and

The classical Hermite-Hadamard inequality, which was first discovered by Hermite in 1883 and independently proved in 1893 by Hadamard, involves a fundamental characteristic of convex functions and it states that if is a convex real valued function on the interval , then

As a generalization of the Hermite-Hadamard inequality, Pal et al. [1] produced the following new inequality where for a convex Riemann’s integrable function , and . A refinement of inequality (6) was given by Furuichi and Minculete [2] and some interesting related integral inequalities can be found in [3], [4–6], and [7, 8].

A motivating problem related to the Hermite-Hadamard inequality is the refinements of this inequality. One of these refinements, due to Feng [9], asserts that

On the other hand, Burqan [10] constructed a refinement of the first inequality in the Hermite-Hadamard inequality as follows:

In this work, we apply the above refinements of the Hermite-Hadamard inequality to obtain a refinement of the inequalities given in (6) that produces improvements of the inequalities in (1) and (2).

2. Main Results

In this section, we obtain the refined inequalities for (6) by using the refinements of the Hermite-Hadamard inequality given in (8) and (9).

Theorem 1. For every convex Riemann’s integrable function and , we have where

Proof. Applying the refinements (8) and (9) of the Hermite-Hadamard inequality on two intervals and , we obtain, respectively. Multiplying inequalities in (12) and (13) by and , respectively, and summing each side, we obtain Which is equivalent to Now, by replacing the variables such as in the first integral and in the second integral given in (15), respectively, we have From (16) and (17) inequality, (15) becomes Finally, the proof is completed by noting that

In the following results, we get refinements of some inequalities that introduced by Raïssouli and Furuichi [11].

Corollary 2. For and we have

Proof. Applying Theorem 1 on the convex function , we get for . By elementary calculations, we get Replacing and with and , respectively, we obtain the inequalities (20) for and
Dividing in all sides of the obtaining inequalities and putting , we have Putting and in (23) and then multiplying by to all sides to get for
By elementary calculations, we have the inequalities for

Therefore, we complete the proof by putting for any in (25) and then multiplying all sides by .

Corollary 3. For and we have

Proof. Applying Theorem 1 on the convex function , we get for By elementary calculations, we have Now, the calculations of the middle part of the above inequality (28) are given by Substituting (28) in (29) and using the monotonicity function, we get the required result.
Our obtained results in the previous section can be extended to the operator inequalities, indicate the space of bounded linear operators on a the Hilbert space by . For we write to mean is positive definite, particularly, denotes that is positive definite. The definitions of means in the scalar case can be used to raise these definitions to the operator level. For and positive definite operators , the weighted geometric operator mean and arithmetic operator mean are defined as follows: It is known that an operator mean is associated with the representing function with a mean for positive numbers The weighted logarithmic operator mean is defined through the representing function for and the weighted identric operator mean is defined through the representing function for Now, we are ready to present operator inequalities related to the logarithmic and identric means. The proofs are passed on the obtained scalar inequalities and the monotonicity property for operators which asserts that if is self-adjoint with a spectrum and if and are continuous functions such that for all then .

Theorem 4. Let be such that and . Then,

Proof. Let and Then, we have and so Corollary 2 implies that where By monotonicity property, we have By putting and multiplying both sides by , we get In the following theorem, we introduce new operator inequalities related to the identric mean.

Theorem 5. Let be such that and . Then,

Proof. Let and Then, we have and so Corollary 3 implies that where By monotonicity property, we have By putting and multiplying both sides by . After simple computations, we get Thus, the proof of Theorem 5 is completed.

3. Conclusions

Mean functions are effective tools to obtain new approaches to inequalities and give estimations in numerical integration, statistics, and also real world problems. The concept of inequalities that will shed light on some unknown points about approximation theory and convex programming is a novelty in applied sciences as well as in mathematics. In addition, it has increased its contribution to the literature with its applications in areas such as engineering, statistics, and information theory. In this research, refined inequalities for a convex Riemann’s integrable function using refinements of the classical Hermite-Hadamard were introduced and implemented to get improvements of the weighted logarithmic and identric mean inequalities. In recent years, progress and achievements in this field have been very rich; we hope this research is one of these achievements that may open new doors for researchers to work on convex Riemann’s integrable functions in different directions.

Data Availability

No underlying data was collected or produced in this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.