Advances in Mathematical Inequalities and Applications

Why do we study inequalities [...]


Introduction
Why do we study inequalities?The answer to this question was given by Bellman in [1], in a very concrete and elegant fashion: "There are three reasons for the study of inequalities: practical, theoretical and aesthetic.In many practical investigations, it is necessary to bind one quantity to another.The classical inequalities are very useful for this purpose.From the theoretical point of view, very simple questions give rise to entire theories.For example, we may ask when the nonnegativity of one quantity implies that to another.This simple question leads to the theory of positive operators and theory of differential inequalities.Another question which gives rise to much interesting research is that of finding equalities associated with inequalities.We use the principle that every inequality should come from an equality which makes the inequality obvious.Along these lines, we may also look for representation which makes inequalities obvious.Often, these representations are the maxima or minima of certain quantities.Finally, let us turn to aesthetic aspects.As has been pointed out, beauty is in the eyes of the beholder.However, it is generally agreed that certain pieces of music, art or mathematics are beautiful.There is an elegance to inequalities that makes them very attractive".
In this Special Issue, we present new results related to classical inequalities, such as the Jensen inequality, Jensen-Steffensen inequality, Jessen inequality, Grüss inequality, Chebyshev inequality, etc.They have various applications in various branches of mathematics, among which are numerical analysis, probability and statistics, as well as in other sciences, such as information theory.

Statistics of the Special Issue
A total of 30 papers were submitted for this Special Issue, of which 10 were published (33.33%) and 20 were rejected (66.67%), indicating a rigorous peer review process. . .,xn) were defined.These suitable extensions were applied to systems of logical equations.Specifically, the system of m logical equations is first transformed in Rn into an equivalent system of m smooth rational equations (SmSRE) in a constructive way, without adding any equations (field equations or otherwise), such that the solution of SmSRE can be reduced to the problem minimization of the objective function, and any numerical optimization methods can be applied, since the objective function will be infinitely differentiable.Transforming SmSRE into an equivalent system of m polynomial equations (SmPE) means that any symbolic https://www.mdpi.com/2227-7390/10/6/947. Motivated by recent investigations relating the sharpness of the Jensen inequality, this paper concerns with the sharpness of the converse of the Jensen inequality.These results are then used for deriving new inequalities for different types of generalized fdivergences.As divergences measure the differences between probability distributions, these new inequalities are then applied on the Zipf-Mandelbrot law as a special kind of a probability distribution.

5.
Alomari The Jensen inequality is considered one of the most consequential inequalities, finding a variety of applications within various science fields.For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature on mathematical inequalities.The main goal of this paper is to find some new bounds for the Jensen inequality using certain classes of doubly differentiable convex functions.The authors obtain the proposed bounds by using the power mean and the Hölder inequality, the notion of convexity, and the Jensen inequality for concave functions.The authors derive several inequalities for power and quasi-arithmetic means as an outcome of the main results.They also establish several improvements of the Hölder inequality and present some applications of the main results in information theory.

Acknowledgments to the Authors and Reviewers
As the Guest Editor of the Special Issue "Advances in Mathematical Inequalities and Applications", I am grateful to all authors who contributed.I would also like to thank all the reviewers for their careful work and valuable comments, which have helped improve the quality of the submitted papers.

Conclusions
The goal of this Special Issue was to present some new results in the field of mathematical inequalities and their applications.It is my hope that these selected research papers will be recognized by the international scientific community as interesting and significant, and that they may form the basis for further research in the field of mathematical inequalities and their applications.
The authors ascertain a variant of the Jessen-type inequality for a semigroup of positive linear operators, defined on a Banach lattice algebra.The corresponding mean value theorems lead to a new family of mean operators.