Reversing Jensen’s Inequality for Information-Theoretic Analyses

In this work, we propose both an improvement and extensions of a reverse Jensen inequality due to Wunder et al. (2021). The new proposed inequalities are fairly tight and reasonably easy to use in a wide variety of situations, as demonstrated in several application examples that are relevant to information theory. Moreover, the main ideas behind the derivations turn out to be applicable to generate bounds to expectations of multivariate convex/concave functions, as well as functions that are not necessarily convex or concave.

Application of the Pick Function in the Lieb Concavity Theorem for Deformed Exponentials

The Lieb concavity theorem, successfully solved in the Wigner–Yanase–Dyson conjecture, is an important application of matrix concave functions. Recently, the Thompson–Golden theorem, a corollary of the Lieb concavity theorem, was extended to deformed exponentials. Hence, it is worthwhile to study the Lieb concavity theorem for deformed exponentials. In this paper, the Pick function is used to obtain a generalization of the Lieb concavity theorem for deformed exponentials, and some corollaries associated with exterior algebra are obtained.

Bounds for the Differences between Arithmetic and Geometric Means and Their Applications to Inequalities

Refining and reversing weighted arithmetic-geometric mean inequalities have been studied in many papers. In this paper, we provide some bounds for the differences between the weighted arithmetic and geometric means , using known inequalities. We improve the results given by Furuichi-Ghaemi-Gharakhanlu and Sababheh-Choi. We also give some bounds on entropies, applying the results in a different approach. In Section 4, we explore certain convex or concave functions, which are symmetric functions on the axis t=1/2.

Fractional Integral Inequalities via Atangana-Baleanu Operators for Convex and Concave Functions

Recently, many fractional integral operators were introduced by different mathematicians. One of these fractional operators, Atangana-Baleanu fractional integral operator, was defined by Atangana and Baleanu (Atangana and Baleanu, 2016). In this study, firstly, a new identity by using Atangana-Baleanu fractional integral operators is proved. Then, new fractional integral inequalities have been obtained for convex and concave functions with the help of this identity and some certain integral inequalities.

Creating new problems on proving inequalities, finding maximum and minimum values based on the critical properties and tangent inequalities of convex and concave functions

In this paper, we present some ideas and methods to create new problems of proving inequalities, problems of finding maximum and minimum values. Based on the maximum and minimum properties and tangent inequalities of convex and concave functions, we propose some ideas and methods to create new problems. We make all ideas and methods to be real via many specific functions. Especially, we combine the ideas and methods with equivalent transforms, Cauchy-Schwarz inequality, and inequality of arithmetic and geometric means to create new hard problems. New proposed examples, they have showed that our ideas and methods are important and efficient to lecturers at high schools and universities in giving questions in examinations, especially in examinations of selecting good students at levels, in Olympic examinations for high school and university students.

Stochastic forms of functional isoperimetric inequalities

The Brunn-Minkowski and Prekopa-Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions, so it seems that convexity has no special significance. On the other hand, it was recently shown that the Brunn-Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. In addition, a number of other geometric inequalities for convex sets arising from Brunn's concavity principle have recently been shown to yield local stochastic formulations, e.g., the Blaschke-Santalo inequality. In the first part of this dissertation, we study reverse inequalities for functionals of polar convex bodies invariant under the general linear group. We strengthen planar isoperimetric inequalities; we do that by attaching a stochastic model to some classical ones, such as Mahler's Theorem, and a reverse Lutwak-Zhang inequality, for the polar of L[subscript p] centroid bodies. In particular, we obtain the dual counterpart to a result of Bisztriczky-Boroczky. For the rest of the dissertation, we initiate a systematic study of stochastic isoperimetric inequalities for random functions. We show that for the subclass of log-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prekopa-Leindler inequality. Ultimately, we extend the latter result by developing stochastic geometry of s-concave functions. In this way, we establish local versions of dimensional forms of Brunn's principle a la Borell, Brascamp-Lieb, and Rinott. To do so, we define shadow systems of sconcave functions and revisit Rinott's approach in the context of multiple integral rearrangement inequalities.

Some generalizations of the Hermite–Hadamard integral inequality

In this article we give two possible generalizations of the Hermite–Hadamard integral inequality for the class of twice differentiable functions, where the convexity property of the target function is not assumed in advance. They represent a refinement of this inequality in the case of convex/concave functions with numerous applications.