Refinements of the Converse Hölder and Minkowski Inequalities

<abstract><p>In this paper, we introduce novel extensions of the reversed Minkowski inequality for various functions defined on time scales. Our approach involves the application of Jensen's and Hölder's inequalities on time scales. Our results encompass the continuous inequalities established by Benaissa as special cases when the time scale $ \mathbb{T} $ corresponds to the real numbers (when $ \mathbb{T = R} $). Additionally, we derive distinct inequalities within the realm of time scale calculus, such as cases $ \mathbb{ T = N} $ and $ q^{\mathbb{N}} $ for $ q > 1 $. These findings represent new and significant contributions for the reader.</p></abstract>

In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb inequality for multiple power concave functions via relative asymmetry.

Preface. Organization of the Book. Notations. I. Convex Functions and Jensen's Inequality. II. Some Recent Results Involving Means. III. Bernoulli's Inequality. IV. Cauchy's and Related Inequalities. V. Hoelder and Minkowski Inequalities. VI. Generalized Hoelder and Minkowski Inequalities. VII. Connections Between General Inequalities. VIII. Some Determinantal and Matrix Inequalities. IX. Cebysev's Inequality. X. Gruss' Inequality. XI. Steffensen's Inequality. XII. Abel's and Related Inequalities. XIII. Some Inequalities for Monotone Functions. XIV. Young's Inequality. XV. Bessel's Inequality. XVI. Cyclic Inequations. XVII. The Centroid Method in Inequalities. XVII. Triangle Inequalities. XVIII. Norm Inequalities. XIX. More on Norm Inequalities. XX. Gram's Inequality. XXI. Frejer-Jackson's Inequalities and Related Results. XXII. Mathieu's Inequality. XXIII. Shannon's Inequality. XXIV. Turan's Inequality from the Power Sum Theory. XXV. Continued Fractions and Pade Approximation Method. XXVI. Quasilinearization Methods for Proving Inequalities. XXVIII. Dynamic Programming and Functional Equation Approaches to Inequalities. XXIX. Interpolation Inequalities. XXX. Minimax Inequalities. Name Index.

Classical and new inequalities in analysis : D.S. Mitrinovic,J.E. Pecaric and A.M. Fink,Kluwer, Dordrecht, 1992. 725 pp., Dfl.425, US$285, UK£170.50, ISBN 0-792-32064-6

We introduce the interval Darboux delta integral (shortly, the $ID$ $\Delta$-integral) and the interval Riemann delta integral (shortly, the $IR$ $\Delta$-integral) for interval-valued functions on time scales. Fundamental properties of $ID$ and $IR$ $\Delta$-integrals and examples are given. Finally, we prove Jensen's, H\"{o}lder's and Minkowski's inequalities for the $IR$ $\Delta$-integral. Also, some examples are given to illustrate our theorems.

Convexity of solutions and Brunn–Minkowski inequalities for Hessian equations in [formula omitted

Mass transportation proofs of free functional inequalities, and free Poincaré inequalities

The Brunn–Minkowski inequality gives a lower bound of the Lebesgue measure of a sum-set in terms of the measures of the individual sets. It has played a crucial role in the theory of convex bodies. This topic has many interactions with isoperimetry or functional analysis. Our aim here is to report some recent aspects of these interactions involving optimal mass transport or the Heat equation. Among other things, we will present Brunn–Minkowski inequalities for flat sets, or in Gauss space, as well as local versions of the theorem which apply to the study of entropy production in the central limit theorem.

Support functions and analytic inequalities

Some inequalities for superadditive functionals defined on convex cones in linear spaces are given, with applications for various mappings associated with the Jensen, Hölder, Minkowski and Schwarz inequalities.

Brunn–Minkowski inequalities for variational functionals and related problems

Inverse Hölder inequalities in one and several dimensions

The Brunn–Minkowski and Prékopa–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 signficance. 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. We show that for the subclass of [Formula: see text]-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prékopa–Leindler inequality.

Hölder-Minkowski type inequality for generalized Sugeno integral

Dimensional variance inequalities of Brascamp–Lieb type and a local approach to dimensional Prékopaʼs theorem