Example 1.2. a) Let X = R and take d(x, y) = |x − y|. This is the most basic and important example. b) More generally, let N ≥ 1, let X = R , and take d(x, y) = ||x − y|| = √∑N i=1(xi − yi). It is very well known but not very obvious that d satisfies the triangle inequality. This is a special case of Minkowski’s Inequality, which will be studied later. c) More generally let p ∈ [1,∞), let N ≥ 1...

The arithmetic-geometric mean inequality in short, AG inequality has been widely used in mathematics and in its applications. A large number of its equivalent forms have also been developed in several areas of mathematics. For probability and mathematical statistics, the equivalent forms of the AG inequality have not been linked together in a formal way. The purpose of this paper is to prove th...

Hölder’s inequality states that ‖x‖p ‖y‖q − 〈x, y〉 ≥ 0 for any (x, y) ∈ Lp(Ω) × Lq(Ω) with 1/p + 1/q = 1. In the same situation we prove the following stronger chains of inequalities, where z = y|y|q−2: ‖x‖p ‖y‖q − 〈x, y〉 ≥ (1/p) [( ‖x‖p + ‖z‖p )p − ‖x + z‖p ] ≥ 0 if p ∈ (1, 2], 0 ≤ ‖x‖p ‖y‖q − 〈x, y〉 ≤ (1/p) [( ‖x‖p + ‖z‖p )p − ‖x + z‖p ] if p ≥ 2. A similar result holds for complex valued fun...

In this paper we extend some theorems published lately on the relationship between convexity/concavity, and subadditivity/superadditivity. We also generalize inequalities of compound functions that refine Minkowski inequality.

We consider a different L-Minkowski combination of compact sets in R than the one introduced by Firey and we prove an L-BrunnMinkowski inequality, p ∈ [0, 1], for a general class of measures called convex measures that includes log-concave measures, under unconditional assumptions. As a consequence, we derive concavity properties of the function t 7→ μ(t 1 pA), p ∈ (0, 1], for unconditional con...

We consider the second variation for the volume of convex bodies associated with the Lp Minkowski-Firey combination and obtain a Poincaré-type inequality on the Euclidean unit sphere Sn−1 . Mathematics subject classification (2010): 52A20.

We prove two kinds of Lyapunov type inequalities for pseudo-integrals. One discusses pseudo-integrals where pseudo-operations are given by a monotone and continuous function g. The other one focuses on the pseudo-integrals based on a semiring 0; 1 ½ Š; sup; ð Þ , where the pseudo-multiplication is generated. Some examples are given to illustrate the validity of these inequalities. As a generali...

In this paper, we establish an Orlicz dual of the log-Aleksandrov–Fenchel inequality, by introducing two new concepts mixed volume measures, and using newly established Aleksandrov–Fenchel inequality. The log-Aleksandrov– Fenchel inequality in special cases yields classical some logarithmic Minkowski type inequalities, respectively. Moreover, is therefore also derived.

The sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body (from below) by the volume of the star body itself is the Busemann-Petty centroid inequality. A decade ago, the Lp analogue of the classical BusemannPetty centroid inequality was proved. Here, the definition of the centroid body is extended to an Orlicz centroid body of a star body, and the corr...