A direct approach is used to establish both Ball and Barthe’s reverse isoperimetric inequalities for the unit balls of subspaces of Lp. This approach has the advantage that it completely settles all the open uniqueness questions for these inequalities. Affine isoperimetric inequalities generally have ellipsoids as extremals. The so called reverse affine isoperimetric inequalities usually have s...

It is well-known that the Hölder-Rogers inequality implies the Minkowski inequality. Infantozzi [6] observed implicitely and Royden [15] proved explicitely that the reverse implication is also true. In this note we discuss and give a new proof of this perhaps surprising fact. Mathematics subject classification (2000): 26D15.

For a real-valued nonnegative and log-concave function f defined in R, we introduce a notion of difference function ∆f ; the difference function represents a functional analog on the difference body K + (−K) of a convex body K. We prove a sharp inequality which bounds the integral of ∆f from above, in terms of the integral of f and we characterize equality conditions. The investigation is exten...

In this research article, we investigate reverse Radon's inequality, Bergström's the weighted power mean Schlömilch's Bernoulli's inequality and Lyapunov's with Specht's ratio on time scales. We also present Rogers--Holder's logarithmic The scale dynamic inequalities unify extend some continuous their corresponding discrete quantum versions.

In this paper our primary concern is with the establishment of weighted Hardy inequalities in L(Ω) and Rellich inequalities in L(Ω) depending upon the distance to the boundary of domains Ω ⊂ R with a finite diameter D(Ω). Improved constants are presented in most cases.

Inequalities for convex functions on the lattice of partitions of a set partially ordered by refinement lead to multivariate generalizations of inequalities of Cauchy and Rogers-Hölder and to eigenvalue inequalities needed in the theory of population dynamics in Markovian environments: If A is an n× n nonnegative matrix, n > 1, D is an n× n diagonal matrix with positive diagonal elements, r(·) ...

Gardner and Zhang defined the notion of radial pth mean body (p > –1) in the Euclidean space Rn. In this paper, we obtain inequalities for dual quermassintegrals of the radial pth mean bodies. Further, we establish dual quermassintegrals forms of the Zhang projection inequality and the Rogers-Shephard inequality, respectively. Finally, Shephard’s problem concerning the radial pth mean bodies is...

We develop the theory of twisted L-cohomology and twisted spectral invariants for flat Hilbertian bundles over compact manifolds. They can be viewed as functions on H(M, R) and they generalize the standard notions. A new feature of the twisted L-cohomology theory is that in addition to satisfying the standard L Morse inequalities, they also satisfy certain asymptotic L Morse inequalities. These...

Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the BakryEmery curvature condition, the subordinate semigroup with power α satisfies a dimension-free Harnack inequality provided α ∈ ` 1 2 , 1 ́ , and it satisfies the log-Harnack inequality for all α ∈ (0, 1). Some infinite-dimensional examples are also presen...