The author’s research devoted to the Hilbert’s double series theorem and its various further extensions are focus of a recent survey article. sharp version inequality result is extended in case not exhaustively investigated non-homogeneous kernel, which mutually covers homogeneous kernel cases as well. Particularly, novel results presented, include upper bounds built exclusively with non-weight...

We describe a framework to build distances by measuring the tightness of inequalities, and introduce the notion of proper statistical divergences and improper pseudo-divergences. We then consider the Hölder ordinary and reverse inequalities, and present two novel classes of Hölder divergences and pseudo-divergences that both encapsulate the special case of the Cauchy-Schwarz divergence. We repo...

Quantum f -divergences are a quantum generalization of the classical notion of f divergences, and are a special case of Petz’ quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f -divergences; special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that t...

Abstract We present ten different characterizations of functions satisfying a weak reverse Hölder inequality on an open subset metric measure space with doubling measure. Among others, we describe these as class $$A_\infty $$ A ∞ weights, which is generalization Muckenh...

We consider the series expansion of [Formula: see text]-Hardy inequality [G. Barbatis, S. Filippas and A. Tertikas, Series for text] Hardy inequalities, Indiana Univ. Math. J. 52 (2003) 171–190], in particular case where distance is taken from an interior point a bounded domain text]. For we improve it by adding as remainder term optimally weighted critical Sobolev norm, generalizing result [S....

Keywords: Nonadditive measure Sugeno integral Comonotone functions Chebyshev's inequality Minkowski's inequality Hölder's inequality a b s t r a c t In this paper strengthened versions of the Minkowski, Chebyshev, Jensen and Hölder inequalities for Sugeno integral and T-(S-)evaluators are given. As an application, some equivalent forms and some particular results have been established. The theo...

Abstract. We show that for each weakly majorizing measure there is a natural metric with respect to which sample paths of stochastic processes are Hölder continuous and their Hölder norm satisfies a strong integrability condition. We call such metric a minorizing metric. The class of minorizing metrics is minimal among all metrics assuring sample Hölder continuity of processes satisfying certai...

The paper addresses probabilistic aspects of the KPZ equation and stochastic Burgers by providing a solution theory that builds on energy [15, 19, 20, 23]. perspective we adopt is to study writing its as [22] plus term can be studied with deterministic PDE considerations. One motivation universality equations for certain class growth models, first in [28]. For this, prove SPDEs general nonlinea...

This paper sheds new light on regularity of multifunctions through various characterizations of directional Hölder/Lipschitz metric regularity, which are based on the concepts of slope and coderivative. By using these characterizations, we show that directional Hölder/Lipschitz metric regularity is stable, when the multifunction under consideration is perturbed suitably. Applications of directi...