Hardy–Hilbert type inequality in reproducing kernel Hilbert space: its applications and related results

Abstract

ABSTRACTA reproducing kernel Hilbert space is a Hilbert space of complex-valued functions on a (non-empty) set Ω, which has the property that point evaluation is continuous on for all . Then the Riesz representation theorem guarantees that for every there is a unique element such that for all . The function is called the reproducing kernel of and the function is the normalized reproducing kernel in . The Berezin symbol of an operator A on a reproducing kernel Hilbert space is defined by The Berezin number of an operator A on is defined by The so-called Crawford number is defined by We also introduce the number defined by It is clear that By using the Hardy–Hilbert type inequality in reproducing kernel Hilbert space, we prove Berezin number inequalities for the convex functions in Reproducing Kernel Hilbert Spaces. We also prove some new inequalities between these numerical characteristics. Some other related results are also obtained.