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

A reproducing kernel Hilbert space is a Hilbert space H = H(Omega) of complex-valued functions on a (non-empty) set Omega, which has the property that point evaluation f -> f (lambda) is continuous on H for all lambda is an element of Omega. Then the Riesz representation theorem guarantees that for every lambda is an element of Omega there is a unique element k(lambda) = k (center dot, lambda) is an element of H such that f (lambda) = < f, k(lambda)> for all f is an element of H. The function k(lambda) is called the reproducing kernel of H and the function (k) over cap (lambda) := k lambda/parallel to k(lambda)parallel to(H) is the normalized reproducing kernel in H. The Berezin symbol of an operator A on a reproducing kernel Hilbert spaceH is defined by (A) over tilde (lambda) = < A (k) over cap (lambda), (k) over cap (lambda)>(H). The Berezin number of an operator A on H is defined by ber (A) = sup {vertical bar(A) over tilde (lambda)vertical bar : lambda is an element of Omega}. The so-called Crawford number c (A) is defined by c (A) = inf {vertical bar < Ax, x >vertical bar : x is an element of H and parallel to x parallel to = 1}. We also introduce the number (c) over tilde (A) defined by (c) over tilde (A) = inf{vertical bar(A) over tilde(lambda)vertical bar : lambda is an element of Omega}. It is clear that c (A) <= (c) over tilde (A) <= ber (A). 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.