Refined Berezin number inequalities via superquadratic and conxex functions

Abstract

In this paper, we generalize and refine some Berezin number inequalities for Hilbert space operators. Namely, we refine the Hermite-Hadamard inequality and some other recent results by using the concept of superquadraticity and convexity. Then we extend these inequalities for the Berezin number. Among other inequalities, it is shown that if $S, T\in \mathcal{L(H}(\Omega))$ such that $\mathbf{ber (T)}\leq \mathbf{ber(|S|)}$ and $f$ is a non-negative superquadratic function, then
\begin{align*}
f\left( \mathbf{ber}\left( T \right) \right)\leq \mathbf{ber}( f\left( |S| \right))-\ell_{\mathbf{ber}} \left(f\left( {\left| {|S| - \mathbf{ber}\left( T \right)} \right|} \right)\right).