On the convexity of Berezin range and Berezin radius inequalities via a class of semi-norms

This paper introduces a new family of semi-norms, say ${\sigma }_{\mu }$-Berezin norm on the space of all bounded linear operators $B(\mathcal{H})$ defined on a reproducing kernel Hilbert space $\mathcal{H}$, namely, for each $\mu \in [0,1]$ and $p\ge 1$, $$\mathrm{\parallel }T{\mathrm{\parallel }}_{{\sigma }_{\mu }\text{-ber}}=\underset{\lambda \in \mathrm{\Omega }}{\sup }\left\{{(\mathrm{\mid }⟨T{\widehat{k}}_{\lambda },{\widehat{k}}_{\lambda }⟩{\mathrm{\mid }}^p{\sigma }_{\mu }\mathrm{\parallel }T{\widehat{k}}_{\lambda }{\mathrm{\parallel }}^p)}^{\frac{1}{p}}\right\}$$ where $T\in B(\mathcal{H})$ and ${\sigma }_{\mu }$ is an interpolation path of the symmetric mean $\sigma$. We investigate many fundamental properties of the ${\sigma }_{\mu }$-Berezin norm and develop several inequalities associated with it. Utilizing these inequalities, we derive improved bounds for the Berezin radius of bounded linear operators, enhancing previously known estimates. Furthermore, we study the convexity of the Berezin range of a class of composition operators and weighted shift operators on both the Hardy space and the Bergman space.