Convexity of the Berezin range of finite rank operators

For a bounded linear operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}(\mathrm{\Omega })$ over a nonempty set $\mathrm{\Omega }$, the Berezin range of $T$ is defined by $$\mathrm{B}\mathrm{e}\mathrm{r}(T)=\{⟨T{\widehat{k}}_{\lambda },{\widehat{k}}_{\lambda }{⟩}_{\mathcal{H}}:\lambda \in \mathrm{\Omega }\}$$ and the Berezin radius is given by $$\mathrm{b}\mathrm{e}\mathrm{r}(T)=\sup \{\mathrm{\mid }\gamma \mathrm{\mid }:\gamma \in \mathrm{B}\mathrm{e}\mathrm{r}(T)\},$$ where ${\widehat{k}}_{\lambda }$ denotes the normalized reproducing kernel at $\lambda \in \mathrm{\Omega }$. In this paper, we study the convexity of the Berezin range of finite rank operators on the Hardy space and the Bergman space over the unit disc $\mathbb{D}$. We present applications of some scalar inequalities to get some operator inequalities. A characterization of closure of the numerical range of reproducing kernel Hilbert space operator in terms of convex hull of its Berezin range is also discussed.