A functional Hilbert space is the Hilbert space ℋ of complex-valued functions on some set $\Theta \subseteq \mathbb{C}$ such that the evaluation functionals $\varphi _{\tau }\left ( f\right ) =f\left ( \tau \right ) $ , $\tau \in \Theta $ , are continuous on ℋ. The Berezin number of an operator $X$ is defined by $\mathbf{ber}(X)=\underset{\tau \in {\Theta } }{\sup }\big \vert \widetilde{X}(\tau )\big \vert = \underset{\tau \in {\Theta } }{\sup }\big \vert \langle X\hat{k}_{\tau },\hat{k}_{\tau }\rangle \big \vert $ , where the operator $X$ acts on the reproducing kernel Hilbert space ${\mathscr{H}}={\mathscr{H}(}\Theta )$ over some (non-empty) set $\Theta $ . In this paper, we introduce a new family involving means $\Vert \cdot \Vert _{\sigma _{t}}$ between the Berezin radius and the Berezin norm. Among other results, it is shown that if $X\in {\mathscr{L}}({\mathscr{H}})$ and $f$ , $g$ are two non-negative continuous functions defined on $[0,\infty )$ such that $f(t)g(t) = t,\,(t\geqslant 0)$ , then \(\begin{aligned} \Vert X\Vert ^{2}_{\sigma }\leqslant \textbf{ber}\left (\frac{1}{4}(f^{4}( \vert X\vert )+g^{4}(\vert X^{*}\vert ))+\frac{1}{2}\vert X\vert ^{2} \right ) \end{aligned}\) and \(\begin{aligned} \Vert X\Vert ^{2}_{\sigma }\leqslant \frac{1}{2}\sqrt{\textbf{ber} \left (f^{4}(\vert X\vert )+g^{2}(\vert X\vert ^{2})\right ) \textbf{ber}\left (f^{2}(\vert X\vert ^{2})+g^{4}(\vert X^{*}\vert ) \right )}, \end{aligned}\) where $\sigma $ is a mean dominated by the arithmetic mean $\nabla $ .
