A functional Hilbert space is defined as the Hilbert space $\mathcal{K}$ of complex-valued functions defined on a set Θ. In this space, the evaluation functionals $\psi _{\varepsilon}(h) = h(\varepsilon )$ , for $\varepsilon \in \Theta $ , are continuous on $\mathcal{K}$ . In this paper, we introduce the mean-Berezin norm $\| \cdot \|_{\rho _{t}}$ , defined as follows: \( \| G \|_{M_{\rho _{t}}} = \sup _{\kappa , \xi \in \Theta} \left \{ \left (\big| \langle G \hat{k}_{\kappa}, \hat{k}_{\xi} \rangle \big| \right ) \rho _{t} \left ( \big| \langle G^{*} \hat{k}_{\kappa}, \hat{k}_{\xi} \rangle \big| \right ) \right \}, \quad \text{for } 0 \leqslant t \leqslant 1, \) where $G \in \mathcal{O}(\mathcal{K})$ and $\rho _{t}$ represents an interpolation path of a symmetric mean ρ. Using the definition of the mean-Berezin norm, we explore some related new inequalities For instance, if $G\in \mathcal{O}(\mathcal{K})$ , then \( \frac{1}{2}\sqrt{\left \Vert \vert G\vert ^{2r}+\vert G^{*}\vert ^{2(1-r)} \right \Vert _{\textbf{ber}}}\geqslant \Vert G\Vert _{M_{\rho}}, \) where $0\leqslant r\leqslant 1$ and ρ is a mean such that $\rho \leqslant \nabla $ .
