A Dirichlet norm inequality and some inequalities for reproducing kernel spaces

Abstract:

Let $f$ be analytic and of finite Dirichlet norm in the unit disk $\Delta$ with $f(0) = 0$. Then, for any $q > 0$, \[ \left \| {\exp f} \right \|_q^2 \leqslant \exp \left \{ {\frac {1} {{\pi q}}{{\int _\Delta {\left | {f’(z)} \right |} }^2}d\sigma (z)} \right \}\quad (d\sigma (z) \equiv (i/2)\;dz \wedge d\bar z).\] Equality holds if and only if $f(z) = - q\log (1 - z\bar \zeta )$ for some $\zeta \in \Delta$. Here, for $g(z) = \Sigma _{n = 0}^\infty {b_n}{z^n}$, analytic in $\Delta$, \[ \left \| g \right \|_q^2 \equiv \sum \limits _{n = 0}^\infty {\frac {{n!}} {{{{(q)}_n}}}{{\left | {{b_n}} \right |}^2},} \] where ${(q)_0} = 1\;{\text {and}}\;{(q)_n} = q(q + 1) \cdots (q + n - 1)$ for $n \geqslant 1$. This also extends with a substantially easier proof, a result of Saitoh concerning the case of $q \geqslant 1$. In addition, a sharp norm inequality, valid for two functional Hilbert spaces whose reproducing kernels are related via an entire function with positive coefficients, is established.