Image areas and $H_{2}$ norms of analytic functions

Abstract:

For an analytic function $f$ in the unit disc $U$ with $f(0) = 0$, the inequality $\left \| f \right \|_2^2 \leqslant \frac {1}{\pi }$ area $\{ f(U)\}$ is shown, where an equality occurs if and only if $f$ is a constant multiple of an inner function. As a corollary, it is shown that for an analytic function in a general domain the square of its ${H_2}$ norm is bounded by its Dirichlet integral, with the equality condition being settled