An integral criterion for normal functions

Abstract:

A new characterization for normal functions is given. It is shown that a function $f$ meromorphic in the unit disk is a normal function if and only if for each $\delta > 0$ and each $p > 2$ there exists a constant ${K_f}(\delta ,p)$ such that, for each hyperbolic disk $\Omega$ with hyperbolic radius $\delta$, \[ \iint _\Omega {{{(1 - {{\left | z \right |}^2})}^{p - 2}}{{({f^ \ne }(z))}^p}dA(z) \leq {K_f}(\delta ,p)},\] where ${f^ \ne }(z)$ denotes the spherical derivative of $f$ and $dA(z)$ is the Euclidean element of area. It is shown by example that this characterization is not valid for $p = 2$.