Applications of T(r)

Abstract

Theorem. If f is holomorphic and $M(r) = \sup [\{ |f(z)|:|z| \leqslant r\}$, then for any R>r

$$ T(r) \leqslant {{\log }^{ + }}M(r) \leqslant \frac{{R + r}}{{R - r}}T(R). $$