Lindelöf’s Principle and Estimates for Holomorphic Functions Involving Area, Diameter or Integral Means

Abstract

Suppose that $f$ is a holomorphic function in the unit disk. We provide bounds for the distance of $f$ from its linearization $f(0)+f^\prime (0)z$; the bounds involve the area, the diameter or the logarithmic capacity of the image $f({\mathbb D})$ of $f$. These results are motivated by a problem posed by Burckel, Marshall, Minda, Poggi-Corradini, and Ransford. We also prove that if, in addition, $f(0)=0$, then the $H^2$ norm of $f$ is bounded by a $\mathrm{Area}f^{\circledcirc }({\mathbb D})/\pi$, where $f^{\circledcirc }$ is a univalent function constructed via the symmetric decreasing rearrangement of the real part of $f$ on the unit circle. The above estimate is stronger than a well-known inequality due to Alexander, Taylor, and Ullman. We give a description of the equality cases in Lindelöf’s principle for the Green function. We prove that the $H^p$ norm of $f$ is smaller or equal to the $H^p$ norm of the universal covering map onto the circular ($0<p<\infty$) or Steiner ($0<p\le 2$) symmetrization of the range of $f$.