Growth of Harmonic Mappings and Baernstein Type Inequalities

Abstract

Seminal works of Hardy and Littlewood on the growth of analytic functions contain the comparison of the integral means $M_p(r,f)$, $M_p(r,f')$, $M_q(r,f)$. For a complex-valued harmonic function f in the unit disk, using the notation $\vert \nabla f \vert =(\vert f_z{\vert }^2+\vert f_{\bar{z}}{\vert ^2})^{1/2}$ we explore the relation between $M_p(r,f)$ and $M_p(r,\nabla f)$. We show that if $\vert \nabla f \vert$ grows sufficiently slowly, then f is continuous on the closed unit disk and the boundary function satisfies a Lipschitz condition. We also prove that for $1 \le p < q \le \infty$, it is possible to give an estimate on the growth of $M_q(r,f)$ whenever the growth of $M_p(r,f)$ is known. We notably obtain Baernstein type inequalities for the major geometric subclasses of univalent harmonic mappings such as convex, starlike, close-to-convex, and convex in one direction functions. Some of these results are sharp. A growth estimate and a coefficient bound for the whole class of univalent harmonic mappings are given as well.