Growth of Planar Harmonic Mappings

Abstract

Seminal works of Hardy and Littlewood [32] on the growth of analytic functions contain the comparison of the integral mean Mp(r,f) with Mp(r,f′) and Mq(r,f). For a complex-valued harmonic function f in the unit disk D, using the notation |∇f| = (|fz|2+|f¯z|2)1/2, we explore the relation between Mp(r,f) and Mp(r,∇f). We show that if |∇f| grows slowly, then f is continuous on the closed unit disk, and the boundary function satisfies a Lipschitz condition. We also discuss the comparative growth of the integral means Mp(r,f) and Mq(r,f). The growth of univalent harmonic functions is studied explicitly. We give an order of growth for these functions, which consequently leads to a coefficient bound. Then we explore the membership of univalent harmonic functions in the harmonic Hardy space hp. Interestingly, our ideas extend to certain classes of locally univalent harmonic functions. As a result, we obtain a “best possible” coefficient estimate for univalent and locally univalent harmonic functions with some nice properties. We produce Baernstein type extremal results for the integral means of univalent harmonic functions, which was earlier unexplored, to the best of our knowledge. In particular, sharp Baernstein type inequalities for the classes of convex and close-to-convex harmonic functions are obtained, which lead to integral mean estimates for the respective classes. We also propose a harmonic analogue of the logarithmic coefficients of an analytic univalent function, and establish a sharp inequality involving these coefficients. Finally, we compare the integral of |f|p, for f harmonic, along certain curves. In particular, we present a Riesz-Fej´er type inequality which compares the integral along a circle to the same along a pair of its diameters. As a consequence, a result pertaining to real sequences is obtained which generalizes a famous inequality of Hilbert. Several of the results turn out to be sharp. We also pose a couple of open problems, one of which, in particular, could lead to a significant progress on the harmonic analogue of the Bieberbach conjecture, due to Clunie and Sheil-Small [15].