Estimates of integral means of harmonic mappings

Generalized harmonic Koebe functions

“…all these functions k a are locally univalent in the unit disk. Moreover, the limit (in compact subsets of D) of the functions k a as a → 0 coincides with (10) k 0 (z) :…”

“…Indeed, vertical shears of k 1 with dilatation ω(z) = −ηz (where η is a complex number of modulus 1) coincide with the functions F (z, η) defined on [10, p. 158], which turn out to be extreme points of an important family related to functions mapping the unit disk onto the half-plane H = {z : Re{z} > −1/2}. Harmonic shears of k 0 have been proved to have also interesting properties (see [10,Sec. 2]).…”

“…with k 0 as in (10). Note that this function k 0 maps the unit disk onto a convex domain (which is convex in the 0 direction).…”