Study On Certain Problems Involving Planar Harmonic Mappings And Minimal Surfaces

Harmonic mappings in the plane have attracted the serious attention of complex analy-sis only recently after the appearance of a basic paper by Clunie and Sheil-Small in 1984,it has become an aroused general interest of complex analysis.Univalent harmonic mappings have long been used in the representation of minimal surfaces,and minimal surface is one of important surfaces in the differential geometry.It has also applications in the seemingly diverse fields of geometry,algebra and topology etc.,minimal surface is widely used and important significance in the theoretical research and engineering technology.The main aim of this thesis is to study the univalency of the convolutions of two har-monic mappings,we construct some minimal surfaces by harmonic mappings,we obtain the sharp radius of fully starlikeness and fully convexity of the harmonic linear differential operators,we also get some basic properties of log-harmonic mappings,such as coefficients estimates,the growth theorem and distortion theorem and so on.This thesis consists of six chapters and the arrangement is as follows.In Chapter one,we provide some notations and concepts.Moreover,we introduce the background of the studied problems and the statement of our main results.In Chapter two,we first obtain several sufficient conditions of the convolution of half-plane harmonic mappings.Furthermore,we consider the convolution of half-plane harmon-ic mappings with respective dilatations?z+a?/?1+az?and e^i?zⁿ,where-1<a<1 and??R,n?N.We prove that such convolutions are locally univalent for n=1,which partly solves an open problem of Dorff et.al^[21].Finally,we provide some numerical computations to illustrate that such convolutions are not univalent for n?2.In Chapter three,we prove the convolutions of generalized harmonic right half-plane mappings with harmonic vertical strip mappings are univalent and convex in the horizontal direction by use Cohn's Rule and Mathematical Induction.Furthermore,we use Gauss-Lucas Theorem to prove conjectures about the convolutions of harmonic right half-plane mappings with harmonic vertical strip mappings,which show that the recent conjecture proposed by Kumar et al.is true.In Chapter four,We construct some sense-preserving univalent harmonic mappings which map the unit disk onto a domain which is convex in the horizontal direction,but with varying dilatation.Also,we obtain minimal surfaces associated with such harmonic map-pings.This solves also a recent problem proposed by Dorff and Muir.When the dilatation is the square of an analytic function,we illustrate mappings together with their minimal surfaces pictorially with the help of Mathematica software.In Chapter five,let f=h+??? be a normalized harmonic mapping in the unit disk D.We first obtain the sharp radius of the harmonic differential operator D_f^?=z f_z-?z f_z?|?|=1?for fully starlikeness and fully convexity of order?.Generally,we explore the radius of univalence,full starlikeness and fully convexity of the harmonic linear differential operator F_??z?=?1-??f+??z f_z+?z f_z??|?|=1?that the coefficients of h and g satisfy some conditions.Some of the results are motivated by the work of David Kalaj et al.^[79].In Chapter six,we consider the class univalent log-harmonic mappings of the form ??? defined in D.Firstly,we obtain the necessary and sufficient conditions for some special complex-valued continuous functions are starlike or convex in D.We construc-t log-harmonic right half-plane mapping,log-harmonic Koebe mapping and log-harmonic two-slits mapping and show precisely the image ranges of these mappings.Moreover,coef-ficients estimates of starlike log-harmonic mappings are obtained,the growth theorem and distortion theorem for the special subclass log-harmonic mappings are studied also.Finally,we propose the log-harmonic Bieberbach conjecture and log-harmonic covering theorem.