On The Study Of P-logharmoic Mappings

Harmonic mappings are the generalization of analytic functions. These mappings are widely applied in Fluid Mechanics, Science of Electricity, Mag-netics, Medical Science and other branches of Mathematics. Hence the study of harmonic mappings has already become an attractive topic in complex anal-ysis.In this paper, we mainly study a kind of generalized harmonic map-pings, which are p-logharmonic mappings, where logharmonic mappings (i.e. 0-logharmoinc mappings) are included. This thesis consists of three Chapters. In Chapter 1, we introduce some necessary notations, background of our study problems and our main results.In Chapter 2, we define the Schwarzian derivative of logharmonic map-pings and obtain several equivalent conditions on which the Schwarzian deriva-tive is analytic. Then we establish the Schwarz lemma for logharmonic map-pings. Based on the Schwarz lemma, two versions of Landau Theorem of logharmonic mappings are obtained.In Chapter 3, we introduce the concept of p-logharmonic mappings (p> 0), which is the generalization of logharmonic mappings, and some properties of these mappings are discussed, which are as follows. (1) Dirichlet problem: We prove the existence of the solutions of this problem and find the equiva-lent condition of univalence of the solutions; (2) Starlikeness:Some relations among the corresponding analytic functions, logharmonic mappings and p-logharmonic mappings are got; (3) Moment of order q:We find a lower bound.