On The Problems Of Quasi-conformality For Some Harmonic Mappings And The Univalent Radius For Harmonic Mappings Under Differential Operator

Recently,the research on the properties for univalent harmonic mappings takes many scholars’attention in the function theory field. Harmonic mapping is different from analytical function,its real and imaginary parts do not satisfy with the Cauchy-Riemann equations. However, harmonic mapping theory has many similar important results as univalent function theory, and there exist many unsolved problems. Harmonic mappings have many relations with quasi-conformal mappings, analytical functions, and possess with their own characteristics. Such as, they all have univalent radii problems, Bloch constant estimates, coefficient estimation problems, and boundary condition problems,etc. In recent years, many scholars devoted to explore the conditions for univalent harmonic mapping to be harmonic K-q.c; the Bieberbach conjecture for univalent harmonic mapping; the various properties of harmonic mappings under the differential operator L. As a result, the researches on the properties for harmonic mappings become one of active topics in Complex analysis.This paper focuses on the following parts:Firstly, considering on univalent harmonic mappings CH2(λ) on the unit disk D={z||z|<1} defined by Kalaj, we find one sufficient condition for functions belonging to CH2(λ) to be harmonic quasi-conformal mappings and give their coefficient estimates for their analytical and conjugate parts. Next, under some coefficient bound conditions that the coefficient bound, we find the sharp estimates for their univalent close-to-convex and star-like radius. Our results improve and generalize the one made by Kalaj.Secondly, we study the univalent radius for harmonic mappings under differential operator. To the class of univalent harmonic mapping SH0defined on the unit disk D, if there exist the following two conjectures n≥1. If f(z) is also univalent convex, then: Kalaj raises that:If the coefficient of harmonic functions satisfy the above conjecture, try to find the univalent radii for such functions. If f(z) is a harmonic mapping on the univalent disk and its coefficients satisfying each of two famous conjecture bounds, we all obtain its sharp univalent radius for L(f)=zfz-zfz-Furthermore, with the condition that the coefficients bound satisfying one general expression, we also obtain the similar sharp result.Thirdly, we obtain several sharp univalent radii estimates for the harmonic mappings that are obtained by multiple differential operators L acting on the given harmonic mapping f(z) with its coefficients satisfying anyone of two famous conjectures or some general coefficient bound condition.