Bloch Constant And Landau Theorem For Harmonic Mappings

The geometry function theory of univalent functions is an important research branch in the theory of complex analysis. The estimate of the Bloch constant and Landau theorem of complex valued mappings is one of the important research topics in the field of the theory of univalent functions. As a kind of generalization of holomorphic functions, harmonic mappings were widely used both in many applied subjects such as medicine, electromagnetics, fluid mechanics and elastic problems, and in many other mathematical branches such as partial differential equations, differential geometry and Teichmuller space. So it is of significance to study the theory of a harmonic mapping. Estimating the univalent radii of a given class of harmonic mappings is one of the important problems of the harmonic mapping theory. In this paper, we mainly study the asymptotically sharp estimate for Bloch constant and the Landau theorem of harmonic mappings.Firstly, we use the refined Schwarz lemma of harmonic mappings and the Bloch constant Bhol of holomorphic functions to obtain an asymptotically sharp estimate of the Bloch constant of the subclass of open harmonic mappings. We also get an estimate of the Bloch constant of K-quasiregular harmonic mappings, which is expressed by the coefficient|b1|and Bhol. Both results improve the corresponding ones obtained by H. H. Chen and P. M. Gauthier, recently.Secondly, we study the Landau theorem of harmonic mappings. Combining the Koebe theorem of bounded univalent functions and the Schwarz lemma of harmonic mappings, we obtain an asymptotically sharp estimate of the Landau constant of a harmonic mapping. These results also improve the ones given by H.H. Chen and P. M. Gauthier, recently.Thirdly, we study the implication relations of the boundedness of|f|,|L(f)|and Λf. We show that if|L(f)|is bounded then|f|is bounded, but Λfis not necessary to be bounded; if|g|is bounded then|L(f)|and Λf are not necessary to be bounded; if Λf is bounded then|f|and|L(f)|are both bounded. Using the method of coefficient estimate, we obtain an asymptotically sharp estimate of the Landau theorem of harmonic mappings under the condition that|L(f)|is bounded.