| Harmonic mapping is a kind of generalization of analytic function,which is closely related to other theories of complex analysis such as quasiconformal mappings,univalent functions and Teichmüller spaces etc.In recent years,many scholars have studied the properties of harmonic mappings and the weighted harmonic mappings,such as their univalence,Schwarz lemma and Bloch theorem.Based on the results of scholars,we mainly investigate the Schwarz lemma for harmonic functions and the weighted harmonic mappings in this paper.Part one,we investigate the Schwarz lemma for real harmonic functions of the unit ball into a general interval.Based on the results obtained by Burgeth,Partyka and Sakan,we use the mean-value theorem of harmonic functions to obtain the Schwarz lemma for harmonic functions with their image domains being a general interval[a,b].As an application of this result,we improve an upper bound given by Partyka and Sakan.Moreover,a lower bound for this class of harmonic functions is also given.Part two,we study the Schwarz lemma for the weighted harmonic mappings.When?=2,with the fact that 2T-harmonic mappings can be represented by the Poisson integral of the boundary function,we explore the Schwarz lemma for real2T-harmonic mappings which map the unit disk D into the real interval I=?-1,1?.The accurate upper and lower bounds estimates are obtained.Also the extremum function is found.Part three,we explore the properties of the Poisson kernel for T?-harmonic mappings.By studying the properties of kernel functions,we find that the kernel functions are subharmonic in a small disk.An upper bound of its radius is also given. |
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