Some subclasses of harmonic univalent functions (Q2708420)

Let \(U\) be the unit disc of the complex plane. In this paper the author studies the class \(H_k\) consisting of all harmonic functions \(f(z)=z+\sum_{n=2}^{\infty}a_nz^n+\sum_{n=1}^{\infty}a_{-n}\bar{z}^n\) on \(U\) which satisfy the condition \(\sum_{2}^{\infty}n^k(|a_n|+|a_{-n}|) \leq 1-|a_{-1}|\), \(k\in {\mathbb Z}^{+}\), \(|a_{-1}|<1\). For this purpose, the author obtains some distortion and covering theorems and a theorem concerning the uniform convergence of a sequence in \(H_2\). In the third part of the paper, the author uses the technique of Hadamard product to study two subclasses of harmonic close-to-convex functions.