Partial sums of certain harmonic univalent functions

Abstract

Let ϕ(z) be a fixed harmonic functions of the form \(\varphi (z) = z + \sum\nolimits_{k = 2}^\infty {c_k z^k + } \overline {\sum\nolimits_{k = 1}^\infty {d_k z^k } } \) (dk ≥ ck ≥ c2 > 0; k ≥ 2) and SH(ck, dk, δ) be the subclass of harmonic univalent functions f(z) which satisfy the inequality \(\sum\nolimits_{k = 2}^\infty {c_k |a_k | + } \sum\nolimits_{k = 1}^\infty {d_k |b_k | \leqslant \delta } \). In this paper, we establish some interesting results on the ratio of harmonic univalent functions to its sequences of partial sums. Relevant connections of the results presented here with various known results are briefly indicated.