A Unified Class of Harmonic Functions with Varying Argument of Coefficients

Abstract

In this paper we investigate several classes of harmonic functions with varying argument of coefficients which are defined by means of the principle of subordination between harmonic functions. Such properties as the coefficient estimates, distortion theorems, convolution properties, radii of convexity, starlikeness and the closure properties of these classes under the generalized Bernardi-Libera-Livingston integral \\operator $\mathcal{L}_{c}(f),$ ($c>-1$) which is defined by
$ \mathcal{L}_{c}(f)=\mathcal{L}_{c}(h)+\overline{\mathcal{L}_{c}(g)}$ where
\begin{equation*}
\mathcal{L}_{c}(h)(z)=\frac{c+1}{z^{c}}\int\limits_{0}^{z}t^{c-1}h(t) dt \;\;\;%
\mathrm{and} \;\;\; \mathcal{L}_{c}(g)(z)=\frac{c+1}{z^{c}}\int\limits_{0}^{z}t^{c-1}g(t) dt
\end{equation*} are investigated.