A comprehensive class of harmonic functions defined by convolution and its connection with integral transforms and hypergeometric functions

For given two harmonic functions $\Phi$ and $\Psi$ with real coefficients in the open unit disk $\mathbb{D}$, we study a class of harmonic functions $f(z)=z-\sum_{n=2}^{\infty}A_nz^{n}+\sum_{n=1}^{\infty}B_n\bar{z}^n$ $(A_n, B_n \geq 0)$ satisfying \RE \frac{(f*\Phi)(z)}{(f*\Psi)(z)}>\alpha \quad (0\leq \alpha <1, z \in \mathbb{D}); * being the harmonic convolution. Coefficient inequalities, growth and covering theorems, as well as closure theorems are determined. The results obtained extend several known results as special cases. In addition, we study the class of harmonic functions $f$ that satisfy \RE f(z)/z>\alpha $(0\leq \alpha <1, z \in \mathbb{D})$. As an application, their connection with certain integral transforms and hypergeometric functions is established.Comment: 14pages, 1 figur

A generalization of starlike functions of order alpha

For every $q\in(0,1)$ and $0\le \alpha<1$ we define a class of analytic functions, the so-called $q$-starlike functions of order $\alpha$, on the open unit disk. We study this class of functions and explore some inclusion properties with the well-known class of starlike functions of order $\alpha$. The paper is also devoted to the discussion on the Herglotz representation formula for analytic functions $zf'(z)/f(z)$ when $f(z)$ is $q$-starlike of order $\alpha$. As an application we also discuss the Bieberbach conjecture problem for the $q$-starlike functions of order $\alpha$. Further application includes the study of the order of $q$-starlikeness of the well-known basic hypergeometric functions introduced by Heine.Comment: 13 pages, 4 figures, submitted to a journa