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

Applications of Theory of Differential Subordination for Functions with Fixed Initial Coefficient to Univalent Functions

By using the theory of first-order differential subordination for functions with fixed initial coefficient, several well-known results for subclasses of univalent functions are improved by restricting the functions to have fixed second coefficient. The influence of the second coefficient of univalent functions is evident in the results obtained