New Subclass of Convex Functions Concerning Infinite Cone

We introduce a new subclass of convex functions as follows:\[  \mathcal{K}_{IC}:=\left\{f\in \mathcal{A}:{\rm  Re}\left(۱+\frac{zf''(z)}{f'(z)}\right)>\left|f'(z)-۱\right|,\quad  |z|<۱\right\},\]where \mathcal{A} denotes the class of analytic and normalized functions in the unit disk |z|<۱. Some properties of this particular class, including subordination relation, integral representation, the radius of convexity, rotation theorem, sharp coefficients estimate and Fekete-Szeg\"{o} inequality associated with the k-th root transform, are investigated.

Keywords:

Univalent , Convexity , Subordination , Fekete-Szego inequality , Coefficients estimates , Rotation theorem