New Subclass of Convex Functions Concerning Infinite Cone

Abstract

We introduce a new subclass of convex functions as follows:\[ \mathcal{K}_{IC}:=\left\{f\in \mathcal{A}:{\rm Re}\left(1+\frac{zf''(z)}{f'(z)}\right)>\left|f'(z)-1\right|,\quad |z|<1\right\},\]where $\mathcal{A}$ denotes the class of analytic and normalized functions in the unit disk $|z|<1$. 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