On a family of $p$-valently analytic functions missing initial Taylor coefficients

Abstract

For $k\geq0$, $0\leq\gamma\leq1$, and some convolution operator $g$,
the object of this paper is to introduce a generalized family $\mathcal{TU}_p^n(g,\gamma,k,b,\alpha)$
of $p$-valently analytic functions of complex order $b\in\mathbb{C}\setminus\{0\}$ and type $\alpha\in[0,p)$.
Apart from studying certain coefficient, radii and subordination problems, we prove that $\mathcal{TU}_p^n(g,\gamma,k,b,\alpha)$ is convex and derive its extreme points. Moreover, the closedness of this family under the modified Hadamard product is discussed. Several previously established results are obtained as particular cases of our theorems.