Argument estimates for certain analytic functions

Abstract

Let $p(z)$ be analytic in the open unit disk $\mathbf{U}$ with $p(0) = 1$ and $p'(0) = 0$. S. S. Miller and P. T. Mocanu (J. Math. Anal. Appl. 276 (2002)) have shown some interesting subordination theorems for such functions $p(z)$. The object of the present paper is to discuss some sufficient conditions for arguments of $p(z)$ to be $|\arg p(z)| < (\pi/2) \rho$ for $z \in \mathbf{U}$.