Argument Properties of Analytic Functions Associated with the Fixed Coefficients

Abstract

In the present paper, we derive a property of analytic functions $p(z) = 1 + p_{n}z^{n} + \cdots$ with fixed initial coefficients in their series expansion, which satisfy the condition $-\pi \beta /2 < \arg p(z_{1}) < \arg p(z) < \arg p(z_{2}) = \pi \alpha /2$, for some $z_{1}$ and $z_{2}$ with $|z_{1}|=|z_{2}|=r<1$ and for all z with $|z|<r$, where $0<\alpha \le 2$ and $0<\beta \le 2$. Using this property, we obtain some sufficient conditions for normalized analytic functions $f(z) = z + a_{n+1}z^{n+1} + \cdots$ by considering the fixed initial coefficients to satisfy $-\pi \beta /2 < \arg \left\{ zf'(z)/f(z) - \gamma \right\} < \pi \alpha /2$ for all z in the unit disk ${\mathbb {U}}$ on the complex plane, where $0 \le \alpha , \beta < 1$, and $\gamma =0$ or 1 / 2.