Integral operators and univalent functions

Abstract

In this paper, we define two new integral operators $L^k$ and $L_k$ which are iterative in nature. We show that for $f(z)=z+a_2z^2+ \cdots +a_nz^n +\cdots$ with radius of convergence larger than one, $L^kf(z)$ and $L_kf(z)$ when restricted on $E=\{z:|z|<1\}$ will eventually be univalent for large enough $k$. We then show that these are the best possible results by demonstrating that there exists a holomorphic function $T(z)$ in normalized form and with radius of convergence equal to one such that $L^kT(z)$ and $L_kT(z)$ fail to be univalent when restricted to $E$ for every $k\in \mathbb{N}$.