On the Distribution of Zeros of Entire Functions

Abstract

Let $f(z)$ be any transcendental entire function. Let ${r_k}$ denote the absolute value of the zero ${z_k}$ of ${f^{(k)}}(z)$ which is nearest to the origin. à lander, Erdös and Rényi, and Pólya have investigated the relation between ${r_k}$ and the growth of the function $f(z)$. Let ${s_k}$ denote the largest disk about the origin where ${f^{(k)}}(z)$ is univalent. Boas, Levinson, and Pólya have obtained some relations between the growth of the function $f(z)$ and ${s_k}$. Recently Shah and Trimble have sharpened the results of Boas and Pólya. We present here results in a different direction, generalizing the above quoted results. We also present results connecting the zero-free disks and the univalent disks about the origin of the normalized remainders of $f(z)$ with the growth of $f(z)$.