Maximum modulus convexity and the location of zeros of an entire function

Abstract:

Let $f$ be an entire function with non-negative Maclaurin coefficients and let $b\left ( r \right ) = r{\left ( {rf’\left ( r \right )/f\left ( r \right )} \right )’ }$. It is shown that if all the zeros of $f$ lie in the angle $\left | {\arg z} \right | \leq \delta$, where $0 < \delta \leq \pi$, then $\lim {\sup _{r \to \infty }}b\left ( r \right ) \geq \frac {1}{4}{\text {cose}}{{\text {c}}^2}\frac {1}{2}\delta$. In particular, we always have $\lim {\sup _{r \to \infty }}b\left ( r \right ) > \frac {1}{4}$ for such functions.