On the zeros of the successive derivatives of integral functions II

Abstract

In 1936 the author proved [2] the following: Theorem 1. If f(z) (≢ 0) is entire of exponential type δ such that each f^(v)(x) (v=0, 1, …) vanishes somewhere in the interval I₁=[0,½] of the real axis, then

$$\delta \geqslant \pi ,$$

(1)

and the function

$$f(z) = CoS \pi z$$

(2)

shows that π is the best constant, because cos πz satisfies all conditions.