A Property of the Zeros of the Successive Derivatives of Integral Functions

The following result has been proved by Takenaka:

THEOREM 1. Let f(z) be an integral function of order one and type not exceeding σ. Let [α_n] be a sequence such that

lim |α_n|=L<log2.                                                               (1)

Let f(ⁿ)(α_n)=0, n=0, 1, 2 . . .[f⁽⁰⁾(z)=f(z)]. Then f(z)=0.