Critical points of real entire functions and a conjecture of Pólya

Abstract:

Let $f(z)$ be a nonconstant real entire function of genus $1^*$ and assume that all the zeros of $f(z)$ are distributed in some infinite strip $|\operatorname {Im} z|\leq A$, $A>0$. It is shown that (1) if $f(z)$ has $2J$ nonreal zeros in the region $a\leq \operatorname {Re} z \leq b$, and $f’(z)$ has $2J’$ nonreal zeros in the same region, and if the points $z=a$ and $z=b$ are located outside the Jensen disks of $f(z)$, then $f’(z)$ has exactly $J-J’$ critical zeros in the closed interval $[a,b]$, (2) if $f(z)$ is at most of order $\rho$, $0<\rho \leq 2$, and minimal type, then for each positive constant $B$ there is a positive integer $n_1$ such that for all $n\geq n_1$ $f^{(n)}(z)$ has only real zeros in the region $|\operatorname {Re} z|\leq Bn^{1/\rho }$, and (3) if $f(z)$ is of order less than $2/3$, then $f(z)$ has just as many critical points as couples of nonreal zeros.