A Theorem on the Zeros of Entire Functions and Its Application

We consider entire functions of exponential type $\le \sigma$ that are bounded and real on $\mathbb R$ and satisfy the estimate $(-1)^k f({k\pi}/{\sigma} +\tau)\ge0$, $k\in \mathbb{Z}$. It is proved that the zeros of such functions are real and simple with the possible exception of points of the form ${k\pi}/{\sigma}+\tau$, which can be zeros of multiplicity at most 2. These results are applied to specific classes of functions and to the problem of the stability of entire functions. We also refine and supplement a few results due to Pólya.