Some properties of functions of exponential type

and on the real axis ƒ(z) is real and bounded by 1. First it is shown that the function cos \z—f(z) cannot have complex zeros. Moreover its real zeros are simple at the points where the strict inequality \f(z)\ <1 is satisfied. This theorem is then used to find a "best possible" dominant over the complex plane of the class of functions f{z). Finally it is shown that these results contain two theorems of S. Bernstein.