Suppose that f(z) is an entire function such that /(*)=0(«M'l), 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. THEOREM 1.

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... iz) be an entire junction of z = x+iy, real for realz, and satisfying the conditions :