On an Extension of a Theorem of Bernstein to Meromorphic Functions

A theorem of Bernstein [2; pp. 206-2111 asserts that if f(z) is an entire function of exponential type T and satisfies If(x) 1 < M uniformly for co < x < W, then If’(x) ( < Mr. In [3] A. J. Macintyre and S. M. Shah state sufficient conditions (same as (a), (c), and (d) below) to guarantee that meromorphic functions which are bounded on the real axis have derivatives which also are bounded there; moreover, they show by example that condition (d) is necessary. The theorem given here is proved by methods similar to those of Macintyre and Shah and extends their result to a class of meromorphic functions which are 0 (exp (/? ( x 17)) as x -+ f co. THEOREM. Iff(z) is a meromorphicfunction which satisjk: