Derivatives of entire functions of higher order

Abstract

Bounds for the derivative of entire functions of higher order are related to bounds on the functions themselves. S. N. Bernstein proved the following result for the derivative of entire functions of finite type: if f is an entire function of exponential type λ and ¦f(x)¦ ⩽ M for R, then ¦f′(x)¦ ⩽ λM on R. In Volume 58 of the Journal of Approximation Theory, R. A. Zalik raised the problem of extending Bernstein's estimate to entire functions of higher order. But he finally considered another question; namely, he proved: Let f be an entire function, n ⩾ 0, A1, A2, a, b, c, d real numbers and write z = x + iy. If ¦f(z)¦ ⩽ (A1 + A2 ¦z¦n) exp(ax2 + by2 + cx + dy) for all z∈C, then there are numbers C1, C2 ⩾ 0 depending only on n, A1, A2, a, b, c, and d such that ¦f′(z)¦ ⩽ (C1 + C2 ¦z¦n + 1) exp(ax2 + by2 + cx + dy)