On Bernstein type theorems concerning the growth of derivatives of entire functions

Abstract

A subspace X of Ll,,(R) which is invariant under all left translation operators Tt, t E R, is called admissible if X is a Banach space satisfying the following properties: (i) If IIfnlIx 0, then there exists a subsequence (nk) such that fink (s) 0 almost everywhere. (ii) The group TX := {Ttlx : t E R} is a bounded strongly continuous group. In this case, let Cx := sup{HITtIx : t E RI. Typical admissible spaces are Co(R), BUC(R) and all spaces LP(R) for 1 0, e-6Itg(t) E L1 (R). With these definitions our main result goes as follows: Theorem 1. If g is an entire function of exponential type r such that its restriction to the real axis, denoted by gR, is subexponential and belongs to some admissible space X, then the derivative g' is also in X. Moreover, Ia + 9' IIX < (Oa2 + r2)1/2* CX 1lg1IIx for each real oz. This result yields as consequences and in a systematic way many new and old Bernstein type inequalities. 1. THE MAIN THEOREMS Recall that an entire function f is called to be of exponential type r if log M(r) lim Sup =7 r-woo r where M(r) := max{lf(z)I: IzI < r} is the maximum modulus of f. The classical Bernstein theorem [Bo, p.206, Theorem 11.1.2] states that if f is an entire function of exponential type r and is bounded on the real axis by M, then If'(t)I < TFM for all t c IR, or equivalently, IIf'IIIo < TIIflIIo for each entire function f of exponential type r. The latter is referred as to Bernstein's inequality. Many extensions are given, see [Bo, Chap. 11], [D-S], [Le, Chap. IV] and [K-S-T]. It is the purpose of Received by the editors August 16, 1995. 1991 Mathematics Subject Classification. Primary 30D20; Secondary 47D03, 47A10.