On Bernstein's inequality for entire functions of exponential type

Abstract

It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that |f(x)|⩽M for −∞<x<∞, then |f′(x)|⩽Mτ for −∞<x<∞. If p is a polynomial of degree at most n with |p(z)|⩽M for |z|=1, then f(z):=p(eiz) is an entire function of exponential type n with |f(x)|⩽M on the real axis. Hence, by the just mentioned inequality for functions of exponential type, |p′(z)|⩽Mn for |z|=1. Lately, many papers have been written on polynomials p that satisfy the condition znp(1/z)≡p(z). They do form an intriguing class. If a polynomial p satisfies this condition, then f(z):=p(eiz) is an entire function of exponential type n that satisfies the condition f(z)≡einzf(−z). This led Govil [N.K. Govil, Lp inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445–452] to consider entire functions f of exponential type satisfying f(z)≡eiτzf(−z) and find estimates for their derivatives. In the present paper we present some additional observations about such functions.