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On Bernstein's inequality for entire functions of exponential type
Authors:Q.I. Rahman  Q.M. Tariq
Affiliation:a Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec H3C 3J7, Canada
b Department of Mathematics & Computer Science, Virginia State University, Petersburg, VA 23806, USA
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.
Keywords:Polynomials   Functions of exponential type   Bernstein's inequality
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