Computational Methods and Function Theory 7 (2007), No. 1, 167--184
Copyright Heldermann Verlag 2007
Qazi I. Rahman
rahmanqi@dms.umontreal.ca , Université de Montréal, Département de Mathématiques et de Statistique, Montréal, Québec H3C 3J7, Canada.
Qazi M. Tariq
tqazi@vsu.edu , Virginia State University, Department of Mathematics & Computer Science, Petersburg, VA 23806, U.S.A.
Polynomials $p$, satisfying the condition $p (z) \equiv z^n p (1/z)$, have been studied since well over thirty years. If $p$ is such a polynomial, then ${f(z) := p (\me^{{i} z})}$ is an entire function of exponential type $n$ which satisfies the relation ${f(z) \equiv \me^{\mi n z} f(-z)}$. Thus, practically any result about entire functions $f$ of exponential type, which satisfy the condition $f(z) \equiv \me^{\mi \tau z} f(-z)$, can be specialized to obtain a result about polynomials $p$ satisfying the condition $p (z) \equiv z^n p (1/z)$. In the present paper we obtain an extension of a result of Govil and Vetterlein, about the derivative of polynomials $p$ satisfying $p (z) \equiv z^n p (1/z)$, to entire functions $f$ of exponential type satisfying $f(z) \equiv \me^{\mi \tau z} f(-z)$.
Keywords: Bernstein's inequality, entire functions, exponential type, uniformly almost periodic functions.
MSC 2000: 30D15, 41A17, 42A05, 42A16, 42A75.
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