Computational Methods and Function Theory 2 (2002), No. 2, 319--336
Copyright Heldermann Verlag 2002
Dimiter P. Dryanov
dryanovd@dms.umontreal.ca, Départament de Mathématique et de Statistique, Université de Montréal, Montréal (QC) H3C 3J7, Canada.
Mohammed A. Qazi
qazima@tusk.edu, Department of Mathematics, Tuskegee University, Tuskegee, Alabama 36088, U.S.A.
Qazi I. Rahman
rahmanqi@dms.umontreal.ca, Départament de Mathématique et de Statistique, Université de Montréal, Montréal (QC) H3C 3J7, Canada.
We consider entire functions of exponential type $\tau >0$, whose modulus is bounded by a constant $M$ at the extrema of $\sin(\tau z)$, and which vanish at the origin. Extending a result of L.\ H\"ormander, we show that if $f$ is any such function, then $|f(x)| \leq M |\sin(\tau x)|$ for all $x \in(-\pi/(2 \tau),\pi/(2 \tau))$, provided that $f (x) = o(x)$ as $x \to \pm \infty$; furthermore, equality holds at any point $x \in (-\pi/(2 \tau),0) \cup (0,\pi/(2 \tau))$ if and only if $f(z) \equiv e^{i\gamma} \sin(\tau z)$ for some $\gamma \in \mathbb{R}$. This also generalizes a result due to R.\ P.\ Boas Jr.\ about trigonometric polynomials. Besides, we prove some other results for entire functions of order $1$ and type $\tau > 0$, one being an analogue of a result of M.\ Riesz about trigonometric polynomials whose zeros are all real and simple.
Keywords: Entire functions, exponential type, local behaviour.
MSC 2000: 30A10, 30C15, 30D10, 30D15.
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