Computational Methods and Function Theory 10 (2010), No. 1, 61--79
Copyright Heldermann Verlag 2010
Mohammed A. Qazi
qazima@aol.com , Tuskegee University, Department of Mathematics, Tuskegee, Alabama 36088, U.S.A.
Qazi I. Rahman
rahmanqi@dms.umontreal.ca , Université de Montréal, Département de Mathématiques et de Statistique, Montréal (Québec) H3C 3J7, Canada.
Let Pn be the class of all polynomials of degree at most n. It is known that if f ∈ Pn and |f(z)| ≤ 1 on the unit circle, then |f' (z)| ≤ n|z|n-1 outside the unit disk. We present an "extension" of this result to rational functions having all their poles in the open unit disk. Some inequalities involving |f(z)|, |f'(z)| and |f''(z)| are also proved in this paper. The last section contains an L2 inequality for the derivative of a rational function.
Keywords: Polynomials, Bernstein's inequality, rational functions, Pick's inequality.
MSC 2000: 30D30, 41A17, 41A44.
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