Bounds of the Derivative of Some Classes of Rational Functions

2020/01/20 by Nuttapong Arunrat, Arunrat, Nuttapong, Keaitsuda Maneeruk Nakprasit +1
Mathematics · Computer Science · #Mathematical functions and polynomials #Polynomial and algebraic computation #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.2001.09791

Abstract

Let r(z) be a rational function with at most n poles, a₁, a₂, …, a_n, where |a_j| > 1, 1≤ j≤ n. This paper investigates the estimate of the modulus of the derivative of a rational function r(z) on the unit circle. We establish an upper bound when all zeros of r(z) lie in |z|≥ k≥ 1 and a lower bound when all zeros of r(z) lie in |z|≤ k ≤ 1. In particular, when k=1 and r(z) has exactly n zeros, we obtain a generalization of results by A. Aziz and W. M. Shah [Some refinements of Bernstein-type inequalities for rational functions, Glas. Mat., \bf 32(52) (1997), 29--37.].

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