Inequalities for the Derivatives of Rational Functions with Prescribed Poles and Restricted Zeros

Abstract

As a generalization of a result obtained by Dubinin [7], Wali (preprint online) [14] recently proved the following: Let r ∈ ${{\mathcal{R}}_{n}}$, where r has n poles at a₁, a₂, …, a_n and all its zeros lie in |z| ≤ 1, with s-fold zeros at the origin, then for |z| = 1

$$\left| {r{\kern 1pt} '(z)} \right| \geqslant \frac{1}{2}\left\{ {\left| {\mathcal{B}{\kern 1pt} '(z)} \right| + (s + m - n) + \frac{{\left| {{{c}_{m}}} \right| - \left| {{{c}_{s}}} \right|}}{{\left| {{{c}_{m}}} \right| + \left| {{{c}_{s}}} \right|}}} \right\}\left| {r(z)} \right|.$$

In this paper, instead of assuming that r(z) has a zero of order s at the origin as Wali did, we suppose that r(z) has a zero of multiplicity s at any point inside the unit circle and all other zeros are inside or outside a circle of radius k. Further, we prove some results which besides generalizing some inequalities for rational functions include refinements of some polynomial inequalities as special cases.