Generalization of certain well-known inequalities for rational functions


Let P_m be a class of all polynomials of degree at most m and let R_{m,n}=R_{m,n}(d_{1}, ..., d_{n})=\{p(z)/w(z);p\in P_{m}, w(z)=\prod_{j={1}}^n(z-d_{j})~ where  ~ \vert d_{j}\vert\ >1, j=1, ..., n and m\leq n\} denote the class of rational functions. It is proved that if the rational function r(z) having all its zeros in \vert z\vert\leq1, then for \vert z\vert=1

\vert r^{'}(z)\vert\geq \frac{1}{2}\{\vert B^{'}(z)\vert-(n-m)\} \vert r(z)\vert.

The main purpose of this paper is to improve the above inequality for rational functions r(z) having all its zeros in \vert z\vert\leq k\leq1 with t-fold zeros at the origin and some other related inequalities. The obtained results sharpen some well-known estimates for the derivative and polar derivative of polynomials.

DOI Code: 10.1285/i15900932v40n1p1

Keywords: Rational functions; Polynomials; Polar derivative; Inequalities; Poles; Restricted Zeros

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