Growth estimate for rational functions with prescribed poles and restricted zeros

Document Type : Research Paper


Department of Mathematics, University of Kashmir, Srinagar-190006, India


Let $r(z)= f(z)/w(z)$ where $f(z)$ be a polynomial of degree at most $n$ and $w(z)= \prod_{j=1}^{n}(z-a_{j})$, $|a_j|> 1$ for $1\leq j \leq n.$ If the rational function $r(z)\neq 0$ in $|z|< k$, then for $k =1$, it is known that $$\left|r(Rz)\right|\leq \left(\frac{\left|B(Rz)\right|+1}{2}\right) \underset{|z|=1}\sup|r(z)|\,\,\, for \,\,\,|z|=1$$ where $ B(z)= \prod_{j=1}^{n}\left\{(1-\bar{a_{j}}z)/(z-a_{j})\right\}$. In this paper, we consider the case $k \geq 1$ and obtain certain results concerning the growth of the maximum modulus of the rational functions with prescribed poles and restricted zeros in the Chebyshev norm on the unit circle in the complex plane.


[1] N.C. Ankeny and T.J. Rivlin, On a Theorem of S. Bernstein, Pacific J. Math. 5(1955) 849–852.
[2] A. Aziz, Q.M. Dawood, Inequalities, Inequalities for a polynomial and its derivative, J. Approx. Theory 54 (1998) 306-313.
[3] A. Aziz and N.A. Rather, Growth of maximum modulus of rational functions with prescribed poles, J. Math. Inequal. Appl. 2(2) (1999) 165–173.
[4] G.V. Milovanovi´c, D.S. Mitrinovic and Th.M. Rassias, Topics in Polynomials: Extremal Properties, Inequalities, Zeros, World Scientific Publishing Co., Singapore, 1994.
[5] G. P´olya and G. Szeg¨o, Problems and Theorems in Analysis, Vol. I, Springer-Verlag, New York, 1972.
[6] M. Reisz, U¨ber einen satz des herrn Serge Bernstein, Acta. Math. 40 (1916) 337–347.
[7] J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc., Providence, 1969.
Volume 13, Issue 1
March 2022
Pages 247-252
  • Receive Date: 20 January 2021
  • Accept Date: 29 May 2021