Inequalities for a class of rational functions

Document Type : Research Paper

Authors

Department of Mathematics, Central University of Kashmir, Ganderbal-191201, India

Abstract

In this paper, we consider a more general class of rational functions $r(s(z)),$ of degree $mn,$ with $s(z)$ being a polynomial of degree $m.$ Our results not only generalize the results due to Wali and Shah [JOA, \textbf{25} (2017), no. 1, 43-53] but also improve the results obtained by Qasim and Liman [Indian. J. Pure Appl. Math. \textbf{46} (2015), no. 3, 337-348] and Mir [Indian J. Pure Appl. Math. \textbf{50} (2019), no. 2, 315-331].    

Keywords

[1] A. Mir, Inequalities concerning rational functions with prescribed poles, J. Pure Appl. Math. 50 (2019), no. 2, 315–331.
[2] A. Mir, Some inequalities concerning rational functions with fixed poles, J. Contemp. Math. Anal. 55 (2020), no. 2, 105–114.
[3] A. Aziz and Q.M. Dawood, Inequalities for polynomials and its derivative, Indian J.Approx. Theory 54 (1988), 119–122.
[4] S.N. Bernstein, Sur la limitation des derives des polynomes, C. R. Acad. Sci. Paris. 190 (1930), 338–340.
[5] I. Qasim and A. Liman, Bernstein type inequalities for rational functions, Indian. J. Pure Appl. Math. 46 (2015), no. 3, 337–348.
[6] R. Osserman, A sharp Schwarz inequality on the boundary for functions regular in disk, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3513–3517.
[7] P. Tur´an, Uber die ableitung von polynomen, ¨ Compos. Math. 7 (1939), 89–95.
[8] S.L. Wali and W.M. Shah, Some applications of Dubinin’s lemma to rational functions with prescribed poles, J. Math. Anal. Appl. 450 (2017), 769–779.
[9] S.L. Wali and W.M. Shah, Applications of the Schwarz lemma to inequalities for rational functions with prescribed poles, J. Anal. 25 (2017), no. 1, 43–53.
[10] X. Li, R.N. Mohapatra and R.S. Rodriguez, Bernstein-type inequalities for rational functions with prescribed poles, J. London Math. Soc. 51 (1995), 523–53
Volume 13, Issue 2
July 2022
Pages 609-617
  • Receive Date: 09 October 2021
  • Revise Date: 11 December 2021
  • Accept Date: 15 December 2021