Inequalities for complex rational functions

  • M. Bidkham Dep. Math., Semnan Univ., Iran
  • E. Khojastehnezhad Dep. Math., Semnan Univ., Iran
Keywords: Rational functions, Polynomial, Polar derivative, Inequality, Restricted Zeros

Abstract

UDC 517.5

For the rational function $r(z)=p(z)/H(z)$ having all its zeros in $|z|\leq 1,$ it is known that
\begin{equation*}
|r'(z)|\geq\dfrac{1}{2}|B'(z)||r(z)|\quad \text{for}\quad |z|=1,
\end{equation*}
where $H(z)=\prod_{j=1}^n(z - c_j),$ $|c_j|>1,$ $n$ is a positive integer, $B(z)=H^*(z)/H(z),$ and $H^*(z)=z^n\overline{H(1/\overline{z})}.$
In this paper, we improve the above mentioned inequality for the rational function $r(z)$ with all zeros in $|z|\leq 1$ and a zero of order $s$ at the origin.
Our main results refine and generalize some known rational inequalities.

References

A. Aziz, W. M. Shah, Inequalities for a polynomial and its derivative, Math. Inequal. Appl., 7, 379 – 391 (2004), https://doi.org/10.7153/mia-07-39 DOI: https://doi.org/10.7153/mia-07-39

K. K. Dewan, S. Hans, Generalization of certain well-known polynomial inequalities, J. Math. Anal. and Appl., 363, 38 – 41 (2010), https://doi.org/10.1016/j.jmaa.2009.07.049 DOI: https://doi.org/10.1016/j.jmaa.2009.07.049

K. K. Dewan, A. Mir, Inequalities for the polar derivative of a polynomial, J. Interdisciplinary Math., 10, 525 – 531 (2007), https://doi.org/10.1080/09720502.2007.10700512 DOI: https://doi.org/10.1080/09720502.2007.10700512

X. Li, A comparison inequality for rational functions, Proc. Amer. Math. Soc., 139, 1659 – 1665 (2011), https://doi.org/10.1090/S0002-9939-2010-10624-X DOI: https://doi.org/10.1090/S0002-9939-2010-10624-X

X. Li, R. N. Mohapatra, R. S. Rodriguez, Bernstein type inequalities for rational functions with prescribed poles, J. London Math. Soc., 51, 523 – 531 (1995), https://doi.org/10.1112/jlms/51.3.523 DOI: https://doi.org/10.1112/jlms/51.3.523

R. Osserman, A sharp Schwarz inequality on the boundary for functions regular in the disk, Proc. Amer. Math. Soc., 12, 3513 – 3517 (2000), https://doi.org/10.1090/S0002-9939-00-05463-0 DOI: https://doi.org/10.1090/S0002-9939-00-05463-0

A. Zireh, Generalization of certain well-known inequalities for the derivative of polynomials, Anal. Math., 41, 117 – 132 (2015), https://doi.org/10.1007/s10476-015-0109-2 DOI: https://doi.org/10.1007/s10476-015-0109-2

Published
20.07.2021
How to Cite
BidkhamM., and KhojastehnezhadE. “Inequalities for Complex Rational Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 7, July 2021, pp. 879 -86, doi:10.37863/umzh.v73i7.455.
Section
Research articles