## Inequalities for complex rational functions.(English)Zbl 1435.30021

Summary: In this paper, we consider a class of rational functions $$r(s(z))$$ of degree at most $$mn$$, where $$s(z)$$ is a polynomial of degree $$m$$ and obtain a certain sharp compact generalization of well-known inequalities for rational functions.

### MSC:

 30C10 Polynomials and rational functions of one complex variable 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 30A10 Inequalities in the complex plane
Full Text:

### References:

 [1] A. Aziz and Q. M. Dawood, Inequalities for a polynomial and its derivative, J. Approx. Theory 54 (1988), 119-122. · Zbl 0663.41019 [2] A. Aziz and N. A. Rather, A refinement of a theorem of Paul Turan concerning polynomials, Math. Inequal. Appl. 1 (1998), 231-238. · Zbl 0911.30002 [3] S. Bernstein, Sur la limitation des derivees des polnomes, C. R. Math. Acad. Sci. Paris. 190 (1930), 338-341. · JFM 56.0301.02 [4] P. Borwein and T. Erdelyi, Polynomial Inequalities, Springer-Verlag. New York. (1995). · Zbl 0840.26002 [5] P. Borwein and T. Erdelyi, Sharp extension of Bernstein inequalities to rational spaces, Mathematika. 43 (1996), 413-423. · Zbl 0869.41010 [6] S. Hans, D. Tripathi, A. A. Mogbademu and Babita Tyagi, Inequalities for rational functions with prescribed poles, Jornal of Interdisciplinary Mathemathics. 21(1) (2018), 157-169. [7] Idrees Qasim and A. Liman, Bernstein type Inequalities for rational functions, Indian J. Pure Appl. Math. 46 (2015), 337-348. · Zbl 1345.30004 [8] P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509-513. · Zbl 0061.01802 [9] X. Li, A comparison inequality for rational functions, Proc. Amer. Math. Soc. 139 (2011) 1659-1665. · Zbl 1243.26009 [10] X. Li, R. N. Mohapatra and R. S. Rodrigues, Bernstein type inequalities for rational functions with prescribed poles, J. Lond. Math. Soc. 51 (1995) 523-531. · Zbl 0834.30002 [11] Q. I. Rahman and G. Schmeisser, Analytic theory of polynomials, Oxford University Press. New York. (2002). · Zbl 1072.30006 [12] P. Turán, Uber die ableitung von polynomen, Compositio Math. 7 (1939) 89-95. · JFM 65.0324.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.