Growth Estimates for Certain Class of Meromorphic Functions
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Authors: A. HUSSAIN
DOI: 10.46793/KgJMat2607.1049H
Abstract:
In this paper, we establish some lower bound estimates for a certain class of rational functions on a disk with prescribed poles and restricted zeros. The results obtained strengthen some known results for rational functions and in turn produce generalizations of some polynomial inequalities as well.
Keywords:
Rational functions, polynomials, poles, zeros.
References:
[1] N. C. Ankeny and T. J. Rivlin, On a theorem of S. Bernstein, Pacific J. Math. 5 (1955), 849–852.
[2] A. Aziz, Growth of polynomials whose zeros are within or outside a circle, Bull. Austral. Math. Soc. 35 (1987), 247–256.
[3] S. Bernstein, Sur l’ordre de la meilleure approximation des functions continues par des polynomes de degré donné, Mem. Acad. R. Belg. 4 (1912), 1–103.
[4] N. K. Govil, M. A. Qazi and Q. I. Rahman, Inequalities describing the growth of polynomials not vanishing in a disk of prescribed radius, Math. Inequal. Appl. 6 (2003), 453–467.
[5] A. Hussain and A. Wani, Turán-type inequalities for certain class of meromorphic functions, Int. J. Non linear. Anal. 16 (2025), 17–26. https://doi.org/10.22075/ijnaa.2023.29759.4249
[6] P. Kumar and G. V. Milovanović, On sharpening and generalization of Rivlin’s inequality, Turk. J. Math. 46 (2022), 1436–1445. https://doi.org/10.3906/mat-2203-82
[7] M. Marden, Geometry of Polynomials, 2nd Ed. Math. Surveys, No. 3, Amer. Math. Soc., Providence, RI, 1966.
[8] G. V. Milovanović, D. S. Mitrinović and Th. M. Rassias., Topics in Polynomials, Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.
[9] A. Mir and S. Hans, Inequalities concerning rational functions in the complex domain, Sib. Math. J. 63 (2022), 1012–1022. https://doi.org/10.1134/s0037446622050196
[10] A. Mir and T. Fayaz, Inequalities for a rational function with prescribed poles, Sib. Math. J. 65 (2024), 899–908. https://doi.org/10.1134/S0037446624040153
[11] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, 2002.
[12] N. A. Rather, M. Shafi and I. Dar, Growth estimate for rational functions with prescribed poles and restricted zeros, Kragujevac J. Math. 49 (2025), 305–311. https://doi.org/10.46793/KgJMat2502.305R
[13] T. J. Rivlin, On the maximum modulus of polynomials, Amer. Math. Monthly 67 (1960), 251–253.
[14] R. S. Varga, A comparision of the successive overrelaxation method and semi-iterative methods using Chebyshev polynomials, J. Soc. Indust. Appl. Math. 5 (1957), 39–46.
