Geodesic
Inequalities for meromorphic functions with prescribed poles
Ufa mathematical journal, Tome 16 (2024) no. 1, pp. 127-137 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The extremal problems for functions of complex variables, as well as approaches for obtaining classical inequalities on the base of various methods of the geometric function theory, are known for various norms and for many classes of functions such as rational functions with various constraints and for various domains in the complex plane. It is important to mention that different types of Bernstein-type inequalities appeared in the literature in more generalized forms in which the underlying polynomial was replaced by a more general class of functions. One such generalization is the passage from polynomials to rational functions. In this paper, we prove some inequalities for meromorphic functions with prescribed poles and restricted zeros. These results not only generalize some Bernstein-type inequalities for rational functions, but also improve and generalize some known polynomial inequalities. These inequalities have their own importance in the approximation theory.
Keywords: polynomials, Blaschke product, inequalities, rational functions. 
@article{UFA_2024_16_1_a8,
     author = {M. Y. Mir and W. M. Shah and S. L. Wali},
     title = {Inequalities for meromorphic functions with prescribed poles},
     journal = {Ufa mathematical journal},
     pages = {127--137},
     year = {2024},
     volume = {16},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2024_16_1_a8/}
}
TY  - JOUR
AU  - M. Y. Mir
AU  - W. M. Shah
AU  - S. L. Wali
TI  - Inequalities for meromorphic functions with prescribed poles
JO  - Ufa mathematical journal
PY  - 2024
SP  - 127
EP  - 137
VL  - 16
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/UFA_2024_16_1_a8/
LA  - en
ID  - UFA_2024_16_1_a8
ER  - 
%0 Journal Article
%A M. Y. Mir
%A W. M. Shah
%A S. L. Wali
%T Inequalities for meromorphic functions with prescribed poles
%J Ufa mathematical journal
%D 2024
%P 127-137
%V 16
%N 1
%U http://geodesic.mathdoc.fr/item/UFA_2024_16_1_a8/
%G en
%F UFA_2024_16_1_a8
M. Y. Mir; W. M. Shah; S. L. Wali. Inequalities for meromorphic functions with prescribed poles. Ufa mathematical journal, Tome 16 (2024) no. 1, pp. 127-137. http://geodesic.mathdoc.fr/item/UFA_2024_16_1_a8/

[1] S. N. Bernstein, “Sur la limitation des dérivées des polynomes”, C. R. Acad. Sci. Paris, 190:1 (1930), 338–340

[2] P. D. Lax., “Proof of a conjecture of P. Erdös on the derivative of a polynomial”, Bull. Amer. Math. Soc., 50:1 (1944), 509–513 | DOI | MR | Zbl

[3] P. Turán, “Über die Ableitung von Polynomen”, Compos. Math., 7 (1939), 89–95 | MR | Zbl

[4] M. A. Malik, “On the derivative of a polynomial”, J. Lond. Math. Soc., s2:2 (1969), 57–60 | DOI | MR | Zbl

[5] S. R. Garcia, J. Mashreghi and W. T. Ross, Finite Blaschke products and their connections, Springer, Switzerland, 2018 | MR | Zbl

[6] Q.I.Rahman and G.Schmeisser, Analytic Theory of Polynomials, Oxford Scientific Publ., Oxford, USA, 2002 | MR | Zbl

[7] G.V. Milovanović, D.S. Mitrinović and Th.M. Rassias, Topics in polynomials, extremal problems, inequalities, zeros, World Sci. Publ. Co., Singapore, 1994 | MR | Zbl

[8] A. Aziz, “Inequalities for the polar derivative of a polynomial”, J. Approx. Theory, 55:2 (1988), 183–193 | DOI | MR | Zbl

[9] W.M. Shah, “A generalization of a theorem of Paul Turán”, J. Ramanujan Math. Soc., 11:1 (1996), 67–72 | MR | Zbl

[10] A. Aziz and W. M. Shah, “Inequalities for the polar derivative of a polynomial”, Indian J. Pure Appl. Math., 29:1 (1998), 163–173 | MR | Zbl

[11] B. Chanam, “Bernstein type inequalities for polar derivative of polynomial”, Eur. J. Molec. Clinic. Medic, 8:2 (2021), 1650–1655 | MR

[12] X. Li, R. N. Mohapatra and R. S. Rodriguez, “Bernstein-type inequalities for rational functions with prescribed poles”, J. Lond. Math. Soc., 51:3 (1995), 523–531 | DOI | MR | Zbl

[13] X. Li., “A comparison inequality for rational functions”, Proc. Amer. Math. Soc., 139:5 (2011), 1659–1665 | DOI | MR | Zbl

[14] A. Mir, “Comparison inequality between rational functions with prescribed poles”, Revista de la Real Academia de Ciencias Exactas, F{\i}sicas y Naturales. Ser. A. Matemáticas, 115 (2021), 83 | DOI | MR | Zbl

[15] K. K. Dewan and S. Hans, “Generalization of certain well known polynomial inequalities”, J. Math. Anal. Appl., 363:1 (2010), 38–41 | DOI | MR | Zbl

[16] S. Hans, D. Tripathi, A. A. Mogbademu and B. Tyagi, “Inequalities for rational functions with prescribed poles”, J. Int. Math., 21:1 (2018), 157–169

[17] A. Aziz and Q. M. Dawood, “Inequalities for a polynomial and its derivative”, J. Approx. Theory, 54:3 (1988), 306–313 | DOI | MR | Zbl

[18] A. Aziz and W. M. Shah, “Some properties of rational functions with prescribed poles and restricted zeros”, Math. Balkanica, 18:1-2 (2004), 33–40 | MR | Zbl

[19] A. Aziz and W. M. Shah, “Some refinements of Bernstein type inequalities for rational functions”, Glas. Math. III Ser., 52:1 (1997), 29–37 | MR

[20] M. A. Malik and M. C. Vong, “Inequalities concerning the derivative of polynomials”, Rendiconts Del Circolo Matematico Di Palermo Serie II, 34:2 (1985), 422–426 | DOI | MR | Zbl

[21] M. Y. Mir, S. L. Wali and W. M. Shah, “Inequalities for meromorphic functions not vanishing outside the disk”, J. Math. Sci., 266:4 (2022), 526–533 | DOI | MR | Zbl