Geodesic
Inequalities for a class of meromorphic functions whose zeros are within or outside a given disk
Ural mathematical journal, Tome 9 (2023) no. 1, pp. 104-112 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, we consider a class of meromorphic functions $r(z)$ having an $s$-fold zero at the origin and establish some inequalities of Bernstein and Turán type for the modulus of the derivative of rational functions in the sup-norm on the disk in the complex plane. These results produce some sharper inequalities while taking into account the placement of zeros of the underlying rational function. Moreover, many inequalities for polynomials and polar derivatives follow as special cases. In particular, our results generalize as well as refine a result due Dewan et al. [6].
Keywords: rational function, $s$-fold zeros, Bernstein inequality.
Mots-clés : polynomial 
@article{UMJ_2023_9_1_a7,
     author = {Mohd Yousf Mir and Shah Lubn Wali and Wali Mohammad Shah},
     title = {Inequalities for a class of meromorphic functions whose zeros are within or outside a given disk},
     journal = {Ural mathematical journal},
     pages = {104--112},
     year = {2023},
     volume = {9},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UMJ_2023_9_1_a7/}
}
TY  - JOUR
AU  - Mohd Yousf Mir
AU  - Shah Lubn Wali
AU  - Wali Mohammad Shah
TI  - Inequalities for a class of meromorphic functions whose zeros are within or outside a given disk
JO  - Ural mathematical journal
PY  - 2023
SP  - 104
EP  - 112
VL  - 9
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/UMJ_2023_9_1_a7/
LA  - en
ID  - UMJ_2023_9_1_a7
ER  - 
%0 Journal Article
%A Mohd Yousf Mir
%A Shah Lubn Wali
%A Wali Mohammad Shah
%T Inequalities for a class of meromorphic functions whose zeros are within or outside a given disk
%J Ural mathematical journal
%D 2023
%P 104-112
%V 9
%N 1
%U http://geodesic.mathdoc.fr/item/UMJ_2023_9_1_a7/
%G en
%F UMJ_2023_9_1_a7
Mohd Yousf Mir; Shah Lubn Wali; Wali Mohammad Shah. Inequalities for a class of meromorphic functions whose zeros are within or outside a given disk. Ural mathematical journal, Tome 9 (2023) no. 1, pp. 104-112. http://geodesic.mathdoc.fr/item/UMJ_2023_9_1_a7/

[1] Aziz A. “A refinement of an inequality of S. Bernstein”, J. Math. Anal. Appl., 142:1 (1989), 226–235 | DOI | MR

[2] Aziz A., Shah W. M. “Some refinements of Bernstein-type inequalities for rational functions”, Glas. Math., 32:1 (1997), 29–37 | MR

[3] Aziz-Ul-Auzeem A., Zargar B. A. “Some properties of rational functions with prescribed poles”, Canad. Math. Bull., 42:4 (1999), 417–426 | DOI | MR | Zbl

[4] Akhter T., Malik S. A., Zargar B. A. “Turán type inequalities for rational functions with prescribed poles”, Int. J. Nonlinear Anal. Appl., 13:1 (2022), 1003–1009 | DOI

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

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

[7] Garcia S. R., Mashreghi J., Ross W. T. Finite Blaschke Products and Their Connections, Springer, Cham, 2018, 328 pp. | DOI | MR | Zbl

[8] Hans S., Tripathi D., Mogbademu A. A., Babita Tyagi. “Inequalities for rational functions with prescribed poles”, J. Interdiscip. Math, 21:1 (2018), 157–169 | DOI | Zbl

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

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

[11] Turán P. “Über die ableitung von polynomen”, Compos. Math., 7 (1939), 89—95 (in German) | MR | Zbl

[12] Wali S. L., Shah W. M. “Applications of the Schwarz lemma to inequalities for rational functions with prescribed poles”, J. Anal., 25:1 (2017), 43–53 | DOI | MR | Zbl