Inequalities for the derivative of rational functions with prescribed poles and restricted zeros
DOI:
https://doi.org/10.21638/spbu01.2023.309Abstract
In this paper, instead of assuming that a rational function r(z) with prescribed poles has a zero of order s at origin, we suppose that it has a zero of multiplicity s at any point inside the unit circle, whereas the remaining zeros are within or outside a circle of radius k and prove some results which besides generalizing some inequalities for rational functions include refinements of some polynomial inequalities as special cases.Keywords:
inequalities, polynomials, rational functions, poles, zeros
Downloads
Download data is not yet available.
References
Литература
1. Bernstein S. Sur la limitation des d´eriv´ees des polynomes. C. R. Acad. Sci. Paris. 190, 338-340 (1930).
2. Schaeffer A. C. Inequalities of A. Markoff and S. Bernstein for polynomials and related functions. Bull. Amer. Math. Soc. 47, 565-579 (1941).
3. Lax P. D. Proof of a conjecture of P. Erd¨os on the derivative of a polynomial. Bull. Amer. Math. Soc. (N. S.) 50, 509-513 (1944).
4. Tur´an P. Uber die ableitung von polynomen. ¨ Compos. Math. 7, 89-95 (1939).
5. Malik M. A. On the derivative of a polynomial. J. London Math. Soc. 1, 57-60 (1969). https://doi.org/10.1112/jlms/s2-1.1.57
6. Aziz A., Shah W. M. Inequalities for a polynomial and its derivative. Math. Inequal. Appl. 7, 379-391 (2004).
7. Borwein P., Erd´elyi T. Sharp extensions of Bernstein inequality to rational spaces. Mathematika 43, 413-423 (1996).
8. Xin Li, Mohapatra R. N., Rodriguez R. S. Bernstein-type inequalities for rational functions with prescribed poles. J. London Math. Soc. 51, 523-531 (1995).
9. Sheil-Small T. Complex polynomials. Cambridge Stud. Adv. Math. (2002).
10. Aziz A., Shah W. M. Some properties of rational functions with prescribed poles and restricted zeros. Math. Balkanica 18, 33-40 (2004).
11. Wali S. L. Inequalities for Maximum Modulus of Rational functions with Prescribed Poles (preprint). Kragujevac J. Math. 47, 865-875 (2023).
12. Ahanger U. M., Shah W. M. Inequalities for the derivative of polynomial with restricted zeros. J. Analysis 29, 1-8 (2021).
13. Aziz A., Zargar B. A. Some properties of rational functions with prescribed poles. Canad. Math. Bull. 42, 417-426 (1999).
14. Dubinin V. N. Applications of the Schwarz Lemma to inequalities for entire functions with constraints on zeros. Journal of Mathematical Sciences 143, 3069-3076 (2007).
References
1. Bernstein S. Sur la limitation des d´eriv´ees des polynomes. C. R. Acad. Sci. Paris. 190, 338-340 (1930).
2. Schaeffer A. C. Inequalities of A. Markoff and S. Bernstein for polynomials and related functions. Bull. Amer. Math. Soc. 47, 565-579 (1941).
3. Lax P. D. Proof of a conjecture of P. Erd¨os on the derivative of a polynomial. Bull. Amer. Math. Soc. (N. S.) 50, 509-513 (1944).
4. Tur´an P. Uber die ableitung von polynomen. ¨ Compos. Math. 7, 89-95 (1939).
5. Malik M. A. On the derivative of a polynomial. J. London Math. Soc. 1, 57-60 (1969). https://doi.org/10.1112/jlms/s2-1.1.57
6. Aziz A., Shah W. M. Inequalities for a polynomial and its derivative. Math. Inequal. Appl. 7, 379-391 (2004).
7. Borwein P., Erd´elyi T. Sharp extensions of Bernstein inequality to rational spaces. Mathematika 43, 413-423 (1996).
8. Xin Li, Mohapatra R. N., Rodriguez R. S. Bernstein-type inequalities for rational functions with prescribed poles. J. London Math. Soc. 51, 523-531 (1995).
9. Sheil-Small T. Complex polynomials. Cambridge Stud. Adv. Math. (2002).
10. Aziz A., Shah W. M. Some properties of rational functions with prescribed poles and restricted zeros. Math. Balkanica 18, 33-40 (2004).
11. Wali S. L. Inequalities for Maximum Modulus of Rational functions with Prescribed Poles (preprint). Kragujevac J. Math. 47, 865-875 (2023).
12. Ahanger U. M., Shah W. M. Inequalities for the derivative of polynomial with restricted zeros. J. Analysis 29, 1-8 (2021).
13. Aziz A., Zargar B. A. Some properties of rational functions with prescribed poles. Canad. Math. Bull. 42, 417-426 (1999).
14. Dubinin V. N. Applications of the Schwarz Lemma to inequalities for entire functions with constraints on zeros. Journal of Mathematical Sciences 143, 3069-3076 (2007).
Downloads
Published
2023-09-23
How to Cite
Ahanger, U. M., & Shah, W. M. (2023). Inequalities for the derivative of rational functions with prescribed poles and restricted zeros. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 10(3), 554–567. https://doi.org/10.21638/spbu01.2023.309
Issue
Section
Mathematics
License
Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
