Extremal problems of Tur´an-type involving the location of all zeros of a class of rational functions
DOI:
https://doi.org/10.21638/spbu01.2024.206Abstract
In this paper, we prove a Tur´an-type inequality for rational functions and thereby extend it to a more general class of rational functions r(s(z)) of degree mn with prescribed poles, where s(z) is a polynomial of degree m. These results not only generalize some Tur´antype inequalities for rational functions, but also improve as well as generalize some known polynomial inequalities.Keywords:
rational function, polynomials, zeros, polar derivative, inequalities
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References
Литература
1. Bernstein S. N. Sur la limitation des d´eriv´ees des polynomes. C. R. Acad. Sci. Paris 190, 338-340 (1930).
2. Tur´an P. Uber die ableitung von polynomen. ¨ Compos. Math. 7, 89-95 (1939).
3. Jain V. K. Generalizations of certain well known inequalities for polynomials. Glas. Math. 32, 45-51 (1997).
4. Li X., Mohapatra R. N. Rodriguez R. S. Bernstein-type inequalities for rational functions with prescribed poles. J. London Math. Soc. 51, 523-531 (1995).
5. Mir A. Certain estimates of the derivative of a meromorphic function on boundary of the unit disk, Indian. J. Pure Appl. Math. 50 (2), 315-331 (2019).
6. Mir A. Comparison inequalities between rational functions with prescribed poles. RACSAM. (2021). https://doi.org/10.1007/s13398-021-01023-5
7. Arunrat N., Nakprasit K. M. Bounds of the derivative of some classes of rational functions. Mathematics and Mathematical Sciences 52, 1-7 (2020).
8. Aziz A., Shah W. M. Some refinements of Bernstein- type inequalities for rational functions. Glas. Math. 32, 29-37 (1997).
9. Aziz A., Dawood Q. M. Inequalities for a polynomial and its derivative. Journal of Approximation theory 54, 306-313 (1988).
10. Aziz A., Shah W. M. Some properties of rational functions with prescribed poles and restricted zeros. Math. Balkanica 18, 33-40 (2004).
11. Akhtar T., Malik S. A., Zargar B. A. Tur´an-type inequalities for rational functions with prescribed poles. Int. J. Nonlinear Anal. Appl. 13, 1003-1009 (2022).
12. Markov A. On a problem of D. I. Mendeleev. Zapiski Imperatorskoi Akademii nauk 62, 1-24 (1889).
13. Mir M. Y., Wali S. L., Shah W. M. Inequalities for a class of rational functions. Int. J. Nonlinear Anal. Appl. 13 (2), 609-617
References
1. Bernstein S. N. Sur la limitation des d´eriv´ees des polynomes. C. R. Acad. Sci. Paris 190, 338-340 (1930).
2. Tur´an P. Uber die ableitung von polynomen. ¨ Compos. Math. 7, 89-95 (1939).
3. Jain V. K. Generalizations of certain well known inequalities for polynomials. Glas. Math. 32, 45-51 (1997).
4. Li X., Mohapatra R. N. Rodriguez R. S. Bernstein-type inequalities for rational functions with prescribed poles. J. London Math. Soc. 51, 523-531 (1995).
5. Mir A. Certain estimates of the derivative of a meromorphic function on boundary of the unit disk, Indian. J. Pure Appl. Math. 50 (2), 315-331 (2019).
6. Mir A. Comparison inequalities between rational functions with prescribed poles. RACSAM. (2021). https://doi.org/10.1007/s13398-021-01023-5
7. Arunrat N., Nakprasit K. M. Bounds of the derivative of some classes of rational functions. Mathematics and Mathematical Sciences 52, 1-7 (2020).
8. Aziz A., Shah W. M. Some refinements of Bernstein- type inequalities for rational functions. Glas. Math. 32, 29-37 (1997).
9. Aziz A., Dawood Q. M. Inequalities for a polynomial and its derivative. Journal of Approximation theory 54, 306-313 (1988).
10. Aziz A., Shah W. M. Some properties of rational functions with prescribed poles and restricted zeros. Math. Balkanica 18, 33-40 (2004).
11. Akhtar T., Malik S. A., Zargar B. A. Tur´an-type inequalities for rational functions with prescribed poles. Int. J. Nonlinear Anal. Appl. 13, 1003-1009 (2022).
12. Markov A. On a problem of D. I. Mendeleev. Zapiski Imperatorskoi Akademii nauk 62, 1-24 (1889).
13. Mir M. Y., Wali S. L., Shah W. M. Inequalities for a class of rational functions. Int. J. Nonlinear Anal. Appl. 13 (2), 609-617
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Published
2024-08-10
How to Cite
Mir, M. Y., Wali, S. L., & Shah, W. M. (2024). Extremal problems of Tur´an-type involving the location of all zeros of a class of rational functions. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 11(2), 324–331. https://doi.org/10.21638/spbu01.2024.206
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Mathematics
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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.
