Generalized Turan-type Inequalities for Polar Derivative of a Polynomial

Year 2023, , 178 - 191, 25.10.2023

Kshetrimayum Krishnadas , Thangjam Singh Barchand Chanam

https://doi.org/10.36753/mathenot.948462

Abstract

Let $P(z)=a_0+\sum\limits_{\nu=\mu}^na_{\nu}z^{\nu}$, $1\leq\mu\leq n$, be a polynomial of degree $n$ having all its zeros in $|z|\leq k$, $k\geq 1$. We obtain an improvement and a generalization of an inequality in polar derivative proved by Somsuwan and Nakprasit [1]. Further, we also extend a result proved by Chanam and Dewan [2] to its polar version. Besides, our results are also found to generalize and improve some known inequalities.

Keywords

Turan type inequality, Polynomial, Polar Derivative, Maximum Modulus

References

• [1] Somsuwan, J., Nakprasit, K. M.: Some bounds for the polar derivative of a polynomial. International Journal of Mathematics and Mathematical Sciences. 2018, 5034607 (2018).
• [2] Chanam, B., Dewan, K. K.: Inequalities for a polynomial and its derivatives. Journal of Interdisciplinary Mathematics. 11(4), 469-478 (2008).
• [3] Bernstein, S.: Lecons sur les propriétés extrémales et la meilleure approximation desfonctions analytiques d’une variable réelle. Gauthier Villars. Paris (1926).
• [4] Lax, P. D.: Proof of a conjecture of P. Erdös on the derivative of a polynomial. Bull. Amer. Math. Soc. 50, 509-513 (1944).
• [5] Turán, P.: Über die Ableitung von Polynomen. Compositio Mathematica. 7, 89-95 (1939).
• [6] Malik, M. A.: On the derivative of a polynomial. Journal of the London Mathematical Society. 2(1), 57-60 (1969).
• [7] Aziz, A., Zargar, B. A.: Inequalities for a polynomial and its derivative. Mathematical Inequalities and Applications. 1(4), 543-550 (1998).
• [8] Aziz, A., Rather, N. A.: A refinement of a theorem of Paul Turán concerning polynomials. Mathematical Inequalities and Applications. 1(2), 231-238 (1998).
• [9] Dewan, K. K., Upadhye, C. M.: Inequalities for the polar derivative of a polynomial. Journal of Inequalities in Pure and Applied Mathematics. 9(4), 1-9 (2008).
• [10] Gardner, R. B., Govil, N. K., Musukala, S. R.: Rate of growth of polynomials not vanishing inside a circle. Journal of Inequalities in Pure and Applied Mathematics. 6(2), 1-9 (2005).
• [11] Pólya, G., Szegö, G.: Aufgaben and Lehratze ous der Analysis I (Problems and Theorems in Analysis I). Springer-Verlag. Berlin (1925).
• [12] Chanam, B., Dewan, K. K.: Inequalities for a polynomial and its derivatives. J. Math. Anal. Appl. 336, 171-179 (2007).
• [13] Qazi, M. A.: On the maximum modulus of polynomials. Proc. Amer. Math. Soc. 115, 337-343 (1992).