Abstract
In this paper, we shall obtain some inequalities for the polar derivative of polynomial having all zeros in \(|z|\le k, k \ge 1.\) Our results sharpen some well-known results of Turán, Dubinin and others.
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References
Aziz, Abdul: Inequalities for the derivative of a polynomial. Proc. Am. Math. Soc. 89(2), 259–266 (1983)
Aziz, A., Rather, N.A.: Some Zygmund type \(L^q\) inequalities for polynomials. J. Math. Anal. Appl. 289, 14–29 (2004)
Aziz, A., Rather, N.A.: Inequalities for the polar derivative of a polynomial with restricted zeros. Math. Bulk. 17, 15–28 (2003)
Aziz, A., Rather, N.A.: A refinement of a theorem of Paul Turán concerning polynomials. Math. Ineq. Appl. 1, 231–238 (1998)
Dubinin, V.N.: Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros. J. Math. Sci. 143, 3069–3076 (2007)
Govil, N.K.: On the derivative of a polynomial. Proc. Am. Math. Soc. 41, 543–546 (1973)
Govil, N.K., Kumar, P.: On sharpening of an inequality of Turán. Appl. Anal. Discrete Math. 13, 711–720 (2019)
Marden, M.: Geometry of Polynomials, Mathematical Surveys, No. 3, American Mathematical Society, Providence, RI (1966)
Milovanović, G.V., Mitrinović, D.S., Rassias, ThM: Topics in Polynomials: Extremal Properties, Inequalities, Zeros. World scientific Publishing Co., Singapore (1994)
P\(\acute{o}\)lya, G., Szeg\(\ddot{o}\), G.: Problems and Theorems in Analysis, II. Springer, Berlin (1976)
Rather, N.A., Dar, I.: Some applications of the Boundary Schwarz lemma for Polynomials with restricted zeros, Appl. Math. E (to appear)
Rather, N.A., Iqbal, A., Dar, I.: Inequalities concerning the Polar derivatives of Polynomials with restricted zeros. Non linear Funct. Anal. Appl. 24, 813–826 (2019)
Rather, N.A., Dar, I., Iqbal, A.: Inequalities for the derivative of polynomials with restricted zeros, arXiv:submit/2648741 10 April 2019
Turán, P.: Über die Ableitung von Polynomen. Compos. Math. 7, 89–95 (1939)
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Rather, N.A., Dar, I. & Iqbal, A. On a refinement of Turán’s inequality. Complex Anal Synerg 6, 21 (2020). https://doi.org/10.1007/s40627-020-00058-5
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DOI: https://doi.org/10.1007/s40627-020-00058-5