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Inequalities for the generalized polar derivative of a polynomial | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 14، دوره 16، شماره 6، شهریور 2025، صفحه 153-159 اصل مقاله (369.61 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2024.24504.2759 | ||
| نویسندگان | ||
| Nisar Ahmad Rather1؛ Naseer Wani1؛ Tanveer Bhat1؛ Ishfaq Dar* 2 | ||
| 1Department of Mathematics, University of Kashmir, Srinagar-190006, India | ||
| 2Department of Applied Sciences, Institute of Technology, University of Kashmir, Srinagar-190006, India | ||
| تاریخ دریافت: 19 شهریور 1400، تاریخ پذیرش: 16 فروردین 1403 | ||
| چکیده | ||
| Recently Rather et al. [18] considered the generalized polar derivative and studied the relative position of zeros of the generalized polar derivative with respect to the zeros of the polynomial. In this paper, by taking into account the size of the constant term and the leading coefficient of the polynomial $P(z)$, we obtain some lower bound estimates for the generalized polar derivative of certain polynomials, which refine and generalize various results due to Aziz and Rather, Malik, Tur'{a}n, Dubinin, Govil and others. | ||
| کلیدواژهها | ||
| Polynomials؛ Generalized polar derivative؛ Gauss Lucas Theorem؛ Inequalities in the complex domain | ||
| مراجع | ||
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[1] A. Aziz, Inequalities for the derivative of a polynomial, Proc. Amer. Math. Soc. 89 (1983), 259–266. [2] A. Aziz, Inequalities for the polar derivative of a polynomial, J. Approx. Theory 55 (1988), no. 2, 183–193. [3] A. Aziz and N.A. Rather, A refinement of a Theorem of Paul Turan concerning polynomials, Math. Inequal. Appl. 1 (1998), 231–238. [4] A. Aziz and N.A. Rather, On an inequality concerning the polar derivative of a polynomial, Proc. Indian Acad. Sci. (Math. Sci.), 117 (2007), 349–357. [5] F.A. Bhat, N.A. Rather, and S. Gulzar, Inequalities for the generalised derivative of a complex polynomial, Pub. Inst. Math. Nouvelle Ser. Tome 114 (2023), no. 128, 71–80. [6] V.N. Dubinin, Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros, J. Math. Sci. 143 (2007), 3069–3076. [7] N.K. Govil, On the Derivative of a polynomial, Proc. Amer. Math. Soc. 41 (1973), 543–546. [8] I. Dar and A. Iqbal, Some lower bounds for the derivative of certain polynomials, Ann. Univ. Ferrara 66 (2020), 295–300. [9] M.A. Malik, On the derivative of a polynomial, J. London Math. Soc. 2 (1969), no.1, 57–60. [10] M. Marden, Geometry of Polynomials, Math. Surveys, Amer. Math. Soc., 1989. [11] N.A. Rather, L. Ali, and I. Dar, Inequalities for the derivative of polynomials with restricted zeros, Korean J. Math. 28 (2020), no. 4, 931–942. [12] N.A. Rather, L. Ali, M. Shafi, and I. Dar, Inequalities for the generalized polar derivative of a polynomial, Palestine J. Math. 9 (2020), no. 2, 931–942. [13] N.A. Rather and I. Dar, Some applications of the Boundary Schwarz lemma for Polynomials with restricted zeros, Appl. Math. E-Notes 20 (2020), 422–431. [14] N.A. Rather, I. Dar, and A. Iqbal, Some extensions of a theorem of Paul Turan concerning polynomials, Kragujevac J. Math. 46 (2022), no. 6, 969–979. [15] N.A. Rather, I. Dar, and A. Iqbal, On a refinement of Turan's inequality, Complex Anal Synerg. 6 (2020), Article number 21. [16] N.A. Rather, I. Dar, and A. Iqbal, Some inequalities for polynomials with restricted zeros, Ann. Univ. Ferrara 67 (2021), 183–189. [17] N.A. Rather, A. Iqbal, and I. Dar, Inequalities concerning the Polar derivatives of Polynomials with restricted zeros, Nonlinear Funct. Anal. and Appl. 24 (2019), 813–826. [18] N.A. Rather, A. Iqbal, and I. Dar, On the zeros of a class of generalized derivatives, Rend. Circ. Mat. Palermo Ser 2 70 (2021), 1201–1211. [19] J. Sz-Nagy, Verallgemeinerung der Derivierten in de Geometric der Polynome, Acta Univ. Szeged. Sect. Sci. Math. 13 (1950), 169–178. [20] A.C. Schaeffer, Inequalities of A. Markoff and S. N. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc. 47 (1941), 565–579. [21] P. Turan, Uber die Ableitung von polynomen, Composit. Math. 7 (1939), 89–95. | ||
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