### Some Extensions of a Theorem of Paul Turán Concerning Polynomials

Download PDF

**Authors:**N. A. RATHER, I. DAR AND A. IQBAL

**DOI:**10.46793/KgJMat2206.969R

**Abstract:**

In this paper, certain new results concerning the maximum modulus of the polar derivative of a polynomial with restricted zeros are obtained. These estimates strengthen some well known inequalities for polynomial due to Turán, Dubinin and others.

**Keywords:**

Polynomials, polar derivative, inequalities in the complex domain.

**References:**

[1] A. Aziz, Inequalities for the derivative of a polynomial, Proc. Amer. Math. Soc. 89 (1983), 259–266.

[2] A. Aziz and Q. M. Dawood, Inequalities for a polynomial and its derivatives, J. Approx. Theory 54 (1998), 306–313.

[3] A. Aziz and N. A. Rather, Some Zygmund type L^{q} inequalities
for polynomials, J. Math. Anal. Appl. 289 (2004), 14–29.

[4] A. Aziz and N. A. Rather, Inequalities for the polar derivative of a polynomial with restricted zeros, Math. Balkanica 17 (2003), 15–28.

[5] A. Aziz and N. A. Rather, A reﬁnement of a theorem of Paul Turán concerning polynomials, Math. Inequal. Appl. 1 (1998), 231–238.

[6] V. N. Dubinin, Distortion theorems for polynomials on a circle, Sb. Math. 191(12) (2000), 1797–1807.

[7] C. Frappier, Q. I. Rahman and St. Ruscheweyh, New inequalities for polynomials, Trans. Amer. Math. Soc. 288 (1985), 69–99.

[8] N. K. Govil, On the derivative of a polynomial, Proc. Amer. Math. Soc. 41 (1973), 543–546.

[9] P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509–513.

[10] M. Marden, Geometry of Polynomials, Amer. Math. Soc., Math. Surveys 3, Providence, 1966.

[11] G. V. Milovanović, D. S. Mitrinović and Th. M. Rassias, Topics in Polynomials: Extremal Properties, Inequalities, Zeros, World scientiﬁc Publishing Co., Singapore, 1994.

[12] G. Plya and G. Szeg, Problems and Theorems in Analysis, II, Springer-Verlag, Berlin, New York, 1976.

[13] N. A. Rather and S. Gulzar, Generalization of an inequality involving maximum moduli of a polynomial and its polar derivative, Nonlinear Functional Analysis and Applications 19 (2014), 213–221.

[14] N. A. Rather and S. Gulzar, Reﬁnements of some inequalities concerning the polar derivative of a polynomial, Funct. Approx. Comment. Math. 51 (2014), 269–283.

[15] S. Gulzar and N. A. Rather, Inequalities concerning the polar derivative of a polynomial, Bull. Malays. Math. Sci. Soc. 40 (2017), 1691–1700.

[16] P. Turán, Über die ableitung von polynomen, Compos. Math. 7 (1939), 89–95.