Inequalities for the polar derivative of a polynomial with $S$-fold zeros at the origin

Document Type: Research Paper

Authors

Department of Mathematics‎, ‎University of Semnan‎, ‎Semnan‎, ‎Iran.

Abstract

‎Let $p(z)$ be a polynomial of degree $n$ and for a complex number $\alpha$‎, ‎let $D_{\alpha}p(z)=np(z)+(\alpha-z)p'(z)$ denote the polar derivative of the polynomial p(z) with respect to $\alpha$‎. ‎Dewan et al proved‎ ‎that if $p(z)$ has all its zeros in $|z| \leq k,\ (k\leq‎ ‎1),$ with $s$-fold zeros at the origin then for every‎ ‎$\alpha\in\mathbb{C}$ with $|\alpha|\geq k$‎,
‎\begin{align*}‎
‎\max_{|z|=1}|D_{\alpha}p(z)|\geq‎
‎\frac{(n+sk)(|\alpha|-k)}{1+k}\max_{|z|=1}|p(z)|‎.
‎\end{align*} In this paper‎, ‎we obtain a refinement‎ ‎of above inequality‎. ‎Also as an application of our result‎, ‎we extend some inequalities for‎ ‎polar derivative of a polynomial of degree $n$ which‎ ‎does not vanish in $|z|< k$‎, ‎where $k\geq 1$‎, ‎except $s$-fold zeros at the origin‎.

Keywords

Main Subjects

References

M. Ahmadi Baseri, M. Bidkham and M. Eshaghi Gordji, A generating operator of inequalities for polynomials, Bull. Math. Soc. Sci. Math. Roumanie 104 (2013) 151--162.

A. Aziz and N. A. Rather, A refinement of a theorem of Paul Turan concerning polynomials, Math. Inequal. Appl. 1 (1998) 231--238.

A. Aziz and W. M. Shah, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 7 (2004) 379--391.

S. Bernstein, Sur la limitation des derivees des polnomes, C. R. Math. Acad. Sci. Paris 190 (1930), 338--341.

M. Bidkham, M. Shakeri and M. Eshaghi Gordji, Inequalities for the polar derivative of a polynomial, J. Inequal. Appl. 2009 (2009), Article ID 515709, 9 pages.

K.K. Dewan and S. Hans, Generalization of certain well-known polynomial inequalities, J. Math. Anal. Appl. 363 (2010) 38-41.

K.K. Dewan and A. Mir, Inequalities for the polar derivative of a polynomial, J. Interdiscip. Math. 10 (2007) 525--531.

K.K. Dewan, N. Singh and A. Mir, Extension of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl. 352 (2009) 807--815.

N.K. Govil, Some inequalities for derivative of polynomials, J. Approx. Theory 66 (1991) 29--35.

N.K. Govil and G.N. McTume, Some generalization involving the polar derivative for an inequality of Paul Turan, Acta Math. Hungar. 104 (2004) 115--126.

E. Laguerre, Sur le role des emanants dans la theorie des equations numeriques, C. R. Math. Acad. Sci. Soc. R. Can. 78 (1874) 278--280.

P.D. Lax, Proof of a conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc. (N.S.) 50 (1944) 509--513.

A. Liman, R.N. Mohapatra and W.M. Shah, Inequalities for the polar derivative of a polynomial, Complex Anal. Oper. Theory 6 (2012) 1199--1209.

M.A. Malik, On the derivative of a polynomial, J. Lond. Math. Soc. (2) 1 (1969) 57--60.

M. Mardan, Geometry of Polynomials: Mathematical Surveys, Vol. 3, Amer. Math. Soc. Providence, RI, 1966.

Q.I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Clarendon Press, New York, 2002.

P. Turan, Uber die Ableitung von Polynomen, Compos. Math. 7 (1939) 89--95.

A. Zireh, On the maximum modulus of a polynomial and its polar derivative, J. Inequal. Appl. 2011 (2011), no. 111, 9 pages.

History

• Receive Date: 18 August 2016
• Revise Date: 09 January 2017
• Accept Date: 09 January 2017