Let , , be a polynomial of degree n having no zero in , , then Qazi [Proc. Amer. Math. Soc., 115 (1992), 337-343] proved
.
In this paper, we first extend the above inequality to polar derivative of a polynomial. Further, as an application of our result, we extend a result due to Dewan et al. [Southeast Asian Bull. Math., 27 (2003), 591-597] to polar derivative.
References
[1]
Bernstein, S. (1926) Lecons sur les propriétésextrémaleset la meilleure approximation des fonctions analytiques d’une variablereele. Gauthier Villars, Paris.
[2]
Lax, P.D. (1944) Proof of a Conjecture of P. Erdös on the Derivative of a Polynomial. Bulletin of the American Mathematical Society, 50, 509-513. http://dx.doi.org/10.1090/S0002-9904-1944-08177-9
[3]
Malik, M.A. (1969) On the Derivative of a Polynomial. Journal of the London Mathematical Society, 1, 57-60. http://dx.doi.org/10.1112/jlms/s2-1.1.57
[4]
Qazi, M.A. (1992) On the Maximum Modulus of Polynomials. Proceedings of the American Mathematical Society, 115, 337-343. http://dx.doi.org/10.1090/S0002-9939-1992-1113648-1
[5]
Dewan, K.K., Singh, H. and Yadav, R.S. (2003) Inequalities Concerning Polynomials Having Zeros in Closed Exterior or Closed Interior of a Circle. Southeast Asian Bulletin of Mathematics, 27, 591-597.
[6]
Laguerre, E. (1898) Oeuvres. 1 Gauthier-Villars, Paris.
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Marden, M. (1949) Geometry of the Zeros of Polynomials in a Complex Variable. Math. Surveys, No.3. Amer. Math. Soc., Providence, R.I.
[9]
Aziz (1988) Inequalities for the Polar Derivative of a Polynomial. Journal of Approximation Theory, 55, 183-193. http://dx.doi.org/10.1016/0021-9045(88)90085-8
[10]
Dewan, K.K. and Singh, B. (2006) Inequalities for the Polar Derivative of a Polynomial. Journal of Combinatorics, Information and System Sciences, 31, 317-324.
[11]
Govil, N.K., Rahman, Q.I. and Schmeisser, G. (1979) On the Derivative of a Polynomial. Illinois Journal of Mathematics, 23, 319-329.
[12]
Mir, A. and Dar, B. (2014) On the Polar Derivative of a Polynomial. Journal of the Ramanujan Mathematical Society, 29, 403-412.
[13]
Govil, N.K. and Rahman, Q.I. (1969) Functions of Exponential Type Not Vanishing in a Half-Plane and Related Polynomials. Transactions of the American Mathematical Society, 137, 501-517. http://dx.doi.org/10.1090/S0002-9947-1969-0236385-6
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, ![\"\"](\"Edit_4e5b47b0-74c6-4f13-b471-ff7a3f36300c.jpg\")
, be a polynomial of degree n having no zero in ![\"\"](\"Edit_2c04c073-4106-4b2a-abd0-ddefd781eb0a.jpg\")
, ![\"\"](\"Edit_02da3529-097c-4a25-8ac0-2c158907f231.jpg\")
, then Qazi [Proc. Amer. Math. Soc., 115 (1992), 337-343] proved
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.
