Sharpening of Tur´an-type inequality for polynomials

Abstract

For the polynomial P(z) = n \sum j=0 cjzj of degree n having all its zeros in | z| \leq k, k \geq 1, V. Jain in “On the derivative of a polynomial”, Bull. Math. Soc. Sci. Math. Roumanie Tome 59, 339–347 (2016) proved that max | z| =1 | P \prime (z)| \geq n \biggl( | c0| + | cn| kn+1 | c0| (1 + kn+1) + | cn| (kn+1 + k2n) \biggr) max | z| =1 | P(z)| . In this paper we strengthen the above inequality and other related results for the polynomials of degree n \geq 2.