Sharpening of Turán-Type Inequality for Polynomials

For the polynomial P(z) = Sigma(n)(j=0)c(j)z(j) of degree n having all its zeros in |z| <= k, k >= 1, 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'(z)| >= n(|c(0)|+|c(n)|k(n+1)/|c(0)|(1+k(n+1))+|c(n)|(k(n+1)+k(2n)))max(|z|=1) |P(z)|. In this paper we strengthen the above inequality and other related results for the polynomials of degree n >= 2.