Turán-Type Inequalities of Polynomials

Robinson Soraisam, Barchand Chanam, Mahmood Bidkham

If $p(z)=\sum_{\nu=0}^na_{\nu}z^{\nu}$ is a polynomial of degree $n$ having all its zeros in $|z|\leq k$, $k\geq 1$, V.\,K. Jain [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. extbf{59} (2016), 339--347] proved \[ \max_{|z|=1}|p'(z)|\geq\frac{neft(|a_0|+|a_n|k^{n+1}\right)}{|a_o|eft(1+k^{n+1}\right)+|a_n|eft(k^{n+1}+k^{2n}\right)}\max_{|z|=1}|p(z)|. \] We first obtain a generalization as well as improvement of the above inequality. Further, we extend our first result to a more generalized result which yields improved results of some known inequalities as particular case.