Extension of Turán-Type Inequalities for Polar Derivatives of Polynomials Into Integral Mean Version


Barchand Chanam




Let $p(z)$ be a polynomial of degree $n$, let $D_{\alpha}p(z)=np(z)+(\alpha-z)p'(z)$ denote the polar derivative of the polynomial $p(z)$ with respect to a real or complex number $\alpha$. If $p(z)$ is a polynomial of degree $n$ having all its zeros in $|z|\leq k$, $k\geq1$, then for a real or complex number $\alpha$ with $|\alpha|\geq k$, Aziz and Rather [J. Math. Ineq. Appl. extbf{1} (1998), 231--238] proved \[ \max_{|z|=1}|D_{lpha}p(z)|\geq n\Big(\frac{|lpha|-k}{1+k^n}\Big)\max_{|z|=1}|p(z)|. \] We first extend the above inequality into integral mean without applying subordination property. As an application of our result, we prove another integral mean inequality. Our results have interesting consequences to the earlier well-known inequalities.