Some integral inequalities for the polar derivative of polynomials

Prasanna Kumar

As a generalization of well-known result due to Turán \cite{T} for polynomials having all their zeros in $|z|\leq1$, Malik \cite{M} proved that, if $P(z)$ is a polynomial of degree $n$, having all its zeros in $|z|\leq1$, then for any $\delta>0$, \[ n\bigg\{\int_0^{2i}|P(e^{iheta})|^{ẹlta}dheta\bigg\}^{1/ẹlta}eq\bigg\{\int_0^{2i}|1+e^{iheta}|^{ẹlta}dheta\bigg\}^{1/ẹlta}\max_{|z|=1}|P'(z)|. \] We generalize the above inequality to polar derivatives, which as special cases include several known results in this area. Besides the paper contains some more results that generalize and sharpen several results known in this direction.