On L inequalities involving polar derivative of a polynomial.

We prove that if P( z) is a polynomial of degree n with zeros z , that satisfy $${|z_m| \geq K_m \geq 1}$$ , $${1 \leq m \leq n}$$ , then for any p > 0, and for every complex number α, with $${|\alpha| \geq 1}$$ , we have where $${G_{p}=\big\{\frac{2\pi}{\int_0^{2\pi}|t_0+e^{i\theta}|^{p}\,d\theta}\big\}^{{1}/{p}}}$$ , and $${t_0=\big\{1+\frac{n}{\sum_{m=1}^n\frac{1}{K_m-1}}\big\}}$$ if $${K_{m}>1}$$ $${(1\leq m \leq n)}$$ , and t = 1 if K = 1 for some m, $${1\leq m\leq n}$$ . Our results generalize and sharpen several of the known results. [ABSTRACT FROM AUTHOR]