New Inequalities on $L_{\gamma}$-Spaces

Abstract

We consider for a fixed $\mu$, the class of polynomials $$P_{n,\mu,s}:=\Bigg\{ P(z) =z^s(a_nz^{n−s}+ \sum^{n-s}_{j=\mu}a_{n−j}z^{n−j−s}); 1≤\mu≤n−s\Bigg\}$$ of degree $n$, having all zeros in $|z|≤k,$ $k≤1$, with $s$-fold zeros at the origin. In this paper, we have obtained inequalities in the reverse direction for the above class of polynomials. Besides, extensions of some Turan-type inequalities for the polar derivative of polynomials have been considered.