(untitled) (Analysis in Theory and Applications, 2014)
Let P_n be the class of polynomials of degree at most n. Rather and Shah [15] proved that if PP_n and P(z) eq 0 in |z| < 1, then for every R > 0 and 0 ≤ q < ∞, | B[P(Rz)]|_q ≤ | R^nB[z^n] +λ_0 |_q | 1+z^n|_q | P(z)|_q,where B is a B_n-operator. In this paper, we prove some generalization of this result which in particular yields some known polynomial inequalities as special. We also consider an operator D_α which maps a polynomial P(z) into D_α P(z) := n P(z) + ( α - z ) P' (z) and obtain extensions and generalizations of a number of well-known L_q inequalities.