On an operator preserving Lp inequalities between polynomials

If P(z) is a polynomial of degree at most n which does not vanish in |z|<1, then it was recently claimed by Shah and Liman [W.M. Shah, A. Liman, Integral estimates for the family of B-operators, Oper. Matrices, 5 (2011) 79-87] that for every R≥1, p≥1, ‖B[P∘ρ](z)‖p≤Rn|Λ|+|λ0|‖1+z‖p‖P(z)‖p, where B is a Bn-operator with parameters λ0,λ1,λ2 in the sense of Rahman and Schmeisser (2002) [5], ρ(z)=Rz and Λ=λ0+λ1n22+λ2n3(n−1)8. Unfortunately the proof of this result is not correct. In this paper, we present certain sharp Lp-inequalities for Bn-operators which not only provide a correct proof of the above inequality and other related results but also extend these inequalities for 0≤p<1 as well.