On an operator preserving 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 ≤ R n | Λ | + | λ 0 | ‖ 1 + z ‖ p ‖ P ( z ) ‖ p , where B is a B n -operator with parameters λ 0 , λ 1 , λ 2 in the sense of Rahman and Schmeisser (2002) [5], ρ ( z ) = R z and Λ = λ 0 + λ 1 n 2 2 + λ 2 n 3 ( n − 1 ) 8 . Unfortunately the proof of this result is not correct. In this paper, we present certain sharp L p -inequalities for B n -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.