2013/06/04 by N. A. Rather, Rather, N. A., Suhail Gulzar +1
Mathematics · #26D10 #41A17 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:26D10 #msc:41A17
paper · pdf · doi:10.48550/arxiv.1306.0714
16 Pages
arxiv created 2013/06/04 · arxiv updated 2013/06/05
If P(z) be a polynomial of degree at most n which does not vanish in |z| < 1, it was recently formulated by Shah and Liman \cite[Integral estimates for the family of B-operators, Operators and Matrices, 5(2011), 79 - 87]wl that for every R≥ 1, p≥ 1, ‖B[P∘σ](z)‖p ≤\fracRn|Λn|+|λ0|‖1+z‖p‖P(z)‖p, where B is a Bn-operator with parameters λ0, λ1, λ2 in the sense of Rahman \citeqir, σ(z)=Rz and Λn=λ0+λ1\fracn22 +λ2\fracn3(n-1)8. Unfortunately the proof of this result is not correct. In this paper, we present a more general sharp Lp-inequalities for Bn-operators which not only provide a correct proof of the above inequality as a special case but also extend them for 0 ≤ p <1 as well.