Integral estimates for the family of B-Operators W.M.Shah Post Graduate Department of Mathematics, Govt.College,Baramulla,Kashmir-193101 India. e-mail: [email protected]
ABSTRACT Let P(z) be a polynomial of degree n.In this paper, we discuss inequalities for the family of operators B = B(λ0 , λ1 , λ2 , .) This family was introduced by Rahman and Maps.A polynomial 2 00 nz nz P (z) B[P (z)] = λ0 P (z) + λ1 P 0 (z) + λ2 2 2 2! n(n−1)(n−2) where λ0 , λ1 , λ2 are such that all the zeros of U (z) = λ0 + n(n−1) λ2 z 2 lie 2! λ1 z + 3! in the half plane |z| ≤ |z| − n2 .
Particular choice of λ0 s gives classical inequalities such as inequality of Zygmund,Hardy and their generalizations.