Bohr and Rogosinski inequalities for operator valued holomorphic functions

Abstract

For any complex Banach space X and each p?[1,?), we introduce the p-Bohr radius of order N(?N) is R�p,N(X) defined by R�p,N(X)=sup?{r?0:?k=0N?xk?prpk??f?Hp}, where f(z)=?k=0?xkzk?H?(D,X). Here D={z?C:|z|<1} denotes the unit disk. We also introduce the following geometric notion of p-uniformly C-convexity of order N for a complex Banach space X for some N?N. For p?[2,?), a complex Banach space X is called p-uniformly C-convex of order N if there exists a constant ?>0 such that (?x0?p+??x1?p+?2?x2?p+?+?N?xN?p)1/p?max??[0,2?)??x0+?k=1Nei?xk? for all x0, x1, �, xN ?X. We denote Ap,N(X), the supremum of all such constants ? satisfying (0.1). We obtain the lower and upper bounds of R�p,N(X) in terms of Ap,N(X). In this paper, for p?[2,?) and each N?N, we prove that a complex Banach space X is p-uniformly C-convex of order N if, and only if, the p-Bohr radius of order N R�p,N(X)>0. We also study the p-Bohr radius of order N for the Lebesgue spaces Lq(?) for 1?p<q<? or 1?q?p<2. Finally, we prove an operator valued analogue of a refined version of Bohr and Rogosinski inequality for bounded holomorphic functions from the unit disk D into B(H), where B(H) denotes the space of all bounded linear operator on a complex Hilbert space H. � 2022 Elsevier Masson SAS