Bohr and Rogosinski inequalities for operator valued holomorphic functions

For any complex Banach space Xand each p.[1, 8), we introduce the p-Bohr radius of order N(. N) is Rp, N(X) defined by R p,N (X) = sup r = 0 : N k=0 xkp rpk = f p H 8(D,X) , where f(z) = 8 k=0xkzk. H8( 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 Nfor a complex Banach space Xfor some N. N. For p.[2, 8), a complex Banach space Xis called p-uniformly C-convex of order Nif there exists a constant. > 0such that x0 p +. x1 p +.2 x2 p + center dot center dot center dot +.N xN p 1/ p = max..[0,2p) x0 + N k=1 ei.xk (0.1) 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 Rp, N(X) in terms of Ap,N(X). In this paper, for p.[2, 8) and each N. N, we prove that complex Banach space Xis p-uniformly C-convex of order Nif, and only if, the p-Bohr radius of order N Rp, N(X) > 0. We also study the p-Bohr radius of order Nfor the Lebesgue spaces Lq(mu) for 1 = p < q<8 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 Dinto B(H), where B(H) denotes the space of all bounded linear operator on a complex Hilbert space H. (C) 2022 Elsevier Masson SAS. All rights reserved.