LIPSCHITZ CONDITIONS FOR THE NORM OF A VECTOR VALUED ANALYTIC FUNCTION

Abstract. We prove some quantitative versions of the Thorp-Whitley max-imum modulus principle as well as extend to vector-valued functions a the-orem of Dyakonov [3] on Lipschitz conditions for the modulus of an analytic functions. Let X be a complex Banach space, and let D be the open unit disk in the complex plane C. Let f: D 7 → X be an analytic function. The function ‖f(·)‖ is subharmonic in D and therefore satisfies the maximum modulus principle: if ‖f(z) ‖ attains its maximum at a point z0 ∈ D, then ‖f(z) ‖ = const. However, the implication (1) ‖f(z) ‖ = const. = ⇒ f(z) = const. need not be true. For example, if X = ` ∞ and f(z) = (1, z, 0,...), then ‖f(z) ‖ ≡ 1. In connection with this problem, Thorp and Whitley [13] introduced the notion of strict c-convexity (or complex strict convexity) as a complex analogue of strict convexity for real Banach spaces. A complex Banach space X is said to be strictly c-convex if for every x, y ∈ X there holds the implication (2) max |z|≤1 ‖x+ zy ‖ ≤ ‖x ‖ = ⇒ y = 0. In other words, X is strictly c-convex if implication (1) holds for complex-linear functions. The result of Thorp and Whitley states the following: 2000 Mathematics Subject Classification. 46B20, 46L52. Key words and phrases. Complex uniform convexity, moduli of c-convexity, Lipschitz condi-tions, maximum principle. The authors are supported by MNZŽS, Grant No. 144010