Domination on sets and in norm

W.K. Hayman (Imperial College London)

Suppose that f and g are analytic in Δ : |z| < 1. Let B be a space with norm ∥ · ∥_{B}. Let E be a subset of Δ.

• When does

|f(z)| ≤ |g(z)|  on E  (1)

imply

∥f∥_{B} ≤ ∥g∥_{B}?  (2)

In this case, E is called dominating.

• If E is not dominating, can ∥f∥_{B} be arbitrary subject to (1)?