Extremal problems in BMO and VMO involving the Garsia norm

Given an L² function f on the unit circle T, we put Φf(z):= P(|f|²)(z)-| Pf(z)|², z∈ D, where D is the open unit disk and P is the Poisson integral operator. The Garsia norm \|f\|G is then defined as z∈ D(Φf(z))1/2, and the space BMO is formed by the functions f∈ L² with \|f\|G<∞. If \|f\|²G=Φf(z₀) for some point z₀∈ D, then f is said to be a norm-attaining BMO function, written as f∈ BMOₙₐ. Note that BMOₙₐ contains VMO, the space of functions with vanishing mean oscillation. We study, first, the functions f in L^∞ (as well as in L^∞∩ BMOₙₐ) with the property that \|f\|G=\|f\|_∞. The analytic case, where L^∞ gets replaced by H^∞, is discussed in more detail. Secondly, we prove that every function f∈ BMOₙₐ with \|f\|G=1 is an extreme point of ball\,( BMO), the unit ball of BMO with respect to the Garsia norm. This implies that the extreme points of ball\,( VMO) are precisely the unit-norm VMO functions. As another consequence, we arrive at an amusing "geometric" characterization of inner functions.