Extremal functions and the Chang-Marshall inequality

Valentin V. Andreev and Alec Lane Matheson

Vol. 162 (1994), No. 2, 233–246

DOI: 10.2140/pjm.1994.162.233

Abstract

Answering a question of J. Moser, S.-Y. A. Chang and D. E. Marshall proved the existence of a constant C such that ∫ ₀^2πe^|f(ei𝜃)|2 d𝜃 ≤ C for all functions f analytic in the unit disk with f(0) = 0 and Dirichlet integral not exceeding one. We show that there are extremal functions for the functionals Λ_α(f) = ∫ ₀^2πe^{α|f(ei𝜃)|2} d𝜃 when 0 ≤ α < 1. We establish a variational condition satisfied by extremal functions. We show that the identity function f(z) = z is a local maximum in a certain sense for the functionals Λ_α and conjecture that it is a global maximum.

Mathematical Subject Classification 2000

Primary: 30D55

Milestones

Received: 20 February 1992

Revised: 7 October 1992

Published: 1 February 1994