Phragmén-Lindelöf Principle

Abstract

This chapter continues the bounded type theory for the upper half-plane II. In Theorem 6.13 we rewrite the inner-outer factorization 5.14(ii) in the form

$$F(z)=e^-i\tau z B(z)\frac{S+(z)}{S-(z)}G(z)$$

, where τ is a real number called the mean type of F(z), B(z) is a Blaschke product, S±(z) are singular inner functions (stripped of factors of the form e^iaz, which go into the factor e^−iτz), and G(z) is an outer function. Formulas for τ are derived in terms of the behavior of F(z) on the imaginary axis and the average behavior of F(z) on semicircles (Theorem 6.15).