Some characterizations of semi-Bloch functions

Abstract

A function f f analytic in the unit disk is called a semi-Bloch function if, for each complex number λ \lambda , the function g λ ( z ) = exp ⁡ ( λ f ( z ) ) {g_\lambda }(z) = \exp (\lambda f(z)) is a normal function. We give both an analytic and a geometric characterization of semi-Bloch functions, together with some examples to show that semi-Bloch functions are not closed under either addition or multiplication.