On products of half-plane mappings

Abstract

Let 𝔻 = {𝑧 ∈ ℂ: |𝑧| < 1} and let 𝑓 and 𝑔 be functions analytic in 𝔻. Then 𝑓 is said to be subordinate to 𝑔 if 𝑓(𝑧) = 𝑔(𝜑(𝑧)) for 𝑧 ∈ 𝔻, where 𝜑: 𝔻 ⟶ 𝔻 is analytic in 𝔻 with 𝜑(0) = 0. This is denoted by 𝑓 ≺ 𝑔. A trivial modification of the Herglotz representation formula for functions subordinate to half-plane mappings implies that if [symbols] where where |𝑐| = 1, then 𝑓(𝑧)= [symbols] 𝜇 is a probability measure on the unit circle 𝜕𝔻.

In 1989, Koepf considered the class of functions p normalized by 𝑝(0) = 1 and [symbols] for some |𝑐| = 1 and proved that each function of the form [symbols], where |𝑥ₖ| = |𝑐| = 1, 𝜆ₖ > 0 for 𝑘 = 1, 2, … , 𝑛 and [symbols] has a representation of the form [symbols] where |𝑥ₖ| = |𝑘| = 1 for 𝜆ₖ =, 2, … , 𝑛 and [symbols].

and arg 𝑥₁ < arg 𝑦₁ < arg 𝑥₂ < arg 𝑦₂ < ⋯ < arg 𝑥𝑛 < arg 𝑦𝑛 < arg 𝑥₁ + 2𝜋. (∗∗)

In this study we first give a new proof of the above product representation using the following known representation for finite Blaschke products:

If 𝐵 is a finite Blaschke product with 𝐵(0) = 0, then [symbols] where |𝑥ₖ| = 1, 𝜆ₖ > 0 for 𝑘 = 1, 2, … , 𝑛 and [symbols].

We then considered the question of whether each function of the form (∗∗) has a representation of the form (∗). We were able to prove it for 𝑛 = 2 directly. Since the computation becomes tedious for 𝑛 = 3 with the direct method, we employed the Herglotz representation formula to prove it. Based on the above results and verification for some more cases using Mathematica, we conjecture that each function of the form 𝑝𝑛(𝑧) = [symbols], where |𝑥ₖ| = |𝑦ₖ| = 1 for 𝑘 = 1, 2, … , 𝑛 and arg 𝑥₁ < arg 𝑦₁ < arg 𝑥₂ < arg 𝑦₂ < ⋯ < arg 𝑥𝑛 < arg 𝑦𝑛 < arg 𝑥₁ + 2𝜋 has a representation of the form 𝑝𝑛(𝑧) [symbols] where |𝑥ₖ| = |𝑐| = 1, 𝜆ₖ > 0 for 𝑘 = 1, 2, … , 𝑛 and [symbols].