THE REAL PART OF AN OUTER FUNCTION AND A HELSON-SZEG¨O WEIGHT

Abstract

Suppose $F$ is a nonzero function in the Hardy space $H^1$. We study the set $\{f; f \mbox{ is outer and } |F| \le \mbox{Re } f \, \mbox{ a. e. on } \partial D\}$, where $\partial D$ is the unit circle. When $F$ is a strongly outer function in $H^1$ and $\gamma$ is a positive constant, we describe the set $\{f; f \mbox{ is outer and } |F| \le \gamma \mbox{ Re } f \mbox{ and } |F^{-1}| \le \gamma \mbox{ Re } \, \, (f^{-1}) \, \, \mbox{ a. e. on } \partial D\}$. Suppose $W$ is a Helson-Szeg¨o weight. As an application, we parametrize real-valued functions $v$ in ${\mathcal L}^\infty(\partial D)$ such that the difference between $\log W$ and the harmonic conjugate function $\tilde{v}$ of $v$ belongs to ${\mathcal L}^\infty(\partial D)$ and $||v||_\infty$ is strictly less than $\pi/2$ using a contractive function $\alpha$ in $H^\infty$ such that $(1 + \alpha)=(1 - \alpha)$ is equal to the Herglotz integral of $W$.