Some $H^{\infty }$-interpolating sequences and the behavior of certain of their Blaschke products

Abstract:

Let $f$ be a strictly increasing continuous real function defined near ${0^ + }$ with $f(0) = f’(0) = 0$. Such a function is called a $K$-function if for every constant $k,f(\theta + kf(\theta ))/f(\theta ) \to 1/$ as $\theta \to {0^ + }$. The curve in the open unit disc with corresponding representation $1 - r = f(\theta )$ is called a $K$-curve. Several analytic and geometric conditions are obtained for $K$-curves and $K$-functions. This provides a framework for some rather explicit results involving parts in the closure of $K$-curves, ${H^\infty }$-interpolating sequences lying on $K$-curves and the behavior of their Blaschke products. In addition, a sequence of points in the disc tending upper tangentially to 1 with moduli increasing strictly to 1 and arguments decreasing strictly to 0 is proved to be interpolating if and only if the hyperbolic distance between successive points remains bounded away from zero.