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

Abstract

Let f f be a strictly increasing continuous real function defined near 0 + {0^ + } with f ( 0 ) = f ′ ( 0 ) = 0 f(0) = f’(0) = 0 . Such a function is called a K K -function if for every constant k , f ( θ + k f ( θ ) ) / f ( θ ) → 1 / k,f(\theta + kf(\theta ))/f(\theta ) \to 1/ as θ → 0 + \theta \to {0^ + } . The curve in the open unit disc with corresponding representation 1 − r = f ( θ ) 1 - r = f(\theta ) is called a K K -curve. Several analytic and geometric conditions are obtained for K K -curves and K K -functions. This provides a framework for some rather explicit results involving parts in the closure of K K -curves, H ∞ {H^\infty } -interpolating sequences lying on K 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.