Vol. 51, No. 1, 1974

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Inner functions under uniform topology

Domingo Antonio Herrero

Vol. 51 (1974), No. 1, 167–175

The structure of the space of all inner functions in the unit disc D = {z : |z| < 1} under the metric topology induced by the H-norm is considered. It is proven that if two inner functions p and q belong to the same component of , then the variation of p∕q on each open arc of ∂D (the boundary of D in the complex plane C) where they can be continued analytically is bounded by a constant C = C(p,q), independent of the arc. This criterion is used to show that a component of can contain nothing but Blaschke products with infinitely many zeroes, exactly one (up to a constant factor) singular inner function or infinitely many pairwise coprime singular inner functions.

Mathematical Subject Classification 2000
Primary: 46J15
Secondary: 30A76
Received: 9 October 1972
Revised: 23 April 1973
Published: 1 March 1974
Domingo Antonio Herrero