Degree and holomorphic extendibility

作者：Leban Darja Govekar^*

来源：Proceedings of the Royal Society of Edinburgh: Section A Mathematics , 2013, 143(1): 129-139.

DOI：10.1017/S0308210511000497

Recently it was shown that if D is a bounded domain in C whose boundary consists of a finite number of pairwise disjoint simple closed curves, then a continuous function f on bD extends holomorphically through D if and only if, for each g is an element of A(D) such that f + g has no zero on bD, the degree of f + g is non-negative (which, for these special domains, is equivalent to the fact that the change of argument of f + g along bD is non-negative). Here A(D) is the algebra of all continuous functions on (D) over bar which are holomorphic on D. This fails to hold for general domains, and generalizing to more general domains presents a major problem that often requires a much larger class of functions g. It is shown that the preceding theorem still holds in the case when D is a bounded domain in C such that (D) over bar is finitely connected and such that D is equal to the interior of D.

• 出版日期2013-2