Analyticity of functions analytic on circles

TL;DR

This paper proves that a continuous function on the closed unit disc, which extends holomorphically from certain circles, must be holomorphic inside the disc, revealing new conditions for analyticity.

Contribution

It establishes that extending holomorphically from circles centered at the origin and passing through a boundary point implies interior holomorphicity.

Findings

Function is holomorphic inside the disc under given conditions.

Extending holomorphically from specific circles ensures interior analyticity.

Provides new criteria for analyticity based on circle extensions.

Abstract

Let U be the closed unit disc in C and let p be a point on the unit circle. Let f be a continuous function on U which extends holomorphically from each circle contained in U and centered at the origin, and from each circle contained in U and passing through the point p. Then f is holomorphic in the interior of U.