An analogue of Polya's theorem for piecewise holomorphic functions

A well-known result due to Polya for a function given by its holomorphic germ at $z=\infty$ is extended to the case of a piecewise holomorphic function on an arbitrary compact set in $\overline{\mathbb C}$. This result is applied to the problem of the existence of compact sets that have the minimum transfinite diameter in the external field of the logarithmic potential of a negative unit charge among all compact sets such that a certain multivalued analytic function is single-valued and piecewise holomorphic on their complement.
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