On a planar bordered
Riemann surface W, a weakly λ-valent function is one whose every image point has
at most λ antiimages. In this note, extremal properties characterizing weakly λ-valent
principal functions are developed. The functionals extremized are, in a rather natural
way, analogous to those of the univalent cases. However, the class of competing
functions consists not only of weakly λ-valent analytic functions on W, but of all
analytic functions which are λ-valent near an interior point ζ ∈ W and near the
isolated border γ of W, and are of arbitrary finite valence elsewhere. Such competing
classes contain the λ-th powers of competing univalent functions, as would be
expected. That these classes contain functions of arbitrary finite valence perhaps
would not be anticipated.
An interpretation is given for that situation in which the competing classes
consist of those analytic functions which are λ-valent near two isolated border
components.
