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Exceptional values in holomorphic families of entire functions

Alexandre Eremenko
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Abstract

We study Picard's exceptional values of holomorphic one-parametric families of entire functions. Our first result shows that the set of parameter values for which zero is a Picard value can be an arbitrary closed set of zero logarithmic capacity. This answers a question of Julia. Second, we show that if a function has a Picard exceptional value for all values of parameter in some region of the plane then this Picard value is a holomorphic function in the complement of some discrete set E, tending to infinity as the papameter tends to E.

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Alexandre Eremenko. 2005-03-31. Exceptional values in holomorphic families of entire functions. https://arxiv.org/abs/math/0503750

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