On Exceptional Values of Meromorphic Functions with the Set of Singularities of Capacity Zero

Abstract

LetEbe a compact set in thez-plane and letΩbe its complement with respect to the extendedz-plane. Suppose thatEis of capacity zero. ThenΩis a domain and we shall consider a single-valued meromorphic functionw=f(z) onΩwhich has an essential singularity at each point ofE. We shall say that a valuewis exceptional forf(z)at a point ζ ∈Eif there exists a neighborhood of C where the functionf(z)does not take this valuew.

Some Note on Exceptional Values of Meromorphic Functions

“…In our paper [7], we showed, by using its Theorem 1, that there is a set E such that all f(z) possessing E as the set of essential singularities have at most three exceptional values and some one of them indeed has just three exceptional values at each point of E. But the condition…”

On totally ramified values

“…Let a ι be any one of the w r points in Δ and let 7 (1) 6 {7} be an arc joining a ι to one of the w 2 -points, being denoted by &!. As mentioned just above, there is another 7 (2) Φ 7 (1) in {7} ending at & l β We denote by a 2 the other end point of 7 (2) which is a Wj-point. If a 2 = a lf we may take 7 (1) -7 (2) as β.…”

“…As mentioned just above, there is another 7 (2) Φ 7 (1) in {7} ending at & l β We denote by a 2 the other end point of 7 (2) which is a Wj-point. If a 2 = a lf we may take 7 (1) -7 (2) as β. If a 2 Φ a ίf we consider the curve 7 α> _ 7 (2) + Ύ w 9 7 (8) ]3 e j ng an arc in { T } w hich differs from 7 (2) and starts from a 2 .…”

“…If a 2 = a lf we may take 7 (1) -7 (2) as β. If a 2 Φ a ίf we consider the curve 7 α> _ 7 (2) + Ύ w 9 7 (8) ]3 e j ng an arc in { T } w hich differs from 7 (2) and starts from a 2 . Let b 2 denote the other end point of 7 (3) .…”

Generalizations of Picard’s theorem for Riemann surfaces

“…On the other hand, it was shown by Matsumoto [7] that there is no general Picard's theorem for singularities of capacity zero. More precisely: for every compact plane set K of capacity zero there is a compact plane set E of capacity zero and a meromorphic function / in the complement of E such that / omits K and has an essential singularity at each point of E.…”