Baker omitted value

Abstract

We define Baker omitted value, in short bov, of an entire or meromorphic function f in the complex plane as an omitted value for which there exists such that for each ball centred at a and with radius r satisfying , every component of the boundary of is bounded. The existence and some dynamical implications of bov is investigated in this article. The bov of a function is the only asymptotic value. An entire function has bov if and only if the image of every unbounded curve is unbounded. It follows that an entire function has bov whenever it has a Baker wandering domain. A sufficient condition for existence of bov of meromorphic functions is also proved. Functions with bov have at most one completely invariant Fatou component. Some counter examples are provided and problems are proposed for further investigation.