On Bungee Set of Composition of Transcendental Entire Functions

Abstract

Let $f$ be a transcendental entire function. For $n \in \mathbb{N},$ let $ f^{n}$ denote the $n^{th}$ iterate of $f$. Let $ I(f) = \{z \in \mathbb{C} : f^n \rightarrow \infty $ as $ n \rightarrow \infty \} $ and $ K(f) = \{z: \textrm{ there exists } R > 0 \textrm{ such that } | f^n(z) | \leq R \textrm{ for } n \geq 0 \}. $ Then the set $ \mathbb{C}\ \setminus (I(f) \cup K(f)) $ denoted by $ BU(f) $ is called Bungee set of $f$. In this paper we give an alternate definition for $ BU(f)$ which is very easy to work with, and we illustrate it by proving some properties of Bungee sets of composite transcendental entire functions and also of Bungee sets of permutable transcendental entire functions.