# On Julia set of permutable transcendental entire functions

Date:
18.06.12
Times:
11:30 to 12:30
Speaker:
Anand Singh
University:
University of Jammu

Abstract:

If $f$ and $g$ are transcendental entire functions such that $f \circ g = g \circ f$ it is an open question whether $J(f) = J(g)$. Some partial results in affirmation are known. Here we give some more results on this topic.For instance apart from the other theorems, we shall prove: Let $f$ and $g$ be permutable transcendental entire functions. Fix $R > \min _{z \in J(f)}|z|$. Let $A(f) = \{ z \in \mathbb{C}$ : $\textrm {there exists } L \in \mathbb{N} \textrm{ such that } \mid f^n(z) \mid > \max_{|z|=R}||f^{n-L}(z)|$ $\textrm{ for } n > L \}.$ If the exterior of $A(f)$ viz.,$A(f)^e \not= \phi$ and if $A( f \circ g )^e$ be connected, then it is shown that either $J(f) = J( f \circ g )$ or $J(g) = J( f \circ g )$ or for every $w \in J(f \circ g ) \setminus (J(f) \cup J(g))$, there exists a disk $D_w = ( \mid z - w \mid < r )$ such that $D_w \cap A( f \circ g ) ^ e = \phi$.