On Julia set of permutable transcendental entire functions [ Back ]

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 $.