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 ot= 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 $.