I was confused by the following question for a long time:


Does there exists a transcendental entire function $f$ such that
$J(f)\cap J(f')=\emptyset$ ?


where $J(f)$, ($J(f')$) is the Julia set of $f$ $(f').$ 


Edit: Following the same style, one may also ask 
Does there exists a transcendental entire function $f$ such that
$I(f)\cap I(f')=\emptyset$ ?


where $I(f)$, ($I(f')$) is the escaping set of $f$ $(f').$ $I(f)\neq \emptyset$ was first proved by Eremenko using Wiman-Valiron method. See $<http://www.math.purdue.edu/~eremenko/dvi/banach.pdf> for details.


Any comments will be appreciated.



A question on $J(f)$ and $J(f')$
In my opinion, the question is completely arbitrary. There is no reason to expect relationships between the dynamics of $f$ and that of its derivative. Their relationship will even change under affine conjugacy.


This being said, the answer to the question is positive in that such a function $f$ does indeed exist; i.e. the conjecture discussed in the comments below the questionis false. 


Indeed, for a counterexample consider $\newcommand{\eps}{\varepsilon}f(z)=\eps e^{-z}$. Then $f'(z) = -\eps e^{-z}=-f(z)$. For sufficiently small $\eps$, the right half plane $H\_R$ belongs to the basin of attraction of an attracting fixed point, for both maps. Hence the Julia sets are both contained in the left half plane $H\_L$. However, clearly $f^{-1}(H\_L)\cap(f')^{-1}(H\_L)=f^{-1}(H\_L)\cap f^{-1}(H\_R)=\emptyset$. Hence $J(f)\cap J(f')=\emptyset$. In particular also $I(f)\cap I(f')=\emptyset$.


**EDIT:** If you are willing to let $f$ be a polynomial, then here is an even simpler example. Take $f(z)=z^3/3 + c$. Then $f'(z)=z^2$, so $J(f')$ is the unit circle. For large $c$, clearly the closed unit disc belongs to the basin of $\infty$ of $f$, and hence does not intersect $J(f)$. (A simple calculation shows that $c=4$ will suffice.) 


Of course, the same argument will work for *any* family of the form $f(z)=p(z)+c$, with $p$ a fixed polynomial of degree at least $3$. (The latter assumption is there to ensure that dynamics of $f'$ is nontrivial.)


