Fatou proved a very interesting result: for a transcendental entire function $f$, the second itarate $f^{2}$ has at least has one fixed point. (Using the technique of Picard theorem)

This result has been generalized by Rosenbloom: $f^{n}$ has at least has one fixed point for every $n\geq 2$ (Using the technique of Nevanlinna theory)

Bergweiler proved a very strong version: $f$ has infinitely many points of minimal period $n$, for each $n\geq 2$ (quite technical)

There may be little hope to expect a elementary proof for the general version.

I wonder whether there exist a elementary proof (without using Nevanlinna theory) for the following theorem: $f$ has at least one point of minimal period $2$.

Any reference and comments will be appreciated.