Let $A(z)$ be a real-entire function.
$$ A(z) = a_0 + a_1 z + a_2 z^2 + \cdots $$
Also for real $x$ we have
$$ A’(x) > 0 .$$
Let $f$ be the functional inverse: $ f(A(z)) = A(f(z)) = z $. More specific $f$ is the main branch inverse.
An example could be $A(z) = \exp(z) $ and $f(z) = \ln(z) $ (not $ \ln(z) + 2 \pi i $ !!).
Ok that was just an example.
So now the question.
Consider the (fixed point) equation
$$ f(z) = z .$$
I'm looking for examples where there are NO solutions for any $z$ on the Riemann Sphere (so complex or complex infinity) or a proof that none exist.
We know for instance that there is always a solution to $ \ln(z) = z \ln(b) $ (the logarithm with base $b$ always has a fixed point). So that is NOT an example.
I also wonder about a description for all these without a solution.
I thought about Riemann surfaces intersecting with the identity function but without result.
There is no such thing as "the main branch".
Let $g(z)=A(z)-z$. If $g(c) =0$ then let $n$ be the order of the zero of $A(z)-c$ at $c$, there is a branch of $f$ such that $f(z)^n$ is analytic and $f(A(z))=z$ near $c$ $$A(c)=c\implies c=f(c)$$
That $A'(x)>0$ for $x\in\Bbb{R}$ means that there is a branch of $f$ analytic on $A(\Bbb{R})$ such that $f(A(x))=x$. But there is no canonical way to continue analytically this function. It will have some branch points at every $s$ such that $A(z_1)=A(z_2)=s$, moreover the branch points may move and appear disappear when rotating around another one, the set of all such branch points doesn't need to be isolated. But locally it is.
Iff $g$ doesn't vanish then $g(z)=\exp(h(z))$ and $A(z)=\exp(h(z))+z$ where $h$ is entire. In that case whatever the branch $f(z)-z$ doesn't vanish.
All the branches of $f$ are the analytic continuations of the same function, analytic continuations along different curves.