# Greatest lower bound for subordination

Consider the set $X$ of all analytic functions $f$ in the unit disk $U$ satisfying $f(0)=0, f'(0)\neq 0$. We say that $f\prec g$ if there exists $\phi\in X$ which maps $U$ into itself, and $f=g\circ\phi.$

This is called subordination. There are hundreds of papers with "subordination" in the title, but I could not find the answer to the following question:

Does the greatest lower bound with respect to $\prec$ exist for arbitrary pair of functions in $X$ ?

Precisely: Let $f,g$ be elements of $X$. Does there exist $h\in X$ such that $h\prec f,\; h\prec g$, and for every $p$ such that $p\prec f,\; p\prec g$, we have $p\prec h$?

I am especially interested in the subclass $X_0\subset X$ of locally univalent functions, $f'(z)\neq 0$ for all $z\in U.$

Remark. For univalent (injective) functions, the g.l.b, corresponds to the intersection of the image domains. That's why I conjecture that g. l. b. must always exist.

EDIT. For meromorphic functions, the answer is negative. Let $f$ be a conformal map of the unit disk onto the disk $|z-1|<2\;$, $f(0)=0$ and $g$ be a conformal map onto the disk $\{ |z-1|>1/2 \}\cup\{\infty\},\;$ $g(0)=0$. As the intersection of the two images is a ring, there is evidently no g.l.b. For holomorphic functions it remains open.