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.


It turns out that the greatest lower bound may not exist. The counterexample can be constructed using the example of Milnor explained in the paper by Poenaru, Extension des immersions en codimension 1, Séminaire Bourbaki, 10 (1966-1968), Exp. No. 342. Milnor's example was brought to my attention in a comment by Misha to my other question Uniqueness theorem for conformal maping

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