A property shared by continuous linear functions and holomorphic functions

Abstract: In this note, we continue to highlight some applications of Theorem 1 of [3]. Here is a sample: Let $X$ be an open set in ${\bf C}^n$, $\Omega$ an open convex set in ${\bf C}$ and $f, g : X\to {\bf C}$ two holomorphic functions such that $f(X)\cap\Omega\neq\emptyset$, $f(X)\setminus\Omega\neq \emptyset$ and $g(X)\subseteq \Omega$. Then, there exists a set $A$ in $[0,1]$ with the following properties:$(a)$ for each $x\in X$, there exists $\lambda\in A$ such that $\lambda g(x)+(1-\lambda)f(x)\in\Omega$\ ; $(b)$ for each finite set $B$ in $A$, there exists $u\in X$ such that $\mu g(u)+(1-\mu)f(u)\in {\bf C}\setminus\Omega$ for all $\mu\in B$.