A property shared by continuous linear functions and holomorphic functions

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 ${\mathbf{C}}^n$, $\mathrm{\Omega }$ an open convex set in $\mathbf{C}$ and $f,g:X\to \mathbf{C}$ two holomorphic functions such that $f(X)\cap \mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$, $f(X)\setminus \mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$ and $g(X)\subseteq \mathrm{\Omega }$. 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 $\lambda g(x)+(1-\lambda )f(x)\in \mathrm{\Omega }$\ ; $(b)$ for each finite set $B$ in $A$, there exists $u\in X$ such that $\mu g(u)+(1-\mu)f(u)\in {\bf C}\setminus\Omega$forall$\mu\in B$.