Complex mean value theorem: Let $g$ be a holomorphic function defined on an open convex subset $D_{g}$ of $ℂ$. Let $v$ and $u$ be two distinct points in $D_{g}$. Then there exist $z₁,z₂∈(u,v)$ such that
$Re(g′(z₁))=Re(((g(u)-g(v))/(u-v)))$,
$Im(g′(z₂))=Im(((g(u)-g(v))/(u-v)))$, i.e.,
$g(u)=g(v)+(u-v)(Re(g′(z₁))+iIm(g′(z₂)))$.
In fact, we have $z₁=v+t₁(u-v),t₁∈(0,1)$ and $z₂=v+t₂(u-v),t₂∈(0,1)$ from the proof of this Theorem.
My question is about the uniqueness of $t₁,t₂∈(0,1)$ . The proof of the result indicate only their existence.