Complex Mean Value Theorem. Let $f$ be a holomorphic function defined on an open convex subset $D_{f}$ of $ℂ$ (or we can assume that $f$ is entire). Let $a$ and $b$ be two distinct points in $D_{f}$. Then there exist $z_1,z_2∈(a,b)$ such that $$\begin{aligned}\operatorname{Re}f'(z_1)&=\operatorname{Re}\frac{f(b)-f(a)}{b-a},\\ \operatorname{Im}f'(z_2)&=\operatorname{Im}\frac{f(b)-f(a)}{b-a}.\end{aligned}$$
My questions are:
(1) Can we deduce that: $f(b)=f(a)+(b-a)\bigl(\operatorname{Re}f'(z_1)+i\operatorname{Im}f'(z_2)\bigr)$?
(2) Let $b=s$ is any complex number different from $a$, can we deduce that the function $$ f(a)+(s-a)\bigl(\operatorname{Re}f'(z_1)+i\operatorname{Im} f'(z_2)\bigr)$$ is entire? Here $z_1$ and $z_2$ are given in function of $s$: $z₁=a+t₁(s-a)$ and $z₂=a+t₂(s-a)$ where $t₁,t₂∈(0,1)$
Your equation in question (1) can be rewritten as $$\frac{f(b)-f(a)}{b-a}=\operatorname{Re}f'(z_1)+i\operatorname{Im}f'(z_2),$$ which is clearly true since $w=\operatorname{Re}w+i\operatorname{Im}w$ for any complex number $w$.
I am not sure I understand your question (2), since it seems to merely ask whether the function $f$, which is assumed to be entire, is entire.