Apologies if this is a trivial result - I'm trying to prove the following: If $F,G,H$ are holomorphic on some open connected set $U \subset \mathbb{C}$, and $$ \frac{\partial G}{\partial z} = F = \frac{\partial H}{\partial z} $$ then $G$ and $H$ differ by a constant.
It seems that these hypotheses are meant to imply that $$\frac{\partial }{\partial \bar z} \bigg(\frac{\partial G}{\partial z} \bigg) = \frac{\partial }{\partial \bar z} \bigg(\frac{\partial H}{\partial z} \bigg) = 0$$ i.e. the partials with respect to $z$ of $G$ and $H$ are holomorphic as well, but that doesn't seem to get me any closer to obtaining the conclusion. Is this the right approach, or is there something I'm missing?
Define $f(z) := G-H$ and write $f(z) = u(z)+\mathrm{i} v(z)$, then $$0=f'(z) = u_x(z) + i v_x(z).$$ Thus $u_x \equiv 0$ and also $v_x \equiv 0$. Using the Cauchy–Riemann equations, i.e. $u_x = v_y$ and $u_y = - v_x$ we also conclude that $v_y \equiv 0$ and $u_y \equiv 0$. Hence $Df(z) =0$, where $Df$ is the classic derivate in $\mathbb{R}^2$. This implies that $f$ is constant.