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Define $F(z)=\overline{f(\overline{z})}$. To apply the identity theorem, we first verify that $F$ is holomorphic; mere complex conjugation of the values of $f$ would not suffice. For $h\ne 0$,Thus $F$ is an entire function, just as $f$ is. At each positive integer $n$, the point $1/n$ is real and $f(1/n)$ is real, so $F(1/n)=f(1/n)$. These distinct points accumulate at $0\in \mathbb{C}$. The identity theorem on the connected complex plane givesThis is conjugate reflection of a holomorphic function, with the reality data forcing equality to its reflection.