For $0 < x< \infty$, let $\phi (x)$ be positive and continuously twice differentiable satisfying:
(a) $\phi (x+1) = \phi (x)$
(b) $\phi(\frac{x}{2}) \phi(\frac{x+1}{2}) = d\phi(x),$ where $d$ is a constant.
Prove that $\phi$ is a constant.
I am trying to answer this as the first step in the proof of Euler's reflection formula. I was am given the hint "Let $g(x) = \frac{\mathrm{d}^2}{\mathrm{d}x^2} \log \phi (x)$ and observe that $g(x+1)=g(x)$ and $\frac{1}{4}(g(\frac{x}{2}) + g(\frac{x+1}{2})) = g(x)$"
Firstly, I don't understand how they get the second bit of the hint, and then even assuming that I'm still not sure what to do. Any help would be much appreciated. Thanks.