Let $u$ be a harmonic function on region $D$, $\phi$ is r a real function on $u(D)$. Prove that $\phi \circ u$ is harmonic on $D$ iff $\phi$ is linear.
I have tried to write an analytic function $f$ with $u$ be its real part and tried to find some properties of $\phi \circ f$. Does this approach correct? What else should I do? Thanks for any help.
We obtain $$g_{xx}=\dfrac{d^2\phi}{du^2}.\left(\dfrac{\partial u}{\partial x}\right)^2+\dfrac{d\phi}{du}.\dfrac{\partial^2u}{\partial x^2}$$ $$g_{yy}=\dfrac{d^2\phi}{du^2}.\left(\dfrac{\partial u}{\partial y}\right)^2+\dfrac{d\phi}{du}.\dfrac{\partial^2u}{\partial y^2}$$ then $g=\phi o u$ is harmonic iff $g_{xx}+g_{yy}=0$ iff $$\dfrac{d^2\phi}{du^2}.\left(\dfrac{\partial u}{\partial x}\right)^2+\dfrac{d\phi}{du}.\dfrac{\partial^2u}{\partial x^2} + \dfrac{d^2\phi}{du^2}.\left(\dfrac{\partial u}{\partial y}\right)^2+\dfrac{d\phi}{du}.\dfrac{\partial^2u}{\partial y^2}=0$$ iff (since $u$ is harmonic) $$\dfrac{d^2\phi}{du^2}[\left(\dfrac{\partial u}{\partial x}\right)^2+\left(\dfrac{\partial u}{\partial y}\right)^2]=0$$ iff $\phi$ be linear.