Linear combination of a real-valued function and its inverse is analytic Implies the real-valued function is analytic.

Question

If $u$ is a real-valued function on a disc $\Delta_R$ such that $u^{-1}+iu$ is analytic on $\Delta_R$, then does this imply that $u$ is analytic on $\Delta_R$?

I am actually trying to prove some other result and that result holds if the above statement is true.

Answer 1

Try the Cauchy-Riemann equations. This implies $u$ is constant on $\Delta_R$.