A problem in analysis (Q2872885)

From author's abstract: Assume that \(D\subset\mathbb{R}^2\) is a bounded domain, diffeomorphic to a disc, star-shaped, with a \(C^{1,\lambda}\)-boundary \(C\), \(\lambda> 0\), which can be represented in polar coordinates as \(r= f(\varphi)\), where \(f> 0\) is a smooth \(2\pi\)-periodic function. Let \(\psi_{\pm n}(\varphi):= e^{\pm in\varphi} f^n(\varphi)\).NEWLINENEWLINE Theorem. Assume that \(\int^{2\pi}_0 \psi_{\pm n}(\varphi) f^2(\varphi)\,d\varphi= 0,\;n=1,2,\dots\), then \(f=\text{const}\).