Let \(\phi\) be non-decreasing on [0,2\(\pi)\) and \(\phi\) (2\(\pi\) -0)-\(\phi\) (0)\(\leq 2\pi\), R: [0,2\(\pi)\to {\mathbb{R}} +\) and \(R(\theta)e^{i\phi (\theta)}\sim\sum^{+\infty}_{n=-\infty}c_ ne^{ni\theta}.\) The author proves interesting inequalities for the Fourier coefficients \(c_ n\), for example: \({1\over2}| c_{-1}|\quad 2+| c_ 0|\quad 2+{1\over2}| c_ 1|\quad 2\geq 27/8\pi\quad 2\) if \(R(\theta)\equiv 1\), \(| c_ 0| +| c_ 1|\geq (4/\pi)(1- 1/\sqrt{2})R_ 0\) if \(R(\theta)\geq R_ 0\) for \(0\leq\theta <2\pi.\) An inequality of this type was derived by E. Heinz in connection with the theory of minimal surfaces.