Problem 1556
Assume that \(\displaystyle \displaystyle\sum_{n=0}^\infty a_nz^n\) is convergent in the unit disc and is injective there. Express the area of the image of the unit disc in terms of the coefficients \(\displaystyle a_n\). Difficulty: 6. |
Problem 1557
Assume that \(\displaystyle f(z)=\displaystyle\sum_{n=1}^\infty a_nz^n\) is convergent on the disc \(\displaystyle |z|<r+\varepsilon\). Prove that \(\displaystyle \frac1{2\pi r}\int\limits_{|z|=r}|f(z)|^2\cdot|\dz|= \sum_{n=0}^\infty|a_n|^2r^{2n}.\) (Parseval-formula for power series) Difficulty: 6. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |