Problem 1636
Let \(\displaystyle f\) be continuous on the closed unit disc and holomorphic inside. Show that the image of the open disc is in the convex hull of the image of the boundary circle. Difficulty: 5. |
Problem 1637
Prove that if \(\displaystyle f\) is holomorphic on an open set, then neither the real part nor the imaginary part of \(\displaystyle f\) has a local extrema. Difficulty: 5. |
Problem 1635
Let \(\displaystyle f\) be continuous on the closed unit disc and holomorphic inside. Let \(\displaystyle A=\max\limits_{0\le t\le\pi}|f(e^{it})|\) and \(\displaystyle {B=\max\limits_{\pi\le t\le2\pi}|f(e^{it})|}\). Show that \(\displaystyle |f(0)|\le \sqrt{AB}\). Difficulty: 7. |
Problem 1638
Let \(\displaystyle 0<r_1<r_2<r_3\) and let \(\displaystyle f\) be holomorphic on \(\displaystyle r_1<|z|<r_3\) with a continuous extension to the boundary. Prove that \(\displaystyle \left(\max_{|z|=r_2}|f(z)|\right)^{\log(r_3/r_1)} \le \left(\max_{|z|=r_1}|f(z)|\right)^{\log(r_3/r_2)} \left(\max_{|z|=r_3}|f(z)|\right)^{\log(r_2/r_1)}. \) (Hadamard) Difficulty: 9. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |