Problem 1623
An entire function \(\displaystyle f(z)\) satisfies \(\displaystyle |f(1/n)|=1/n^2\) for \(\displaystyle n=1,2,\ldots\), and \(\displaystyle |f(i)|=2\). What are the possible values of \(\displaystyle |f(-i)|\)? Difficulty: 3. Solution is available for this problem. |
Problem 1628
Give an example of a function that is holomorphic in the open unit disc and has infinitely many roots there. Difficulty: 5. Solution is available for this problem. |
Problem 1630
Assume that \(\displaystyle f\in O(\CC)\) and \(\displaystyle |f(x)|=1\) for all \(\displaystyle x\in \RR\). Prove that \(\displaystyle \displaystyle\overline{f(\bar{z}})=\frac{1}{f(z)}\). Difficulty: 6. |
Problem 1625
Show that if \(\displaystyle f\) takes only real values on the real and imaginary axes then \(\displaystyle f\) is even. Difficulty: 7. |
Problem 1632
If \(\displaystyle f\in O(|z|>1)\), is bounded and \(\displaystyle f(n)=0 \qquad (n=2, 3, \dots)\) then \(\displaystyle f\equiv 0\). Difficulty: 7. |
Problem 1633
Show that if \(\displaystyle f\in O(\CC)\), \(\displaystyle \displaystyle \left|f\left(\frac 1n\right)\right|<\frac{1}{2^n}\) then \(\displaystyle f\equiv 0\). Can one do better? Difficulty: 7. |
Problem 1634
Given that \(\displaystyle f\in O(\CC)\), \(\displaystyle \displaystyle f\left(\frac{1}{n^2}\right)=\cos\frac 1 n\) find \(\displaystyle f(-1)\). Difficulty: 8. |
Problem 1806
Call an entire function \(\displaystyle f\) ``interesting'', if \(\displaystyle f(z)\) is real along the parabola \(\displaystyle \re z=(\im z)^2\). (b) Prove that if \(\displaystyle f\) is an interesting function then \(\displaystyle f'(-3/4)=0\). (a) Show an example for a non-constant interesting function. CIIM 2014, Costa Rica Difficulty: 9. Solution is available for this problem. |
Supported by the Higher Education Restructuring Fund allocated to ELTE by the Hungarian Government |