Problem 1 (yau-contest 2010)
Section: Analysis and Differential Equations

a) Let $f(z)$ be holomorphic in $D: |z| < 1$ and $|f(z)| \le 1$ $(z \in D)$. Prove that
$$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \le |f(z)| \le \frac{|f(0)| + |z|}{1 - |f(0)||z|}, \quad (z \in D).$$

b) For any finite complex value $a$, prove that
$$\frac{1}{2\pi} \int_{0}^{2\pi} \log |a - e^{i\theta}|\, d\theta = \max\{\log |a|, 0\}.$$

Problem 2 (yau-contest 2010)
Section: Analysis and Differential Equations

Let $f \in C^{1}(\mathbf{R}),\ f(x+1) = f(x)$, for all $x$, then we have
$$\|f\|_{\infty} \le \int_{0}^{1} |f(t)|\, dt + \int_{0}^{1} |f'(t)|\, dt.$$

Problem 3 (yau-contest 2010)
Section: Analysis and Differential Equations

Consider the equation
$$\ddot{x} + (1 + f(t))x = 0.$$
We assume that $\int^{\infty} |f(t)| dt < \infty$. Study the Lyapunov stability of the solution $(x, \dot{x}) = (0, 0)$.

Problem 4 (yau-contest 2010)
Section: Analysis and Differential Equations

Suppose $f : [a, b] \to \mathbf{R}$ be a $L^{1}$-integrable function. Extend $f$ to be $0$ outside the interval $[a, b]$. Let
$$\phi(x) = \frac{1}{2h} \int_{x-h}^{x+h} f.$$
Show that
$$\int_{a}^{b} |\phi| \le \int_{a}^{b} |f|.$$

Problem 5 (yau-contest 2010)
Section: Analysis and Differential Equations

Suppose $f \in L^{1}[0, 2\pi]$, $\hat{f}(n) = \frac{1}{2\pi} \int_{0}^{2\pi} f(x)e^{-inx} dx$, prove that

1) $\sum_{|n|=0}^{\infty} |\hat{f}(n)|^{2} < \infty$ implies $f \in L^{2}[0, 2\pi]$, 

2) $\sum_{n} |n \hat{f}(n)| < \infty$ implies that $f = f_{0}$ a.e., $f_{0} \in C^{1}[0, 2\pi]$, 

where $C^{1}[0, 2\pi]$ is the space of functions $f$ over $[0, 1]$ such that both $f$ and its derivative $f'$ are continuous functions.

Problem 6 (yau-contest 2010)
Section: Analysis and Differential Equations

Let $\Omega$ be a bounded domain of $\mathbf{R}^{n}$ and let $f$ be a smooth function defined in $[0, +\infty)$ such that $f(t)/t$ is strictly decreasing. Assume that $u_{1}$ and $u_{2}$ are positive solutions of
$$\Delta u + f(u) = 0 \text{ in } \Omega, \quad u = 0 \text{ on } \partial \Omega$$
Show that $u_{1} = u_{2}$. (Hint: Calculate $\Delta \log \dfrac{u_{2}}{u_{1}}$ and consider the maximum principle.)