Let be the set $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Show that: $ \text{(1)}\sin\in\Lambda $ $ \text{(2)}\sum_{p\in\mathbb{Z}}\frac{1}{(1+2p)^2} =\frac{\pi^2}{4} $ $ \text{(3)} f\in\Lambda\implies \left| f'(0) \right|\le 1 $

Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$, $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$.

Let be a function $ \xi:\mathbb{R}\to\mathbb{R} $ of class $ C^{\infty } $ such that $ \left| \frac{d^n\xi }{dx^n} \left( x_0 \right) \right|\le 1=\frac{d\xi}{dx}(0) , $ for any real numbers $ x_0, $ and all natural numbers $ n, $ and let be the function $ h:\mathbb{C}\longrightarrow\mathbb{C} , h(z)=1+\sum_{n\in\mathbb{N}} \left(\frac{z^n}{n!}\cdot\frac{d^n\xi }{dx^n} \left( 0 \right)\right) . $ [b]a)[/b] Show that $ h $ is well-defined and analytic. [b]b)[/b] Prove that $ h\bigg|_{\mathbb{R}} =\xi\bigg|_{\mathbb{R}} . $ [b]c)[/b] Demonstrate that $$ \frac{d}{dt}\left( \frac{\xi }{\cos} \right)\left( t_0 \right) =4\sum_{p\in\mathbb{Z}}\frac{(-1)^p\xi\left( \frac{(1+2p)\pi}{2} \right)}{\left( (1+2p)\pi -2t_0\right)^2} , $$ for any $ t_0\in\left( -\frac{\pi }{2} ,\frac{\pi }{2} \right) $ and that $$ \sum_{p\in\mathbb{Z}} \frac{(-1)^p\left(\xi\left( \frac{(1+2p)\pi}{2} \right)\right)^2}{1+2p} =\frac{\pi }{2} . $$ [b]d)[/b] Deduce that $ \xi\left( \frac{(1+2p)\pi}{2} \right)=(-1)^p, $ for any integer $ p, $ and that $$ \frac{d}{dt}\left( \frac{\xi }{\cos} \right)\left( t_0 \right) =\frac{d}{dt}\left( \frac{\sin }{\cos} \right)\left( t_0 \right) , $$ for any $ t_0\in\left( -\frac{\pi }{2} ,\frac{\pi }{2} \right) . $ [b]e)[/b] Conclude that $ \xi\bigg|_\mathbb{R} =\sin\bigg|_\mathbb{R} . $

Let $H(D)$ denote the space of functions holomorphic on the disc $D=\{ z\colon |z|<1 \}$, endowed with the topology of uniform convergence on each compact subset of $D$. If $f(z)=\sum_{n=0}^{\infty} a_nz^n$, then we shall denote $S_n(f,z)=\sum_{k=0}^n a_kz^k$. A function $f\in H(D)$ is called [i]universal[/i] if, for every continuous function $g\colon\partial D\rightarrow \mathbb{C}$ and for every $\varepsilon >0$, there are partial sums $S_{n(j)}(f,z)$ approximating $g$ uniformly on the arc $\{ e^{it} \colon 0\le t\le 2\pi - \varepsilon\}$. Prove that the set of universal functions contains a dense $G_{\delta}$ subset of $H(D)$.

For a function ${u}$ defined on ${G \subset \Bbb{C}}$ let us denote by ${Z(u)}$ the neignborhood of unit raduis of the set of roots of ${u}$. Prove that for any compact set ${K \subset G}$ there exists a constant ${C}$ such that if ${u}$ is an arbitrary real harmonic function on ${G}$ which vanishes in a point of ${K}$ then: \[\displaystyle \sup_{z \in K} |u(z)| \leq C \sup_{Z(u)\cap G}|u(z)|.\]

If the binary representations of the positive integers $k$ and $n$ are $k = \sum_{i=0}^{\infty} k_i 2^i$ and $n = \sum_{i=0}^{\infty} n_i 2^i$, then the logical sum of these numbers is \[ k \oplus n =\sum_{i=0}^{\infty} |k_i-n_i|2^i. \] Let $N$ be an arbitrary positive integer and $(c_k)_{k \in \mathbb{N}}$ be a sequence of complex numbers such that for all $k \in \mathbb{N}$, $ |c_k| \le 1$. Prove that there exist positive constants $C$ and $\delta$ such that \[ \int_{[-\pi,\pi] \times [-\pi, \pi]} \sup_{n<N, n \in \mathbb{N}} \frac{1}{N} \Big| \sum_{k=1}^{n} c_k e^{i(kx+(k \oplus n) y)} \Big| \mathrm d(x,y) \le C \cdot N^{-\delta} \] holds.

Let us assume that the series of holomorphic functions $ \sum_{k=1}^{\infty}f_k(z)$ is absolutely convergent for all $ z \in \mathbb{C}$. Let $ H \subseteq \mathbb{C}$ be the set of those points where the above sum funcion is not regular. Prove that $ H$ is nowhere dense but not necessarily countable. [i]L. Kerchy[/i]

Let $ f:\mathbb{C}\longrightarrow\mathbb{C} $ be an holomorphic function which has the property that there exist three positive real numbers $ a,b,c $ such that $ |f(z)|\geqslant a|z|^b , $ for any complex numbers $ z $ with $ |z|\geqslant c. $ Prove that $ f $ is polynomial with degree at least $ \lceil b\rceil . $

Let $D\subseteq \mathbb{C}$ be a compact set with at least two elements and consider the space $\Omega=\bigtimes_{i=1}^{\infty} D$ with the product topology. For any sequence $(d_n)_{n=0}^{\infty} \in \Omega$ let $f_{(d_n)}(z)=\sum_{n=0}^{\infty}d_nz^n$, and for each point $\zeta \in \mathbb{C}$ with $|\zeta|=1$ we define $S=S(\zeta,(d_n))$ to be the set of complex numbers $w$ for which there exists a sequence $(z_k)$ such that $|z_k|<1$, $z_k \to \zeta$, and $f_{d_n}(z_k) \to w$. Prove that on a residual set of $\Omega$, the set $S$ does not depend on the choice of $\zeta$.

Prove that a harmonic function that is not identically zero in the plane cannot vanish on a two-dimensional positive-measure set.

Let $p(x)=x^5+x$ and $q(x)=x^5+x^2$, Find al pairs $(w,z)\in \mathbb{C}\times\mathbb{C}$, $w\not=z$ for which $p(w)=p(z),q(w)=q(z)$.

Let $D=\{ z\in \mathbb C\colon |z|<1\}$ and $D=\{ w\in \mathbb C \colon |w|=1\}$. Prove that if for a function $f\colon D\times B\rightarrow\mathbb C$ the equality $$f\left( \frac{az+b}{\overline{b}z+\overline{a}}, \frac{aw+b}{\overline{b}w+\overline a} \right)=f(z,w)+f\left(\frac{b}{\overline a}, \frac{aw+b}{\overline b w+\overline a} \right)$$ holds for all $z\in D, w\in B$ and $a, b\in \mathbb C,|a|^2=|b|^2+1$, then there is a function $L\colon (0, \infty )\rightarrow \mathbb C$ satisfying $$L(pq)=L(p)+L(q)\,\,\,\text{for all}\,\,\, p,q > 0$$ such that $f$ can be represented as $$f(z,w)=L\left( \frac{1-|z|^2}{|w-z|^2}\right)\,\,\,\text{for all}\,\,\, z\in D, w\in B$$. [Gy. Maksa]

Let $\Gamma(s)$ denote Euler's gamma function. Construct an even entire function $F(s)$ that does not vanish everywhere, for which the quotient $F(s)/\Gamma(s)$ is bounded on the right halfplane $\{\Re(s)>0\}$.

Let be a natural number $ n, $ denote with $ C $ the square in the complex plane whose vertices are the affixes of $ 2n\pi\left( \pm 1\pm i \right) , $ and consider the set $$ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} $$ Prove the following implications. [b]a)[/b] $ \exists \alpha\in\mathbb{R}_{>0}\quad \forall z\in\partial C\quad \left| \cos z \right|\ge\alpha e^{|\text{Im}(z)|} $ [b]b)[/b] $ \forall f\in\Lambda\quad\frac{1}{2\pi i}\int_{\partial C} \frac{f(z)}{z^2\cos z} dz=f'(0)+\frac{4}{\pi^2}\sum_{p=-2n}^{2n-1} \frac{(-1)^{p+1} f(z-p)}{(1+2p)^2} $ [b]c)[/b] $ \forall f\in\Lambda\quad \sum_{p\in\mathbb{Z}}\frac{(-1)^pf\left( \frac{(1+2p)\pi}{2} \right)}{(1+2p)^2} =\frac{\pi^2 f'(0)}{4} $

Let $p(z)$ be a polynomial of degree $n>0$ with complex coefficients. Prove that there are at least $n+1$ complex numbers $z$ for which $p(z)\in \{0,1\}$.

$P(z)=a_nz^n+\dots+a_1z+z_0$, with $a_n\neq 0$ is a polynomial with complex coefficients, such that when $|z|=1$, $|P(z)|\leq 1$. Prove that for any $0\leq k\leq n-1$, $|a_k|\leq 1-|a_n|^2$. [i]Proposed by Yijun Yao[/i]

Let $ G$ be a locally compact solvable group, let $ c_1,\ldots, c_n$ be complex numbers, and assume that the complex-valued functions $ f$ and $ g$ on $ G$ satisfy \[ \sum_{k=1}^n c_k f(xy^k)=f(x)g(y) \;\textrm{for all} \;x,y \in G \ \ .\] Prove that if $ f$ is a bounded function and \[ \inf_{x \in G} \textrm{Re} f(x) \chi(x) >0\] for some continuous (complex) character $ \chi$ of $ G$, then $ g$ is continuous. [i]L. Szekelyhidi[/i]