The sequence $(a_n)$ of complex numbers is considered in the complex plane, in which is: $$a_0 = 1, \,\,\, a_n = a_{n-1} +\frac{1}{n}(\cos 45^o + i \sin 45^o )^n.$$ Prove that the sequence of the real parts of the terms of $(a_n)$ is convergent and its limit is a number between $0.85$ and $1.15$.

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 $H(D)$.

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

Let us consider sequences of complex numbers that are infinite in both directions $c=(c_k) , k\in Z$ with finite norm $||c||= (\sum_{k \in Z} |c_k|^2)^{1/2}$ Let $T_m-$ this is a shift operation sequences on m ($(T_mc)_k=c_{k-m}$) Prove that: $\lim_{n \to \infty} \frac{\sum_{i=0}^{n-1} T_ic}{n} =0$ (Adding and multiplying a sequence by a number defined component by component)

Denote by $ M(r,f)$ the maximum modulus on the circle $ |z|\equal{}r$ of the transcendent entire function $ f(z)$, and by $ M_n(r,f)$ that of the $ nth$ partial sum of the power series of $ f(z)$. Prove that the existence of an entire function $ f_0(z)$ and a corresponding sequence of positive numbers $ r_1<r_2<...\rightarrow \plus{}\infty$ such that \[ \limsup_{n\rightarrow\infty} \frac{M_n(r_n,f_0)}{M(r_n,f_0)}\equal{}\plus{}\infty\] [P. Turan]

Solve the Dirichlet problem \[\Delta u=0\text{ for }x^2+y^2<1\] \[u=1\text{ for }x^2+y^2=1,\;y>0\] \[u=-1\text{ for }x^2+y^2=1,\;y<0\]

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.

Given a polynomial $P$, assume that $L = \{z \in \mathbb{C}: |P(z)| = 1\}$ is a Jordan curve. Show that the zeros of $P'$ are in the interior of $L$.

Let $ x_0$ be a fixed real number, and let $ f$ be a regular complex function in the half-plane $ \Re z>x_0$ for which there exists a nonnegative function $ F \in L_1(- \infty, +\infty)$ satisfying $ |f(\alpha+i\beta)| \leq F(\beta)$ whenever $ \alpha > x_0$ , $ -\infty <\beta < +\infty$. Prove that \[ \int_{\alpha-i \infty} ^{\alpha+i \infty} f(z)dz=0.\] [i]L. Czach[/i]

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]

prove that if $r_n$ is a rational function whose numerator and denominator have at most degrees $n$, then $$||r_n||_{1/2}+\left\|\frac{1}{r_n}\right\|_2\geq\frac{1}{2^{n-1}}$$ where $||\cdot||_a$ denotes the supremum over a circle of radius $a$ around the origin.

Let $D=\{ z \in \mathbb{C} : \operatorname{Im}(z) >0 , \operatorname{Re}(z) >0 \} $. Let $n \geq 1 $ and let $a_1,a_2,\dots a_n \in D$ be distinct complex numbers. Define $$f(z)=z \cdot \prod_{j=1}^{n} \frac{z-a_j}{z-\overline{a_j}}$$ Prove that $f'$ has at least one root in $D$. [i]Proposed by Géza Kós (Lorand Eotvos University, Budapest)[/i]

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

Let $ f:\mathbb{D}\longrightarrow\mathbb{C} $ be a continuous function, where $ \mathbb{D} $ is the closed unit disk. Suppose that $ f $ is holomorphic on the open unit disk and that $ e^{i\theta } $ are roots, for any $ \theta\in\left[ 0,\pi /4 \right] . $ Show that $ f=0_{\mathbb{D}} . $

Suppose that $ R(z)= \sum_{n=-\infty}^{\infty} a_nz^n$ converges in a neighborhood of the unit circle $ \{ z : \;|z|=1\ \}$ in the complex plane, and $ R(z)=P(z) / Q(z)$ is a rational function in this neighborhood, where $ P$ and $ Q$ are polynomials of degree at most $ k$. Prove that there is a constant $ c$ independent of $ k$ such that \[ \sum_{n=-\infty} ^{\infty} |a_n| \leq ck^2 \max_{|z|=1} |R(z)|.\] [i]H. S. Shapiro, G. Somorjai[/i]

[b]9.[/b] Let $w=f(x)$ be regular in $ \left | z \right |\leq 1$. For $0\leq r \leq 1$, denote by c, the image by $f(z)$ of the circle $\left | z \right | = r$. Show that if the maximal length of the chords of $c_{1}$ is $1$, then for every $r$ such that $0\leq r \leq 1$, the maximal length of the chords of c, is not greater than $r$. [b](F. 1)[/b]

Let $p,q\in \mathbb{R}[x]$ such that $p(z)q(\overline{z})$ is always a real number for every complex number $z$. Prove that $p(x)=kq(x)$ for some constant $k \in \mathbb{R}$ or $q(x)=0$. [i]Proposed by Mohammad Ahmadi[/i]

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 $ 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]

Say that a polynomial with real coefficients in two variable, $ x,y,$ is [i]balanced[/i] if the average value of the polynomial on each circle centered at the origin is $ 0.$ The balanced polynomials of degree at most $ 2009$ form a vector space $ V$ over $ \mathbb{R}.$ Find the dimension of $ V.$

Prove that $$\operatorname{Re}\left(\operatorname{Li}_2\left(\frac{1-i\sqrt3}2\right)+\operatorname{Li}_2\left(\frac{\sqrt3-i}{2\sqrt3}\right)\right)=\frac{7\pi^2}{72}-\frac{\ln^23}8$$ where as usual $$\operatorname{Li}_2(z)=-\int^z_0\frac{\ln(1-t)}tdt,z\in\mathbb C\setminus[1,\infty)$$ [i]Proposed by Paolo Perfetti[/i]

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 $D\subseteq \mathbb{C}$ 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 $\zeta$.

Let $f:\mathbb{C} \to \mathbb{C}$ be an entire function, and suppose that the sequence $f^{(n)}$ of derivatives converges pointwise. Prove that $f^{(n)}(z)\to Ce^z$ pointwise for a suitable complex number $C$.

Let $q{}$ be an arbitrary polynomial with complex coefficients which is not identically $0$ and $\Gamma_q =\{z : |q(z)| = 1\}$ be its contour line. Prove that for every point $z_0\in\Gamma_q$ there is a polynomial $p{}$ for which $|p(z_0)| = 1$ and $|p(z)|<1$ for any $z\in\Gamma_q\setminus\{z_0\}.$