\documentclass[portrait,a0,final,10pt]{a0poster}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 1.8.2018 by Juha-Matti Huusko (C)
% This document is released under licence CC-BY-SA:
% https://creativecommons.org/licenses/by-sa/2.0/
% Further restrictions:
% * This copyright notice cannot be removed.
% * Poster template must be distributed in web page of
% Department of Physics and Mathematics, University of Eastern Finland.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Any questions? Need help?
% Feel free to contact:
% juha-matti.huusko@uef.fi
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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%%%% ---- ENVIRONMENTS ---- %%%%
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%%%% ---- DOCUMENT ---- %%%%
\begin{document}
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\begin{poster}

{\bfseries \sffamily \fontsize{140}   
{0}\selectfont {}{THE POSTER {\it \color{uefturquoise} \textbf{TITLE}}\\[7.5mm] COMES HERE}}

\vspace{2cm}

{\bfseries \sffamily \fontsize{82}   
{0}\selectfont {}{Subtitle}}

\vspace{1cm}

{\bfseries \sffamily \fontsize{44}   
{0}\selectfont {}{\color{uefturquoise} \it \textbf{Name Surname}}}
{\bfseries \sffamily \fontsize{44}   
{0}\selectfont {}{// joint work with Nimi Sukunimi}}}

\vspace*{2cm}

\toprule[2mm]

\vspace*{2cm}

%%%% ---- CONTENT ---- %%%%
\begin{center}

%%%% ---- COLUMN 1 ---- %%%%
\begin{pcolumn}{0.32}

\vspace{-2cm}

%%%% ---- ABSTRACT ---- %%%%
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{}
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{\centering \textsc{Abstract}}
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\bigskip

\lipsum[1]
}

\bigskip

%%%% ---- INTRODUCTION ---- %%%%
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{\centering \textsc{Introduction}}
\end{center}

\lipsum[2-3]


\end{pcolumn}
\hspace*{0.5cm}
%%%% ---- COLUMN 3 ---- %%%%
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%%%% ---- RESULTS ---- %%%%
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%%%% ---- THEOREM ---- %%%%
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\begin{theorem}\label{th:1}
\lipsum[5]
\end{theorem}
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%%%% ---- THEOREM ---- %%%%
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{
\begin{example}
Let $f=f_{C,\zeta}$ be a locally univalent analytic function in $\D$ such that $f(-1)=0$ and
\begin{equation*}
\label{examplef}
    f'(z)=-i\left(\frac{1+z}{1-z}\right)^\frac12e^{\frac{C\z z}2},\quad \z\in\partial\D,\, z\in\D.
\end{equation*}
Then
    $$
    \frac{f''(z)}{f'(z)}=\frac1{1-z^2}+\frac{C\z}{2},
    $$
and $f$ is univalent in $\D$ if $C\le1$ by Becker's univalence criterion. If $f$ is univalent, then we obtain for $\zeta=1$,
		$$
		1\geq \frac{|f'(x)|}{|k'(x)|}=\frac{e^{\frac{Cx}{2}}(1-x)^{5/2}}{(1+x)^{1/2}}
		\sim\frac{1+Cx/2}{1+3x},\quad x\to 0^+,
		$$
where $k(z)=z/(1-z)^2$ is the Koebe function. Therefore, if $C>6$, then $f$ is not univalent.

The boundary curve $\partial f(\D)$ has a cusp at $f(-1)=0$. When $\zeta=-i$, the cusp has its worst behavior, and by numerical calculations the function $f$ is not univalent if $C>2.21$.

Moreover, as $C$ increases, the valence of $f$ increases.

The curve $\set{f(e^{it})\,:\, t\in(0,\pi]}$ is a spiral unwinding from $f(-1)$. We may calculate the valence of $f$ by counting how many times $h(t)=\Real(f(e^{it}))$ changes its sign on $(0,\pi]$. Numerical calculations suggest that the valence of $f$ is approximately equal to $\frac{100}{63}C$.
\end{example}
}

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\hspace*{0.5cm}
%%%% ---- COLUMN 3 ---- %%%%
\begin{pcolumn}{0.32}

%Clearly $T$ defined by~\eqref{eq:T} satisfies all the assumptions of Lemma~\ref{lm:local}. Hence, if we set $g=f\circ T$ then we can study differential equation~\eqref{eq:de2} of $f$ by studying differential equation~\eqref{eq:gk} of $g$. In this case, if we can get a lower bound for the $n$-order of $g$, we get a lower bound for the $n$-order of $f$ by the next lemma.

\vspace{4cm}

%%%% ---- THEOREM ---- %%%%
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{\begin{lemma}
\label{lm:order}
\lipsum[6]
\end{lemma}}
%
\vspace{-1cm}
%
\myfig{fig1a}{0.8}
%
\vspace{-1cm}
%%%% ---- BIBLIOGRAPHY ---- %%%%
\begin{small}
\begin{thebibliography}{99}
\bibitem{Becker:1972} J.~Becker, \emph{L\"ownersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen}, (German) J. Reine Angew. Math.~\textbf{255} (1972), 23--43.

\bibitem{BP:1984} J.~Becker and Ch.~Pommerenke, \emph{Schlichtheitskriterien und Jordangebiete}, J. Reine Angew. Math. \textbf{354}, 74--94 (1984)

\bibitem{BP:2016} J. Becker and Ch. Pommerenke,~\emph{Locally univalent functions and the Bloch and Dirichlet norm}, Comput. Methods Funct. Theory \textbf{16}, 43--52 (2016)

\bibitem{H:1922} E.~Hille, \emph{Oscillation Theorems in the Complex Domain}, Trans. Amer. Math. Soc. \textbf{23} (1922), no.~4, 350--385.

\bibitem{Kraus:1932}  W.~Kraus, \emph{\"Uber den Zusammenhang einiger Characteristiken eines einfach zusammenh\"angenden Bereiches mit der Kreisabbildung}, Mitt. Math. Sem. Giessen~\textbf{21} (1932), 1--28.

\bibitem{N:1949} Z.~Nehari, \emph{The Schwarzian derivative and schlicht functions},
    Bull. Amer. Math. Soc. \textbf{55}  (1949), 545--551.

\bibitem{P:1975} Ch.~Pommerenke, \emph{Univalent functions. With a chapter on quadratic differentials by Gerd Jensen}, Studia Mathematica/Mathematische Lehrb\"ucher, Band XXV. Vandenhoeck \& Ruprecht, G\"ottingen, 1975.

\bibitem{S:1955} B. Schwarz, \emph{Complex nonoscillation theorems and criteria of univalence}, Trans. Amer. Math. Soc. \textbf{80} (1955), 159--186.
\end{thebibliography}
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