\documentclass{standalone}
\usepackage[active,tightpage]{preview}
\usepackage{amsmath}
\usepackage{pgfplots}
\usepgfplotslibrary{fillbetween}
\usetikzlibrary{arrows.meta,intersections}

\usepackage{anyfontsize}
\renewcommand{\normalsize}{\fontsize{14pt}{16pt}\selectfont}

\pgfplotsset{compat=1.18}
\PreviewEnvironment{tikzpicture}

\begin{document}    

\begin{tikzpicture}
\begin{axis}[
  xlabel={$\alpha$},
  ylabel={$\beta$},
  xtick={\empty},
  ytick={\empty},
  xlabel style={at={(axis description cs:0.98,0)}},
  ylabel style={at={(axis description cs:0,0.98)}, rotate=-90},
  ymin=0.5,
  ymax=0.7,
  xmin=0.15,
  xmax=0.35,
  clip=false,
  axis on top,
  scale=3,
  axis equal image,
] 
% The curve
\addplot [domain=3/13:1/3,samples=1000] plot {(1-2*x)/(1-x)};
\addplot [domain=0.15:0.35,samples=1000] plot {1/(2-x)};
\addplot [color=red!40,name path=A,domain=2/11:0.29289,samples=1000]
  plot {
    and(2/11 <= x, x <= 1/5) * 5/8
    + and(1/5 < x, x <= 2/9) * 4/7
    + and(2/9 < x, x <= 1/4) * 2/3
    + and(1/4 < x, x <= 2/7) * 3/5
    + and(2/7 < x, x <= 0.29289) * (1-2*x)/(1-x)
  };
\addplot [very thick,color=red,name path=B,domain=2/11:0.29289,samples=1000] plot {1/(2-x)};
\addplot [color=red!40] fill between[of=A and B];
\addplot [very thick,domain=2/7:0.29289,color=red,samples=100] plot {(1-2*x)/(1-x)};
\draw [dashed] (axis cs: 0.29289, {1/(2-0.29289)}) -- (axis cs: 0.29289, 0.5)
  node[below, xshift=1.25em] {$1-\dfrac{\sqrt{2}}{2}$};
\draw [dashed] (axis cs: 2/7, 3/5) -- (axis cs: 2/7, 0.5) node[below] {$\dfrac{2}{7}$};
\draw [dashed] (axis cs: 0.15, 3/5) node[left] {$\dfrac{3}{5}$} --
  (axis cs: 1/4, 3/5) -- (axis cs: 1/4, 0.5) node[below] {$\dfrac{1}{4}$};
\draw [dashed] (axis cs: 2/9, 4/7) -- (axis cs: 2/9, 0.5) node[below] {$\dfrac{2}{9}$};
\draw [dashed] (axis cs: 0.15, 4/7) node[left] {$\dfrac{4}{7}$} --
  (axis cs: 1/5, 4/7) -- (axis cs: 1/5, 0.5) node[below] {$\dfrac{1}{5}$};
\draw [dashed] (axis cs: 0.15, 11/20) node[left] {$\dfrac{11}{20}$} --
  (axis cs: 2/11, 11/20) -- (axis cs: 2/11, 0.5) node[below] {$\dfrac{2}{11}$};
\draw [dashed] (axis cs: 2/11, 5/8) -- (axis cs: 0.15, 5/8) node[left] {$\dfrac{5}{8}$};
\draw [dashed] (axis cs:  2/9, 2/3) -- (axis cs: 0.15, 2/3) node[left] {$\dfrac{2}{3}$};
\draw [very thick,color=red,dashed]
  (axis cs: 2/11, 11/20) -- (axis cs: 2/11, 5/8)
  (axis cs:  2/9,   4/7) -- (axis cs:  2/9, 2/3);
\draw [very thick,color=red] 
  (axis cs: 2/11, 5/8) node[draw,fill=white,circle,inner sep=-1.5pt] {} -- 
  (axis cs:  1/5, 5/8) -- (axis cs:  1/5, 4/7) -- (axis cs:  2/9, 4/7)
  (axis cs:  2/9, 2/3) node[draw,fill=white,circle,inner sep=-1.5pt] {} -- 
  (axis cs: 1/4, 2/3) -- (axis cs:  1/4, 3/5) -- (axis cs: 2/7, 3/5);
\node [draw,very thick,color=red,fill=white,circle,inner sep=-1.5pt] 
  at (axis cs: 2/11, 11/20) {};
\node [above right] at (axis cs: 0.275, {(1-2*0.275)/(1-0.275)}) 
  {$\beta=\dfrac{1-2\alpha}{1-\alpha}$};
\node [below, yshift=-0.5em] at (axis cs: 0.268, {1/(2-0.268)}) {$\beta=\dfrac{1}{2-\alpha}$};
\end{axis}
\end{tikzpicture}

\end{document}