Problem giving color to a logo

Here is a simple solution. You may adjust the parameters (\anglei, \angleii, \angleiii, and \len) to change the flame.

\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}[line width=0,draw=blue]
  \fill[red] (0,0) circle (2.41);
  \begin{scope}[overlay]
    \fill[white] (0,.65) circle (1.92);
  \end{scope}
  \def\anglei{-65}
  \def\angleii{45}
  \def\angleiii{160}
  \def\len{1.5}
  \fill[red]
  (0,.65) ++(0,-1.92)
  arc[start angle=\anglei,end angle=\angleii,radius=\len]
  arc[start angle=\angleii+180,end angle=\angleiii,radius=\len]
  %
  arc[start angle=\anglei+180,end angle=\angleii+180,radius=\len]
  arc[start angle=\angleii+360,end angle=\angleiii+180,radius=\len]
  ;
\end{tikzpicture}
\end{document}

run with xelatex. The origin is the center of the big circle. \pscustom builds a closed path which can be filled.

\documentclass[pstricks]{standalone}
\usepackage{pstricks}
\begin{document}

\begin{pspicture}(-3,-3)(3,3)
\pscircle*[linecolor=red]{2.41}
\pscircle*[linecolor=white](0,0.65){1.92}
\pscustom[fillstyle=solid,fillcolor=red,linecolor=red]{%
  \psarc(0.65,1.15){1.4}{!2.01 RadtoDeg}{!4.0 RadtoDeg}
  \psarcn(-0.91,-0.79){1.08}{!0.93 RadtoDeg}{!-0.48 RadtoDeg}
  \psarc(-0.58,0){1.43}{!-1.12 RadtoDeg}{!0.68 RadtoDeg}
  \psarcn(1.1,1.94){1.18}{!4.2 RadtoDeg}{!2.72 RadtoDeg}
}
\end{pspicture}

\end{document}

If you want to use pdflatex:

\documentclass{book}
\usepackage{pstricks}
\usepackage{auto-pst-pdf}
\begin{document}
[...] 

and run pdflatex --shell-escape file.tex

Not quite right, but anyway...

\documentclass[tikz,border=5]{standalone}
\begin{document}
\tikz[line join=round]\filldraw [red] (0,0) 
  arc (270:180-30:cos 45 / cos 30) arc (180-45:360+45:1)
  arc (30:-90:cos 45 / cos 30) -- cycle
  .. controls ++(30:1) and ++(210:3/2) .. (0,1.5)
  .. controls ++(210:1) and ++(30:3/2) .. cycle;
\end{document}