Xjyuk

I am trying to shade a polygon with red curved lines as shown in the MWE. I would like to shade the curved polygon enclosed by the points (A,B,C,D), i.e., the polygon ABCD.

MWE

documentclassarticle
usepackagestandalone
usepackagetikz
begindocument
 begintikzpicture
 draw [->] (0, 0) -- (5, 0) node [right] $X$;
 draw [->] (0, 0) -- (0, 3) node [above] $Y$;

 draw [domain=0.66:4.5, thick, blue] plot (x, 2/x);
 draw [domain=1.66:4.5, thick, blue] plot (x, 6/x);

 draw [red, ultra thick] (3, 2) edge [bend right=10] (3.5, 0.57);
 draw [red, ultra thick] (2, 3) edge [bend right=10] (2.5, 0.8);

 draw [domain=2.5:3.5, ultra thick, red] plot (x, 2/x);
 draw [domain=2:3, ultra thick, red] plot (x, 6/x);

 node [above] at (2, 3) $C$;
 node [below] at (2.5, 0.8) $B$;
 node [above] at (3, 2) $D$;
 node [below] at (3.5, 0.57) $A$;
 endtikzpicture
enddocument

How to shade the polygon enclosed by ABCD with tikz?

Using fadings library of TiKz:

documentclass[tikz,margin=5mm]standalone
usetikzlibraryfadings
makeatletter
pgfdeclareverticalshadingpgf@lib@fade@north100bp
color(0bp)=(pgftransparent!0);
 color(5bp)=(pgftransparent!10);
 color(60bp)=(pgftransparent!100); color(80bp)=(pgftransparent!100)%
pgfdeclarefadingmyfade%
 pgfuseshadingpgf@lib@fade@east%

makeatother
usepackagetikz
begindocument
 begintikzpicture
 draw [->] (0, 0) -- (5, 0) node [right] $X$;
 draw [->] (0, 0) -- (0, 3) node [above] $Y$;

 draw [domain=0.66:4.5, thick, blue] plot (x, 2/x);
 draw [domain=1.66:4.5, thick, blue] plot (x, 6/x);

 draw [red, ultra thick] (3, 2)coordinate(D) edge [bend right=10] (3.5, 0.57)coordinate(A);
 draw [red, ultra thick] (2, 3)coordinate(C) edge [bend right=10] (2.5, 0.8)coordinate(B);

 draw [domain=2.5:3.5, ultra thick, red] plot (x, 2/x);
 draw [domain=2:3, ultra thick, red] plot (x, 6/x);

 node [above] at (2, 3) $C$;
 node [below] at (2.5, 0.8) $B$;
 node [above] at (3, 2) $D$;
 node [below] at (3.5, 0.57) $A$;
 fill[red,path fading=myfade] (3,2) to [bend right=10] (3.5,0.57) 
to [bend left=10] (2.5,0.8) to [bend left=10] (2,3) to [bend right=10] (3,2);

 endtikzpicture
enddocument

I don't know how you wanted it shaded, but if you draw it as a single path you can use for example shadedraw[left color=.., right color.., ].

documentclassarticle
usepackagestandalone
usepackagetikz
begindocument
 begintikzpicture
 draw [->] (0, 0) -- (5, 0) node [right] $X$;
 draw [->] (0, 0) -- (0, 3) node [above] $Y$;

 draw [domain=0.66:4.5, thick, blue] plot (x, 2/x);
 draw [domain=1.66:4.5, thick, blue] plot (x, 6/x);

 shadedraw[
 draw=red,
 ultra thick,
 left color=blue,
 right color=red,
 bend angle=10
 ]
 plot[domain=2.5:3.5] (x, 2/x)
 to[bend left] (3,2)
 plot[domain=3:2] (x, 6/x)
 to[bend right] (2.5, 2/2.5);

 node [above] at (2, 3) $C$;
 node [below] at (2.5, 2/2.5) $B$;
 node [above] at (3, 2) $D$;
 node [below] at (3.5, 2/3.5) $A$;
 endtikzpicture
enddocument