Let $A_r$ be a complex annulus of modulus $r>0$ obtained from a $1\times r$ rectangle in $\mathbb C$ with vertices $A=0$, $B=r$, $C=r+i$, $D=i$, by identifying isomterically $AB$ with $DC$.
Let us now chose $0<s<r$ and make a cut of the cylinder $A_r$ along the segment $[0,s]$ (if $s=r$ we get back the rectangle). Denote the obtained annulus by $A_r(s)$. Clearly, the modulus of $A_r(s)$ is less than $r$.
Question. Is there a reasonably looking estimation from above of the modulus $M(A_r(s))$ of cylinder $A_r(s)$? More precisely I wonder if there is a relatively simply looking continuous function, $f(r,s)$ defined on the domain $0\le s\le r$ such that
$$f(r,s)\ge M((A_r(s)),$$ $$f(r,0)=r,\;\;f(r,s_1)>f(r,s_2)\; if \;\; s_2>s_1,\;\; f(r,r)=0.$$