psimax[k_]= ArcCos[1-k^2/2];
m[k_]=(1-Cos[psimax[k]])/2;
arg[theta_,k_]=ArcSin[Sqrt[(1-Cos[theta])/(1-Cos[psimax[k]])]];

y[theta_,k_]=- EllipticF[arg[theta,k],m[k]]+2 EllipticE[arg[theta,k],m[k]];

f[w_]:=Re[FindRoot[2 y[psimax[x],x]-y[Pi/2,x]==w/2,{x,1.7}][[1,2]]]

g[w_]:=Re[FindRoot[y[Pi/2,x]==w/2,{x,1}][[1,2]]]

Plot[g[w]-9 f[w],{w,.1,.2}]

FindRoot[g[soln]-9 f[soln]==0,{soln,.13}]