f=h m+r (m^2)+u (m^4);

s=Solve[D[f,m]==0,m]

ss=Series[1/D[m/.s[[3]],h],{r,0,1},Assumptions->{h>0,u>0,r<0}]

Solve[Normal[ss]==0,h]

Plot3D[Evaluate[D[m/.s[[1]]/.u->1,h]],{r,-1,1},{h,-1,1}]

z0=Integrate[Exp[-n/4 (m^2)]Exp[β (Sqrt[n])RealAbs[m]],{m,-∞,∞},Assumptions->{β>0,n>0}]

Z0=(z0^(n^2))

Z1=β λ y((n^2)(z0^((n^2)-1))Integrate[(1/n-2 (m^2)+(m^4))Exp[-n/4 (m^2)]Exp[β (Sqrt[n])RealAbs[m]],{m,-∞,∞},Assumptions->{β>0,n>0}]+2(n^2)(n-1)(z0^((n^2)-2))Integrate[(m1^2)(m2^2)Exp[-n/4 ((m1^2)+(m2^2))]Exp[β (Sqrt[n])(RealAbs[m1]+RealAbs[m2])],{m1,-∞,∞},{m2,-∞,∞},Assumptions->{β>0,n>0}])

tmp=FullSimplify[Log[Z0+Z1],{n>0,β>0,λ>0}]/(n^2)

Series[tmp,{n,∞,2}]