R[z_]:=R1 z+R2 z^2+R3 z^3+R4 z^4+R5 z^5

u[z_,t_]:=1+t f1[z/t]+t^2 f2[z/t]+t^3 f3[z/t]+t^4 f4[z/t]+t^5 f5[z/t]

s[t_]:=s1 t+s2 t^2+s3 t^3+s4 t^4+s5 t^5(St D[u[z,t],t]-D[u[z,t],z,z]-2R'[z]/R[z]D[u[z,t],z]+2u[z,t]/R[z])/.z->t y

(St D[u[z,t],t]-D[u[z,t],z,z]-2R'[z]/R[z]D[u[z,t],z]+2u[z,t]/R[z])/.z->t y;

Series[%,{t,0,3}]//Simplify

eq=Simplify[% t];

u[s[t],t]-1//Simplify;

Series[%,{t,0,5}]//Simplify

bc1=Simplify[%/t];

(D[u[z,t],z]-1-s'[t]/.z->s[t])//Simplify;

Series[%,{t,0,4}]//Simplify

bc2=Simplify[%];

Simplify[eq/.t->0]

DSolve[%==0,f1,y]

f1[y]/.%[[1]]/.C[1]->0

f1[y_]=%

(* Must be bounded as y->0 *)

{bc1,bc2}/.t->0

Solve[%==0,{C[2],s1}]//Simplify

s1=(s1/.%[[1]]);f1[y_]=(Simplify[f1[y]/.%[[1]]]);

{eq,bc1,bc2}=Simplify[{eq,bc1,bc2}/t];

(* Now repeat procedure *)

Simplify[eq/.t->0];

DSolve[%==0,f2,y]

f2[y]/.%[[1]]/.C[1]->0//Simplify

f2[y_]=%;

{bc1,bc2}/.t->0//Simplify;

Solve[%==0,{C[2],s2}]//Simplify;

s2=(s2/.%[[1]]);f2[y_]=(Simplify[f2[y]/.%[[1]]]);

{eq,bc1,bc2}=Simplify[{eq,bc1,bc2}/t];

Simplify[eq/.t->0];

DSolve[%==0,f3,y]//Simplify

f3[y]/.%[[1]]/.C[1]->0//Simplify;

f3[y_]=%;

{bc1,bc2}/.t->0//Simplify;

Solve[%==0,{C[2],s3}]//Simplify;

s3=(s3/.%[[1]]);f3[y_]=(Simplify[f3[y]/.%[[1]]]);

ss=Series[s[t],{t,0,3}]//Simplify

(* We need s'(t)≥0 and hence R1<1 *)

(* As R1->1 we then need s''(t)≥0 *)

Simplify[D[ss,t,t]/.t->0]

Simplify[%/(1-R1)]/.R1->1

(* Therefore as R1->1 we need R2≥-1/4 *)

(* As R1->1 and R2->-1/4, we need s'''[0]≥0 *)

Simplify[D[ss,t,t,t]/.t->0]

(* R3 must exceed the zero of this coefficient *)

(* Let R2->-1/4 and R1->1 carefully! *)

Solve[Evaluate[D[ss,t,t,t]/.t->0]==0,R3]//Simplify

FullSimplify[(R3/.%[[1]])/.R2->-1/4]

FullSimplify[%/.R1->1]

(* This gives us the required maximal radius profile
(Subscript[R, m])(z) ∼ z - ((z^2)/4) + ((z^3)/24) + ... *)