t0=TimeUsed[];
$HistoryLength=2;
assump=Element[{α,(Sqrt[4α-1]),1/(Sqrt[4 α-1])} ,Reals]&&1≥  α≥7/8 ;
dom=1≥  α≥875/1000;
A[α_]=(({{0, 1}, {-α, 1}}));
 F[{u_,v_}, α_]={0, v(E^(-u)-1)+α(E^(-u)-1+u)};
eigensys[α_]=Eigensystem[A[α]];
eigensystr[α_]=Eigensystem[Transpose[A[α]]];
q[α_]=FullSimplify[eigensys[α][[2,2]],assump];
v1[α_]=FullSimplify[eigensystr[α][[2,1]],assump];(*assistant variable for normalizing p*)
c[α_]=FullSimplify[Conjugate[1/Simplify[Conjugate[v1[α]].q[α]]],assump];
p[α_]=Together[FullSimplify[c[α]*v1[α], assump]]
FullSimplify[Conjugate[p[α]].q[α],assump](*checking*)
μ[α_]=Refine[eigensys[α][[1,2]] ,assump]

FullSimplify[Conjugate[p[α]].{u,v},assump];(*=z[u,v,α]*)
FullSimplify[%*q[α]+Conjugate[%*q[α]],assump&&Element[{u,v},Reals]](*checking*)

g[z_, zconj_,α_]=Simplify[Conjugate[p[α]].F[z*q[α]+zconj*Conjugate[q[α]], α],assump];
gD10[z_,zconj_,α_]=D[g[z,zconj,α],z];
gD01[z_,zconj_,α_]=D[g[z,zconj,α],zconj];
{gD10[0,0,α],gD01[0,0,α]} (*checking*)

gD20[z_,zconj_,α_]=Simplify[D[g[z,zconj,α],{z,2}]];
gD11[z_,zconj_,α_]=Simplify[D[g[z,zconj,α],z,zconj]];
gD02[z_,zconj_,α_]=Simplify[D[g[z,zconj,α],{zconj,2}]];

gD30[z_,zconj_,α_]=Simplify[D[g[z,zconj,α],{z,3}]];
gD21[z_,zconj_,α_]=Simplify[D[g[z,zconj,α],{z,2},zconj]];
gD12[z_,zconj_,α_]=Simplify[D[g[z,zconj,α],z,{zconj,2}]];
gD03[z_,zconj_,α_]=Simplify[D[g[z,zconj,α],{zconj,3}]];

gD40[z_,zconj_,α_]=D[g[z,zconj,α],{z,4}];
gD31[z_,zconj_,α_]=D[g[z,zconj,α],{z,3},zconj];
gD22[z_,zconj_,α_]=D[g[z,zconj,α],{z,2},{zconj,2}];
gD13[z_,zconj_,α_]=D[g[z,zconj,α],z,{zconj,3}];
gD04[z_,zconj_,α_]=D[g[z,zconj,α],{zconj,4}];

h[w_]=w+H20 w^2/2 + H11 *w* Conjugate[w] + H02*Conjugate[w]^2/2 +H30/6 w^3 +  H12/2 w Conjugate[w]^2 + H03/6 Conjugate[w]^3;
G4[z_]=mu z + G20/2z^2+G11 z Conjugate[z] +G02/2 Conjugate[z]^2 + G30/6 z^3 +  G12/2 z Conjugate[z]^2 +G21/2 z^2 Conjugate[z]+ G03/6 Conjugate[z]^3 +G40/24z^4+G31/6z^3Conjugate[z]+G22/4z^2 Conjugate[z]^2+G13/6z Conjugate[z]^3+G04/24Conjugate[z]^4; (*G up to the 4th order terms*)
G4h[w_,conjw_]=Total[Flatten[Table[CoefficientList[FunctionExpand[G4[h[w]]]/. Conjugate[w]-> conjw,{w,conjw}][[i+1,j+1]]w^i conjw^j,{i,0,5},{j,0,5-i}]]];

hinv4par[z_]=z+a20 z^2+a11 z Conjugate[z]+a02 Conjugate[z]^2+a30 z^3+a21 z^2 Conjugate[z] + a12 z Conjugate[z]^2+a03 Conjugate[z]^3+
a40 z^4+a31 z^3 Conjugate[z] + a22 z^2 Conjugate[z]^2+a13 z Conjugate[z]^3 + a04 Conjugate[z]^4;
hinv4hpar[w_,conjw_]=hinv4par[h[w]]//.Conjugate[u_+v_]-> Conjugate[u]+Conjugate[v]//.Conjugate[u_*v_]-> Conjugate[u]*Conjugate[v]/.Conjugate[w]->conjw;
coeffhinv4hpar=CoefficientList[hinv4hpar[w,conjw],{w,conjw}];
solhinv4=Flatten[Solve[{coeffhinv4hpar[[3,1]]==0,coeffhinv4hpar[[2,2]]==0,coeffhinv4hpar[[1,3]]==0,coeffhinv4hpar[[4,1]]==0,coeffhinv4hpar[[3,2]]==0,coeffhinv4hpar[[2,3]]==0,coeffhinv4hpar[[1,4]]==0,coeffhinv4hpar[[5,1]]==0,coeffhinv4hpar[[4,2]]==0,coeffhinv4hpar[[3,3]]==0,coeffhinv4hpar[[2,4]]==0,coeffhinv4hpar[[1,5]]==0},{a20,a11,a02,a30,a21,a12,a03,a40,a31,a22,a13,a04}]];

hinv[z_]=hinv4par[z]/.solhinv4;

hinvG4h[w_,conjw_]=FunctionExpand[hinv[G4h[w,conjw]]];

coeffhinvG4h=CoefficientList[hinvG4h[w,Conjugate[w]]/.Conjugate[w]-> conjw,{w,conjw}];

sol20=Flatten[Solve[coeffhinvG4h[[3,1]]==0,H20]];
sol11=Flatten[Solve[coeffhinvG4h[[2,2]]==0,H11]];
sol02=Flatten[Solve[coeffhinvG4h[[1,3]]==0,H02]];
sol30=Flatten[Solve[coeffhinvG4h[[4,1]]==0,H30]/.sol20 /. sol11/. sol02];
sol12=Flatten[Solve[coeffhinvG4h[[2,3]]==0,H12]/.sol20 /. sol11 /. sol02];
sol03=Flatten[Solve[coeffhinvG4h[[1,4]]==0,H03]/.sol20 /. sol11 /. sol02];

rules={G20->gD20[0,0,α],G11->gD11[0,0,α],G02->gD02[0,0,α],G30->gD30[0,0,α],G21->gD21[0,0,α],G12->gD12[0,0,α],G03->gD03[0,0,α],G30->gD30[0,0,α],G21->gD21[0,0,α],G12->gD12[0,0,α],G03->gD03[0,0,α],G40->gD40[0,0,α],G31->gD31[0,0,α],G22->gD22[0,0,α],G13->gD13[0,0,α],G04-> gD04[0,0,α],mu->μ[α]};
h20[α_]=FullSimplify[H20/. sol20/.rules,assump];
h11[α_]=FullSimplify[H11/. sol11/.rules,assump];
h02[α_]=Together[FullSimplify[FullSimplify[H02/. sol02/.rules,assump]/.Conjugate[(Sqrt[1-4 α])]->-I Sqrt[4α-1],assump]];
h30[α_]=FullSimplify[FullSimplify[H30/. sol30/.rules,assump]/.Conjugate[(Sqrt[1-4 α])]->-I Sqrt[4α-1],assump];
h12[α_]=FullSimplify[FullSimplify[H12/. sol12/.rules,assump]/.Conjugate[(Sqrt[1-4 α])]->-I Sqrt[4α-1],assump];
h03[α_]=Simplify[FullSimplify[H03/. sol03/.rules,assump]/.Conjugate[(Sqrt[1-4 α])]->-I Sqrt[4α-1],assump];

hrules=Flatten[{rules,H20-> h20[α], H11 ->  h11[α], H02-> h02[α], H30-> h30[α] , H12 -> h12[α] , H03 -> h03[α]}];

hα[w_,α_]=w+h20[α] w^2/2 + h11[α] *w* Conjugate[w] + h02[α]*Conjugate[w]^2/2 +h30[α]/6 w^3 +  h12[α]/2 w Conjugate[w]^2 + h03[α]/6 Conjugate[w]^3;

ESTIMATIONS

Estimation of (Subscript[h, α])

Abshα2nd[α_]=FullSimplify[Total[Abs[{h20[α]/2,h11[α],h02[α]/2}]],assump];
h30abs[α_]=FullSimplify[Abs[h30[α]],assump];
h03abs[α_]=FullSimplify[Abs[h03[α]],assump];
Simplify[Re[Refine[ComplexExpand[Refine[h12[α],assump]],assump]],assump];
Simplify[Im[Refine[ComplexExpand[Refine[h12[α],assump]],assump]],assump];
h12abs[α_]=Refine[Sqrt[%^2+%%^2],assump];
Abshα3rd[α_]=h30abs[α]/6+h12abs[α]/2+h03abs[α]/6;

Union[Table[Abshα2nd[Interval[{1-i/1000,1-(i+1)/1000}]]<101/100,{i,0,124}]]
Union[Table[Abshα3rd[Interval[{1-i/1000,1-(i+1)/1000}]]<83/100,{i,0,124}]]

(*NMaxValue[{Abshα2nd[α],dom},α]
NMaxValue[{Abshα3rd[α],dom},α]*)

Showing that (Subscript[h, α]) is injective on (Subscript[B, 1/4]) and (h^(-1)) is valid in (Subscript[B, 1/6]), |z|<1/28 and |u|,|v|<1/13 if |w|<1/30.

2*101/100*(1/4)+3*83/100*(1/4)^2<1
1/6+101/100 *(1/4)^2+83/100*(1/4)^3<1/4

1/30+101/100/30^2+83/100/30^3<1/28

1/13-2/28/(7/8)^(1/2)>0

Estimations on g and (Subscript[R, 1])

Absg2nd[α_]=FullSimplify[Total[Abs[{gD20[0,0,α]/2,gD11[0,0,α],gD02[0,0,α]/2}]],assump];
Union[Table[Absg2nd[Interval[{1-i/1000,1-(i+1)/1000}]]<116/100,{i,0,124}]]
Absg3rd[α_]=Total[FullSimplify[Abs[{gD30[0,0,α]/6,gD21[0,0,α]/2,gD12[0,0,α]/2, gD03[0,0,α]/6}],assump]];
Union[Table[Absg3rd[Interval[{1-i/1000,1-(i+1)/1000}]]<132/100,{i,0,124}]]

Absg4th[α_]=FullSimplify[Total[FullSimplify[Abs[{gD40[0,0,α]/24,gD31[0,0,α]/6,gD22[0,0,α]/4, gD13[0,0,α]/6,gD04[0,0,α]/24}],assump]],assump];
R1part2[α_]=13/12 Sqrt[1/((4α-1)α^3)](4/3/28+32/120/28);
R1[α_]=FullSimplify[Absg4th[α]+R1part2[α],assump];
Union[Table[R1[Interval[{1-i/1000,1-(i+1)/1000}]]<99/100,{i,0,124}]]

(*NMaxValue[{Absg2nd[α],dom},α]
NMaxValue[{Absg3rd[α],dom},α]
NMaxValue[{R1[α],dom},α]
NMaxValue[{Absg4th[α],dom},α]*)

1/(1-101/100/30-83/100/30^2)<104/100

hinvα[z_,α_]=Refine[hinv[z]/.hrules,assump];

R3hpar[w_,conjw_]=w-hinv4hpar[w,conjw];
CoeffR3[α_]=Refine[CoefficientList[R3hpar[w,conjw],{w,conjw}]/.solhinv4/.hrules,assump];
FullSimplify[Table[CoeffR3[α][[i+1,j+1]],{i,0,4},{j,0,4-i}],assump](*check*)

Abshinvα20[α_]=FullSimplify[Abs[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[3,1]]],assump];
Abshinvα11[α_]=FullSimplify[Abs[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[2,2]]],assump];
Abshinvα02[α_]=FullSimplify[Abs[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[1,3]]],assump];
hinvα2nd[α_]=Abshinvα20[α]+Abshinvα11[α]+Abshinvα02[α];
Union[Table[hinvα2nd[Interval[{1-i/1000,1-(i+1)/1000}]]<101/100,{i,0,124}]]

(*NMaxValue[{hinvα2nd[α],dom},α]*)

Refine[Re[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[4,1]]]],assump];
Refine[Im[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[4,1]]]],assump];
Abshinvα30[α_]=Simplify[Sqrt[%^2+%%^2],assump];

Refine[Re[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[3,2]]]],assump];
Refine[Im[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[3,2]]]],assump];
Abshinvα21[α_]=Simplify[Sqrt[%^2+%%^2],assump];

Refine[Re[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[2,3]]]],assump];
Refine[Im[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[2,3]]]],assump];
Abshinvα12[α_]=Simplify[Sqrt[%^2+%%^2],assump];

Refine[Re[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[1,4]]]],assump];
Refine[Im[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[1,4]]]],assump];
Abshinvα03[α_]=Simplify[Sqrt[%^2+%%^2],assump];

hinvα3rd[α_]=Abshinvα30[α]+Abshinvα21[α]+Abshinvα12[α]+Abshinvα03[α];
Union[Table[hinvα3rd[Interval[{1-i/2000,1-(i+1)/2000}]]<153/100,{i,0,249}]]

NMaxValue[{hinvα3rd[α],dom},α]

Refine[Re[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[1,5]]]],assump];
Refine[Im[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[1,5]]]],assump];
Abshinvα04[α_]=Simplify[Sqrt[%^2+%%^2],assump];

Refine[Re[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[2,4]]]],assump];
Refine[Im[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[2,4]]]],assump];
Abshinvα13[α_]=Simplify[Sqrt[%^2+%%^2],assump];

Refine[Re[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[3,3]]]],assump];
Refine[Im[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[3,3]]]],assump];
Abshinvα22[α_]=Simplify[Sqrt[%^2+%%^2],assump];

Refine[Re[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[4,2]]]],assump];
Refine[Im[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[4,2]]]],assump];
Abshinvα31[α_]=Simplify[Sqrt[%^2+%%^2],assump];

Refine[Re[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[5,1]]]],assump];
Refine[Im[ComplexExpand[CoefficientList[hinvα[z,α]/.Conjugate[z]->conjz,{z,conjz}][[5,1]]]],assump];
Abshinvα40[α_]=Simplify[Sqrt[%^2+%%^2],assump];

R3α4th[α_]=Simplify[Abshinvα40[α]+Abshinvα31[α]+Abshinvα22[α]+Abshinvα13[α]+Abshinvα04[α],assump];
Union[Table[R3α4th[Interval[{1-i/10000,1-(i+1)/10000}]]<245/100,{i,0,1249}]]

(*NMaxValue[{R3α4th[α],dom},α]*)

Absh20[α_]=FullSimplify[Abs[h20[α]],assump];
Union[Table[Absh20[Interval[{1-i/1000,1-(i+1)/1000}]]<123/100,{i,0,124}]]
Absh11[α_]=FullSimplify[Abs[h11[α]],assump];
Union[Table[Absh11[Interval[{1-i/1000,1-(i+1)/1000}]]<10/100,{i,0,124}]]
Absh02[α_]=FullSimplify[Abs[h02[α]],assump];
Union[Table[Absh02[Interval[{1-i/1000,1-(i+1)/1000}]]<60/100,{i,0,124}]]
Absh30[α_]=FullSimplify[Abs[h30[α]],assump];
Union[Table[Absh30[Interval[{1-i/1000,1-(i+1)/1000}]]<139/100,{i,0,124}]]

Refine[Re[Refine[ComplexExpand[h12[α]],assump]],assump];
Refine[Im[Refine[ComplexExpand[h12[α]],assump]],assump];
Absh12[α_]=Refine[Sqrt[%^2+%%^2],assump];
Union[Table[Absh12[Interval[{1-i/1000,1-(i+1)/1000}]]<78/100,{i,0,124}]]

Absh03[α_]=FullSimplify[Abs[h03[α]],assump];
Union[Table[Absh03[Interval[{1-i/1000,1-(i+1)/1000}]]<131/100,{i,0,124}]]

(*{NMaxValue[{Absh20[α],dom},α],NMaxValue[{Absh11[α],dom},α],NMaxValue[{Absh02[α],dom},α],NMaxValue[{Absh30[α],dom},α],NMaxValue[{Absh12[α],dom},α],NMaxValue[{Absh03[α],dom},α]}*)

hinvmax[z_]=z+((H20 (z^2))/2)+(1/6) (z^3) (3 (H20^2)+H30+3 H11 Conjugate[H02])+(1/24) (z^4) (+15 (H20^3)+10 H20 H30+30 H11 H20 Conjugate[H02]+3 H02 ((Conjugate[H02])^2)+4 H11 Conjugate[H03]+12 H11 Conjugate[H02] Conjugate[H11])+H11 z Conjugate[z]+(1/2) (z^2) (3 H11 H20+H02 Conjugate[H02]+2 H11 Conjugate[H11]) Conjugate[z]+(1/6) (z^3) (+15 H11 (H20^2)+4 H11 H30+12 (H11^2) Conjugate[H02]+3 H12 Conjugate[H02]+6 H02 H20 Conjugate[H02]+H02 Conjugate[H03]+12 H11 H20 Conjugate[H11]+6 H02 Conjugate[H02] Conjugate[H11]+6 H11 ((Conjugate[H11])^2)+3 H11 Conjugate[H12]+3 H11 Conjugate[H02] Conjugate[H20]) Conjugate[z]+(1/2) H02 ((Conjugate[z])^2)+(1/2) z (2 (H11^2)+H12+H02 H20+2 H02 Conjugate[H11]+H11 Conjugate[H20]) ((Conjugate[z])^2)+(1/4) (z^2) (+12 (H11^2) H20+3 H12 H20+3 H02 (H20^2)+H02 H30+H03 Conjugate[H02]+9 H02 H11 Conjugate[H02]+12 (H11^2) Conjugate[H11]+4 H12 Conjugate[H11]+6 H02 H20 Conjugate[H11]+6 H02 ((Conjugate[H11])^2)+2 H02 Conjugate[H12]+3 H11 H20 Conjugate[H20]+3 H02 Conjugate[H02] Conjugate[H20]+6 H11 Conjugate[H11] Conjugate[H20]) ((Conjugate[z])^2)+(1/6) (+H03+3 H02 H11+3 H02 Conjugate[H20]) ((Conjugate[z])^3)+(1/6) z (+6 (H11^3)+6 H11 H12+H03 H20+9 H02 H11 H20+3 (H02^2) Conjugate[H02]+3 H03 Conjugate[H11]+18 H02 H11 Conjugate[H11]+6 (H11^2) Conjugate[H20]+3 H12 Conjugate[H20]+3 H02 H20 Conjugate[H20]+12 H02 Conjugate[H11] Conjugate[H20]+3 H11 ((Conjugate[H20])^2)+H11 Conjugate[H30]) ((Conjugate[z])^3)+(1/24) (4 H03 H11+12 H02 (H11^2)+6 H02 H12+3 (H02^2) H20+12 (H02^2) Conjugate[H11]+6 H03 Conjugate[H20]+18 H02 H11 Conjugate[H20]+15 H02 ((Conjugate[H20])^2)+4 H02 Conjugate[H30]) ((Conjugate[z])^4);

R3Hat[w_,conjw_]=FunctionExpand[hinvmax[h[w]]]/.Conjugate[w]-> conjw;
R3Hat1var[x_]=R3Hat[w,conjw]/.conjw-> x /. w-> x;
R3HatHigherTo4th=Table[CoefficientList[R3Hat1var[x],x][[i+1]]/30^(i-4),{i,5,12}];
Total[Simplify[R3HatHigherTo4th/. H20-> 123/100 /. H11 ->  10/100 /. H02-> 60/100/. H30-> 139/100 /. H12 -> 78/100 /. H03 -> 131/100]]<202/100

(*Total[Simplify[R3HatHigherTo4th/. H20-> 123/100 /. H11 ->  10/100 /. H02-> 60/100/. H30-> 139/100 /. H12 -> 78/100 /. H03 -> 131/100]]//N*)

202/100*(104/100)^4+245/100<482/100

Estimation of R2 by calculating hinv(G(h))

hmax[w_]=w+101/100 w^2+83/100 w^3;
Gmax[z_]=z+116/100 z^2 + 132/100 z^3+ 99/100 z^4;
hinvmax[z_]=z+101/100 z^2+153/100 z^3+ 482/100 z^4;
Q[w_]=Expand[ hinvmax[Gmax[hmax[w]]]];

Rough estimation on R2

(*Total[Table[Coefficient[Q[w],w,i]/30^(i-4),{i,4,48}]]//N*)

Total[Table[Coefficient[Q[w],w,i]/30^(i-4),{i,5,48}]]<503/100
(*Total[Table[Coefficient[Q[w],w,i]/30^(i-4),{i,5,48}]]//N*)

hinvGh[w_,conjw_]=FunctionExpand[hinv[G4h[w,conjw]]];
coeffhinvGh=CoefficientList[hinvGh[w,Conjugate[w]]/.Conjugate[w]-> conjw,{w,conjw}];

hinvG4h40[α_]=FullSimplify[Refine[coeffhinvGh[[5,1]]/.hrules/.Conjugate[(Sqrt[1-4 α])]->-I Sqrt[4α-1],assump],assump];
Refine[Re[ComplexExpand[hinvG4h40[α]]],assump];
Refine[Im[ComplexExpand[hinvG4h40[α]]],assump];
abshinvG4h40[α_]=Simplify[Sqrt[%^2+%%^2],assump];

hinvG4h31[α_]=FullSimplify[Refine[coeffhinvGh[[4,2]]/.hrules/.Conjugate[(Sqrt[1-4 α])]->-I Sqrt[4α-1],assump],assump];
Refine[Re[ComplexExpand[hinvG4h31[α]]],assump];
Refine[Im[ComplexExpand[hinvG4h31[α]]],assump];
abshinvG4h31[α_]=Simplify[Sqrt[%^2+%%^2],assump];

hinvG4h22[α_]=FullSimplify[Refine[coeffhinvGh[[3,3]]/.hrules/.Conjugate[(Sqrt[1-4 α])]->-I Sqrt[4α-1],assump],assump];
Refine[Re[ComplexExpand[hinvG4h22[α]]],assump];
Refine[Im[ComplexExpand[hinvG4h22[α]]],assump];
abshinvG4h22[α_]=Simplify[Sqrt[%^2+%%^2],assump];

hinvG4h13[α_]=FullSimplify[Refine[coeffhinvGh[[2,4]]/.hrules/.Conjugate[(Sqrt[1-4 α])]->-I Sqrt[4α-1],assump],assump];
Refine[Re[ComplexExpand[hinvG4h13[α]]],assump];
Refine[Im[ComplexExpand[hinvG4h13[α]]],assump];
abshinvG4h13[α_]=Simplify[Sqrt[%^2+%%^2],assump];

hinvG4h04[α_]=FullSimplify[Refine[coeffhinvGh[[1,5]]/.hrules/.Conjugate[(Sqrt[1-4 α])]->-I Sqrt[4α-1],assump],assump];
Refine[Re[ComplexExpand[hinvG4h04[α]]],assump];
Refine[Im[ComplexExpand[hinvG4h04[α]]],assump];
abshinvG4h04[α_]=Simplify[Sqrt[%^2+%%^2],assump];

R24th[α_]=FullSimplify[abshinvG4h40[α]+abshinvG4h31[α]+abshinvG4h22[α]+abshinvG4h13[α]+abshinvG4h04[α],assump];

Union[Table[R24th[Interval[{1-i/1000,1-(i+1)/1000}]]<204/100,{i,0,124}]]
(*NMaxValue[{R24th[α],dom},α]*)

c1[α_]=FullSimplify[coeffhinvG4h[[3,2]]/.sol20 /. sol11 /. sol02/.rules/.Conjugate[(Sqrt[1-4 α])]->-I Sqrt[4α-1],assump];
a[α_]=FullSimplify[Re[ComplexExpand[Conjugate[μ[α]]/Abs[μ[α]]*c1[α]]], assump]

maxa=FullSimplify[MaxValue[{a[α],dom},α]]

(*Estimation on R4*)
Absc1[α_]=FullSimplify[Abs[c1[α]],assump];
R44th[α_]=Simplify[(Absc1[α]^2-a[α]^2)/57α,assump];
Union[Table[R44th[Interval[{1-i/1000,1-(i+1)/1000}]]<4/100,{i,0,124}]]

(*NMaxValue[{R44th[α],dom},α]*)

zminw=MinValue[{w-101/100 w^2-83/100w^3,1/30≤w≤1/4},w];
1/32<zminw

uvEst[α_]=Sqrt[(α^2+2α)/(4α-1)]
MinuvEst=FullSimplify[MinValue[{1/uvEst[α],dom},α]];
eps=MinuvEst 1/32;
1/37<eps

(*eps//N*)

TimeUsed[]-t0