If[!TrueQ[Private`$ev],Block[{Private`$ev=True},Print["Total time: ",First@AbsoluteTiming[ClearAll["Global`*"];
NotebookEvaluate[EvaluationNotebook[],InsertResults->True]]," s"]]]

ClearAll["Global`*"];
(* functions in (A.2)*)
q1[t_]:=21539/93423+127023/185578 *t^2-1/8885055 *t^3+54169/401949 *t^4-3/44981 *t^5+202/73305 *t^6-113/80657 *t^7+293/59051 *t^8-604/151861 *t^9+127/76892 *t^10-28/94431* t^11;
q2[t_]:=18176/78783-21/15850 t+295367/428350 t^2-1415/123249 t^3+31027/204823 t^4-5162/329873 t^5+3025/287391 t^6-17/36388 t^7+2/74523 t^8+5/86563 t^9-1/120831 t^10+1/1183575 t^11;
f1[t_]:=Exp[-t]/t;
p2[t_]:=q2[t]+q1[9/10]-q2[9/10]+(q1'[9/10]-q2'[9/10])(t-9/10);
ggg[t_]:=(2 Exp[t] ExpIntegralEi[-4 t]-Exp[-t] ExpIntegralEi[-2 t])/t;
AAAAAA=(p2[5/2] ggg'[5/2]+p2'[5/2] ggg[5/2])/(f1[5/2] ggg'[5/2]-f1'[5/2] ggg[5/2]) 1/(p2[5/2])^2;
BBBBBB=-(p2[5/2] f1'[5/2]+p2'[5/2] f1[5/2])/(f1[5/2] ggg'[5/2]-f1'[5/2] ggg[5/2]) 1/(p2[5/2])^2;
Q[t_]:=If[t≤9/10,1/q1[t],If[t≤5/2,1/p2[t],((AAAAAA Exp[-t])/t)+BBBBBB ggg[t]]];
ETA[t_]:=7 10^(-5) Exp[-t]/(1+t);
(*(Subscript[Q, *]^2) in (A.3)*)
Qanew[t_]:=-32389/844*t^14+692573/1409*t^13-14647839/5182*t^12+13634891/1418*t^11-31307644/1461*t^10+22954676/703*t^9-1268792/37*t^8+34165996/1415*t^7-12679168/1223*t^6+2124876/1117*t^5+78128/353*t^4+15317/829*t^3-138192/1231*t^2+15295/813;
(* (Subscript[W, 1]) in (A.4)*)
W1[t_]:=-126751 t^22/80621266+998813 t^21/24102701-44784703 t^20/90944085+532787760 t^19/154914809-1478230681 t^18/97178472+1104148971 t^17/27114439-6386998649 t^16/165889120-4563349408 t^15/24620175+12295664789 t^14/11848354-25742069395 t^13/8929817+22711315187 t^12/4190364-21505639879 t^11/2916474+32713853029 t^10/4462850-12946708519 t^9/2496198+3403022733 t^8/1411424-6729001962 t^7/11651297-626483074 t^5/65670795+752248559 t^4/20846520-89936485 t^2/6984383+93423/21539;
(* (Subscript[W, 2]) in (A.5)*)
W2[t_]:=157*t^8/1255706-5215*t^7/1535379+25008*t^6/610441-1118777*t^5/3886821+1230788*t^4/949047-784873*t^3/202656+10232110*t^2/1348117-2084185*t/228972+43251/8233;
(* (Subscript[P, 1]), (Subscript[P, 2]) in (B.2)*)
P1[t_]:=2(W1[t]-75 10^(-6))^2;P2[t_]:=(W1[t]+75 10^(-6))(W1'[t]-42 10^(-5));
(* (Subscript[P, 3]), (Subscript[P, 4]) in (B.3)*)
P3[t_]:=2(W1[t]+75 10^(-6))^2;P4[t_]:=(W1[t]+75 10^(-6))^4;
(*The following function polynomialMin corresponds to formula (1.25) in Lemma 1.4*)
polynomialMin[p_,a_,b_,dx_]:=Module[{d,pValues,m1,m2,lam,Nsub,coeffList},If[!PolynomialQ[p,x],Return["Error: p must be a polynomial in x."];];
coeffList=CoefficientList[p,x];
d=Exponent[p,x];
Nsub=Ceiling[(b-a)/dx];
If[6 Nsub^2-d^2 (d^2-1)<=0,Return["Error: lambda must be positive."];];
pValues=ParallelTable[FromDigits[Reverse[coeffList],a+k*dx],{k,0,Nsub}];
m1=Min[pValues];
m2=Max[Abs[pValues]];
lam=(d^2 (d^2-1))/(6 Nsub^2-d^2 (d^2-1));
m1-lam*m2]

Appendix A

(A .6)

Clear[p];p=Qanew[x];
polynomialMin[p,0,8/5,10^(-4)]>0
Clear[p];p=-Simplify[Qanew'[x]/x];
polynomialMin[p,0,8/5,10^(-4)]>0

(A .8)

(*Check the first inequality in (A.8) via Corollary 1.5*)
Clear[p];p=75*10^(-5)q1[x]^2+(q1[x]^2 Qanew[x]-(1-7*10^(-5)q1[x])^2);
polynomialMin[p,0,9/10,10^(-4)] >0
Clear[p];p=75*10^(-5)q1[x]^2-( q1[x]^2 Qanew[x]-(1-7*10^(-5)q1[x])^2);
polynomialMin[p,0,9/10,10^(-4)] >0

(*Check the second inequality in (A.8) via Corollary 1.5*)
Clear[p];p=75*10^(-5)p2[x]^2+( p2[x]^2 Qanew[x]-(1-7*10^(-5)p2[x])^2);
polynomialMin[p,9/10,8/5,10^(-4)]>0
Clear[p];p=75*10^(-5)p2[x]^2-( p2[x]^2 Qanew[x]-(1-7*10^(-5)p2[x])^2);
polynomialMin[p,9/10,8/5,10^(-4)] >0

(A .9)

(*Check the first inequality in (A.9) via Corollary 1.5*)
Clear[p];p=75*10^(-5)q1[x]^2+( q1[x]^2 Qanew[x]-(1+7*10^(-5)q1[x])^2);
polynomialMin[p,0,9/10,10^(-4)] >0
Clear[p];p=75*10^(-5)q1[x]^2-( q1[x]^2 Qanew[x]-(1+7*10^(-5)q1[x])^2);
polynomialMin[p,0,9/10,10^(-4)] >0

(*Check the second inequality in (A.9) via Corollary 1.5*)
Clear[p];p=75*10^(-5)p2[x]^2+( p2[x]^2 Qanew[x]-(1+7*10^(-5)p2[x])^2);
polynomialMin[p,9/10,8/5,10^(-4)] >0
Clear[p];p=75*10^(-5)p2[x]^2-( p2[x]^2 Qanew[x]-(1+7*10^(-5)p2[x])^2);
polynomialMin[p,9/10,8/5,10^(-4)] >0

(A .10)

(*Check the first inequality in (A.10) via Corollary 1.5*)
Clear[p];p=32*10^(-7)q1[x]+(1-q1[x] W1[x]); 
polynomialMin[p,0,9/10,10^(-4)]>0
Clear[p];p=32*10^(-7)q1[x]-(1- q1[x] W1[x]); 
polynomialMin[p,0,9/10,10^(-4)]>0

(*Check the second inequality in (A .10) via Corollary 1.5*)
Clear[p];p=32*10^(-7)p2[x]+(1-p2[x] W1[x]); 
polynomialMin[p,9/10,5/2,10^(-4)]>0
Clear[p];p=32*10^(-7)p2[x]-(1-p2[x] W1[x]); 
polynomialMin[p,9/10,5/2,10^(-4)]>0

(A .11)

(*Check the first inequality in (A.11) via Corollary 1.5*)
Clear[p];p=57*10^(-6)q1[x]^2+(q1'[x]+q1[x]^2 W1'[x]);
polynomialMin[p,0,9/10,10^(-5)]>0
Clear[p];p=57*10^(-6)q1[x]^2-(q1'[x]+q1[x]^2 W1'[x]);
polynomialMin[p,0,9/10,10^(-5)]>0

(*Check the second inequality in (A.11) via Corollary 1.5*)
Clear[p];p=57*10^(-6)p2[x]^2+(p2'[x]+p2[x]^2 W1'[x]);
polynomialMin[p,9/10,5/2,10^(-4)]>0
Clear[p];p=57*10^(-6)p2[x]^2-(p2'[x]+p2[x]^2 W1'[x]);
polynomialMin[p,9/10,5/2,10^(-4)]>0

(A .13)

(*Check the first inequality in (A.13) via Corollary 1.5*)
Clear[p];p=18*10^(-7)*p2[x]+(1-p2[x]W1[x]);
polynomialMin[p,18/10,5/2,10^(-4)]>0
Clear[p];p=18*10^(-7)*p2[x]-(1-p2[x]W1[x]);
polynomialMin[p,18/10,5/2,10^(-4)]>0

(*Check the second inequality in (A.13) via Corollary 1.5*)
Clear[p];p=29*10^(-6)*p2[x]^2+(p2'[x]+p2[x]^2 W1'[x]);
polynomialMin[p,18/10,5/2,10^(-4)]>0
Clear[p];p=29*10^(-6)*p2[x]^2-(p2'[x]+p2[x]^2 W1'[x]);
polynomialMin[p,18/10,5/2,10^(-4)]>0

(A .14)

Clear[p];p=48*10^(-7)*x+(x W2''[x]+2W2'[x]-x W2[x]+x W2[x]^3);
polynomialMin[p,5/2,7/2,10^(-4)]>0
Clear[p];p=48*10^(-7)*x-(x W2''[x]+2W2'[x]-x W2[x]+x W2[x]^3);
polynomialMin[p,5/2,7/2,10^(-4)]>0

(A .15)

polynomialMin[W2[x],5/2,7/2,10^(-4)]>0
polynomialMin[9/100-W2[x],5/2,7/2,10^(-4)]>0

(A.18)

((48 (10^(-7)))/( (1-2(( 243)/10000)-(81/10000))))<(51/(10^7))

(max_((5/2)≤t≤(7/2)))|η(t)|<(5/(10^6))

((3 (9/100) ((51/(10^7)))^2 + ((51/(10^7)))^3 + 48 (10^(-7)))/(1-3  ((((243/10000)))^2)))<(5/(10^6))

(A.19)

((48 (10^(-7)))/(1-(243/10000)-(81/10000)))<(5/(10^6))

(A.20)

(83/(10^7))+(109/147) ((5/(10^6))+(48/(10^7)))≤(156/(10^7))

footnote 38

((((7/2)))^3)(W2'[(7/2)]-(156/(10^7)))-((((7/2)))^2)(W2[(7/2)]+(5/(10^6)))>-2

(A.21)

{((((17/10)))^3)(W1'[(17/10)]-(42/(10^5)))-((((17/10)))^2)(W1[(17/10)]+(75/(10^6)))>-(31/10),((((17/10)))^2)(1-(((W1[(17/10)]+(75/(10^6))))^2))-2>0}
W1[(17/10)]+(75/(10^6))<(((93/10)-(Sqrt[(((-(93/10)))^2)-4 (867/100)2]))/(2(867/100)))

(A.22)

2-6(W2[(7/2)]+156 10^(-7))>(18525/10000)

(A .23)

Clear[p];p=2+x^3(W1[x]+75*10^(-6))(W1'[x]-42*10^(-5));
polynomialMin[p,0,17/10,10^(-4)]>0

Appendix B

Case 1≤τ≤1.2

(B.4)

Clear[A];A=(3/(10^4)); 1+(10/13)(-(12872/10000))>(98/10000)
((31/100)-2(W2[(7/2)]+5 10^(-6))^2-(1/(2A ((26/10))^2)) (W2[(7/2)]+5 10^(-6))^4)-(A/4)(W2[(7/2)]+5 10^(-6))^2>0

(B.5)

Clear[A];A=(3/(10^4));(31/100)-2(W1[(5/2)]+75 10^(-6))^2-(1/(2A ((((26/10)) )^2)))  (W1[(5/2)]+75 10^(-6))^4>(27/100)

(B.6)

Clear[af,p];
af=1/2441Sum[1/k!(-13/5(x-3))^k,{k,0,3}] ;
p=W2[x]- 5 10^(-6)+2(10/26-630/26 af)(W2'[x]-156 10^(-7))-2 10^(-7);
polynomialMin[p,5/2,7/2,10^(-3)]>0

(1/2)AQ((((8/5)))^4)<(3/(10^6)) before (B.9)

Clear[A];A=(3/(10^4));(1/2)A (W1[(8/5)]+75 10^(-6))^4<(3/(10^6))

(B.9)

Clear[a1f,p,A,pnew];
a1f= 73/15070Sum[1/k!(-13/5(x-41/20))^k,{k,0,5}];
p=P1[x]+ (10/26-630/26a1f)4 P2[x];
A=3 10^(-4);
pnew=31/100-P3[x]-1/(2A)(10/26-630/26 a1f)^2P4[x];
{polynomialMin[p,8/5,5/2, 15/10^6]>5/2 10^(-4),polynomialMin[pnew,8/5,5/2,2/10^5]>2 10^(-2)}

Case 1.2 ≤τ ≤ 2.1

(B.10)

Clear[A];A=(2/(10^3));
{(1/2)A (W1[(5/2)]+75 10^(-6))^4<(1/(10^4)),
(2/10)+(2-(40/14) (380/271))(W1[(5/2)]+75 10^(-6))^2>(1/10) ,
 (171/100)-2(W1[(5/2)]+75 10^(-6))^2-(1/(2A)) ((10/14))^2 (W1[(5/2)]+75 10^(-6))^4>1 }

(1/2)AQ((((11/10)))^4)<(1/(10^3)) before (B.11)

Clear[A];A=(2/(10^3));(1/2)A (W1[(11/10)]+75 10^(-6))^4<(1/(10^3))

(B.11)

Clear[p,pnew,a2f,x,A];a2f=8/100 Sum[1/k! (-7/5(x-9/5))^k ,{k,0,3}]; A=2*10^(-3);
p= 1/5+P1[x]+ (10/14-461/140a2f)4 P2[x]; 
pnew=171/100-P3[x]-1/(2A)(10/14-461/140 a2f)^2P4[x];
{polynomialMin[p,11/10,5/2,2/10^4]>1/10,polynomialMin[pnew,11/10,5/2,1/10^4]>2/100}

Case 2.1≤τ ≤ 5.4

(B.12)

Clear[A];A= (1/100);
{(1/2)A (W1[(5/2)]+75 10^(-6))^4<(1/(10^4)), 
(11/10)+(2-4 (380/271) (5/2))(W1[(5/2)]+75 10^(-6))^2>1 ,
(31/10)-2(W1[(5/2)]+75 10^(-6))^2-1/(2A) (5/2)^2 (W1[(5/2)]+75 10^(-6))^4>1 }

(1/2)AQ((((91/100)))^4)<(9/(10^3)) and  (1/2)((AQ(T))^4)<(12/(10^3)) before (B.14)

Clear[A,T];A=(1/100);T=(856/1000);{(1/2)A (W1[(91/100)]+75 10^(-6))^4<(9/(10^3)), (1/2)A (W1[T]+75 10^(-6))^4<(12/(10^3))}

(B.14) and (B.15)

Clear[p,pnew];p=11/10+P1[x]+4(x-845/1000)P2[x];A=10^(-2);
pnew=31/10-P3[x]-1/(2A)(x-845/1000)^2 P4[x];
 {polynomialMin[p,91/100,5/2,15/10^5]>7/10,polynomialMin[pnew,91/100,5/2,8/10^5]>16/100}
{polynomialMin[p,856/1000,91/100,2/10^6]>3,polynomialMin[pnew,856/1000,91/100,9/10^7]>41*10^(-4)}

Case 5.4 ≤τ ≤ 12.4

({{ ,  (1/2)AQ((((5/2)))^4)<(1/(10^4))
(Subscript[f, 1]) >(44/10)+2((Q((5/2)))^2)+4·(-(380/271))·(5/2)·((Q((5/2)))^2)>1,    ∀t≥(5/2); 
(Subscript[f, 2])>(64/10)-2((Q((5/2)))^2)-(1/(2A))·((((5/2)))^2)·((Q((5/2)))^4)>1,    ∀t≥(5/2). ,  
 }})

Clear[A];A= (2/100);
{(1/2)A (W1[(5/2)]+75 10^(-6))^4<(1/(10^4)), 
(44/10)+(2-4 (380/271) (5/2))(W1[(5/2)]+75 10^(-6))^2>1 ,
(64/10)-2(W1[(5/2)]+75 10^(-6))^2-1/(2A) (5/2)^2 (W1[(5/2)]+75 10^(-6))^4>1 }

(1/2)AQ((((68/100)))^4)<(1/10) and (1/2)((AQ(T))^4)<(12/100) before (B.16)

Clear[A,T];A=(2/100);T=(2/3);{(1/2)A (W1[(68/100)]+75 10^(-6))^4<(1/10), (1/2)A (W1[T]+75 10^(-6))^4<(12/100)}

(B.16) and (B.17)

Clear[p,pnew,A];A=2*10^(-2);p=44/10+P1[x]+4(x-655/1000)P2[x];
pnew=64/10-P3[x]-1/(2A)(x-655/1000)^2P4[x];
{polynomialMin[p,68/100,5/2,52/10^5]>3,polynomialMin[pnew,68/100,5/2,8/10^5]>8/100}
{polynomialMin[p,2/3,68/100,1/(9 10^5)]>98/10,polynomialMin[pnew,2/3,68/100,1/(12 10^5)]>14*10^(-3)}

Case 12.4≤τ ≤ 19.2

({{ ,  (1/2)AQ((((5/2)))^4)<(1/(10^4))
(Subscript[f, 1]) >(114/10)+2((Q((5/2)))^2)+4·(-(380/271))·(5/2)·((Q((5/2)))^2)>1,    ∀t≥(5/2); 
(Subscript[f, 2])>(134/10)-2((Q((5/2)))^2)-(1/(2A))·((((5/2)))^2)·((Q((5/2)))^4)>1,    ∀t≥(5/2). ,  
 }})

Clear[A];A= (32/1000);
{(1/2)A (W1[(5/2)]+75 10^(-6))^4<(1/(10^4)), 
(114/10)+(2-4 (380/271) (5/2))(W1[(5/2)]+75 10^(-6))^2>1 ,
(134/10)-2(W1[(5/2)]+75 10^(-6))^2-1/(2A) (5/2)^2 (W1[(5/2)]+75 10^(-6))^4>1 }

(1/2)AQ((((68/100)))^4)<(2/10) and  (1/2)((AQ(T))^4)<(68/100) before (B.18)

Clear[A,T];A=(32/1000);T=(475/1000);{(1/2)A (W1[(68/100)]+75 10^(-6))^4<(2/10), (1/2)A (W1[T]+75 10^(-6))^4<(68/100)}

(B.18) and (B.19)

Clear[p,pnew,A];A=32/10^3;p=114/10+P1[x]+4(x-46/100)P2[x];
pnew=134/10-P3[x]-1/(2A)(x-46/100)^2P4[x];
{polynomialMin[p,68/100,5/2,2/10^4]>95/10,polynomialMin[pnew,68/100,5/2,1/10^4]>5/100}
{polynomialMin[p,475/1000,68/100,1/10^5]>123/10,polynomialMin[pnew,475/1000,68/100,1/10^5]>59/1000}