###### Conjectured value of $g_r(\rho)$ ######################h:=proc(r,t,rho) factor((t-1)!/((t-r+1)!*(t*(t+1))^(r-1))*(t-(r-1)*sqrt(t*(t-rho*(t+1))))*(t+sqrt(t*(t-rho*(t+1))))^(r-1)); end proc:simplify(h(3,t,rho));###### Derivatives of $h_t(x)$ at the end points #############simplify(subs(x=1-1/t,diff(h(3,t,x),x))); factor(limit(diff(h(3,t,x),x),x=1-1/(t+1))); ###### Via (3.6), Conjecture 1 for $r=4$ implies the same conjecture for $r=3$ #########simplify(h(3,t,a)-(a*(2*a-1)*diff(h(3,t,a),a)+h(4,t,a))/(diff(h(3,t,a),a)+3*a-2));###### Verifying (3.10) ########################################simplify(2/3*a*diff(h(3,t,a),a)-h(3,t,a));###### Verifying the lower bound in (3.14) #####################simplify((2/3*a*diff(h(3,t,a),a)-h(3,t,a)) / (2/3*diff(h(3,t,a),a))^2);theta:= z^3*h(3,s,(z-mu)/z^2):xi:=sqrt(s*(s*z^2-(s+1)*(z-mu))):###### Verifying proof of Claim 3.3 ###########################simplify(diff(diff(theta,z),z) + 3*(4*mu*s-s-1)^2*(s-1)/(2*xi*(2*xi-2*z*s+s+1)^2)) assuming z>=0;###### Verifying proof of (3.20) ################################LHS_of_eq3_22:=simplify(evala(subs(s=t-1,z=2*mu,diff(theta,z)))) assuming mu>=0;simplify(subs(mu=t/(4*(t-1)), LHS_of_eq3_22 - 3/2*(1-2*mu) ));simplify(subs(mu=(t-1)/(4*(t-2)), LHS_of_eq3_22 - 3/2*(1-2*mu) ));simplify(subs(mu=delta+t/(4*(t-1)),t=u+1,diff(diff(LHS_of_eq3_22,mu),mu))) assuming u>=0;###### Verifying proof of (3.21) ################################LHS_of_eq3_23:=diff(theta,z):simplify(LHS_of_eq3_23-3*(s-1)/(s*(s+1)^2)*(z*s+xi)*(s+1-z*s-xi)) assuming z>=0;simplify(subs(mu=(s+1)/(4*s), LHS_of_eq3_23-3/2*(1-2*mu))) assuming z>=0 and 2*z*s<=s+1;simplify(subs(mu=(s+1)/(4*s),xi)) assuming 2*z*s<=s+1;solve(diff((z*s+_xi)*(s+1-z*s-_xi),_xi),_xi);mu:=(t-1)/(4*(t-2))-sigma^2/(4*(t-1)*(t-2)):####### Inverting the function $h_t'(a)$ #######################a:=simplify(solve(diff(h(3,t,a),a)=3/2*A,a));b:=a*A-mu*A^2:####### Checking the value of $\eta_{t-1}$ ##############################simplify(subs(z=(t-1-sigma)/(2*(t-2)), (z-mu)/z^2));eta(t-1):=(t-1-sigma)/(2*(t-2)):####### Computing the polynomial $W(t,A,\sigma)$ #######################LHS_of_eq3_26:=b*(3/2*A+3*a-2)-3/2*a*(2*a-1)*A:RHS_of_eq3_26:=A^3*(3/2*(1-2*mu)*(a/A-eta(t-1))+eta(t-1)^3*(t-2)*(t-3)/(t-1)^2):W:= factor((LHS_of_eq3_26-RHS_of_eq3_26) *8*(t-1)^2*(t-2)^2 / (sigma-1) / A^2);####### Finishing the proof ############################################left_end:=factor(subs(A=2*(t-1)/(t+1),W)*(t+1)/2/(t-1));right_end:=factor(subs(A=2*(t-1)/t,W)/2/(t-1));subs(sigma=0,left_end); factor(subs(sigma=1,left_end)); subs(sigma=0,right_end); subs(sigma=1,right_end);