gpd2Mdell2mpl

# FILE d2Mdell2mpl.txt

# deriving equation eq:d2Mdell2 for

# the second derivative of M w.r.t. ell

phi2 := z-> arctan(1/sqrt(z))/sqrt(z);

factor(diff(phi2(z), z)+phi2(z)/(2*z));

# ell in LaTeX is ell in code below

dMdellsol:= solve(diff(sqrt(M(ell))*tan(exp(ell)*sqrt(M(ell))), ell) = 0, diff(M(ell), ell));

dMdell := simplify(subs(tan(exp(ell)*sqrt(M(ell))) = 1/sqrt(M(ell)), dMdellsol));

d2Mdell2 := simplify(subs(diff(M(ell), ell) = dMdell, diff(dMdell, ell)));

factor(-d2Mdell2/dMdell);

# Remark

# ell = log(phi2(M))

# phi2: = z -> arctan(1/sqrt(z))/sqrt(z)

# Remark

# log(M(ell)) is concave, so M(ell) is not CM (Completely Monotone).

# There is a chance that mu2(c) is CM (in c)

dlogMdell := dMdell/M(ell);

factor(simplify(subs(diff(M(ell), ell) = dMdell, factor(diff(dlogMdell, ell)))));

# the sign of which is that of exp(ell)*(M-1)-1

# use 1/M <= phi2(M)=exp(l)< 1/(M+1)