# FILE muHigherDerivsmpl.txt at google sites, but rmumpl.txt elsewhere
eqmu := mu(c)*tan(c*mu(c))-1/beta;
deqmu := solve(diff(eqmu, c) = 0, diff(mu(c), c));
removeTan := u -> simplify(subs(tan(c*mu(c)) = 1/(beta*mu(c)), u));
d1mu := removeTan(deqmu); lprint(%);
d1mu2 := -2*mu2(c)*(1+beta^2*mu2(c))/(beta+c*(1+beta^2*mu2(c))); lprint(%);
removed1 := u -> simplify(subs(diff(mu(c), c) = -mu(c)*(beta^2*mu(c)^2+1)/(beta+c*mu(c)^2*beta^2+c), u));
removed1mu2 := u -> simplify(subs(diff(mu2(c), c) = -2*mu2(c)*(1+beta^2*mu2(c))/(beta+c*(1+beta^2*mu2(c))), u));
#
d2mu2 := removed1mu2(diff(d1mu2, c));
d3mu2 := factor(removed1mu2(diff(d2mu2, c)));
d4mu2 := factor(removed1mu2(diff(d3mu2, c)));
d5mu2 := factor(removed1mu2(diff(d4mu2, c)));
# visually all consistent with mu2 possibly completely monotone
#
# as before, mu2 is better than log-convex
factor((mu2(c)*d2mu2/d1mu2^2-3/2)/d1mu2); # neat expression, obviously negative
#
# Now mu as opposed to mu2 immediately above
d2mu := removed1(diff(d1mu, c));
d3mu := factor(removed1(diff(d2mu, c)));
d4mu := factor(removed1(diff(d3mu, c)));
d5mu := factor(removed1(diff(d4mu, c)));
# visually all consistent with mu possibly completely monotone
#
factor((mu(c)*d2mu-2*d1mu^2)/d1mu);
# as before, negaative, so mu is better than log-convex
#
factor(d1mu*d3mu/d2mu^2-3/2); # neat expression, obviously positive
factor(d2mu*d4mu/d3mu^2-5/4); # neat expression, obviously positive
factor(d3mu*d5mu/d4mu^2-9/8) # longish, but positive
with(LinearAlgebra);
# Hoped to find nice expressions for Hankel determinants
L := <mu(c), -d1mu, d2mu, -d3mu, d4mu, k, l, m, n>;
H3 := HankelMatrix(L, 3);
factor(Determinant(H3)/d1mu^3); # obviouly negative