A mathematical proof would be best, but Mathematica can provide some evidence against the statement that
f[a_, b_] := ((a - b)/(b*Log[b])) - Log[Log[a]] + Log[Log[b]]
is negative for any integer a, b
.
A straightforward way would be to do
NMinimize[{f[a, b], a > 1, b > 1}, {a, b}, Integers]
{0., {a -> 4, b -> 4}}
Indeed, f[a, b]==0
for a == b
. Dropping the condition Integers
,
NMinimize[{f[a, b], a > 1, b > 1}, {a, b}]
{-1.11022*10^-16, {a -> 3.71536, b -> 3.71536}}
gives a numerical zero.
A ContourPlot
:
ContourPlot[f[a, b], {a, 1, 10}, {b, 1, 10}, PlotLegends -> Automatic]

hints that indeed there are no truly negative values of f[a, b]
for a, b >= 1
.
A contour highlighting f[a, b]==0
:
ContourPlot[f[a, b], {a, 1, 10}, {b, 1, 10}, Contours -> {0}]

confirms that 0
is the smallest value attained by f[a, b]
for positive integer a, b
.
FindInstance[((a - b)/(b*Log[b])) - Log[Log[a]] + Log[Log[b]] < 0 && a > 0 && b > b, {a, b}, Integers]
but Mathematica does not give me any instance. Did you take a look at 3D plots of the function? $\endgroup$ – Mauricio Fernández Nov 17 '16 at 14:22