That proof that for any positive
relies on
, so doesn’t generalize to anything nearly as simple relating
to
. But let’s see what we can do.
Here’s an inequality which, if you’re like me, looks pretty mysterious:
for
and
positive real numbers.
I mean, the difference between and
doesn’t “know” anything about
or
, right? And neither does their ratio. But the difference over the log of the ratio is always between
and
? Weird!
(Okay, maybe you’re not so easily impressed, since if you know and
you can get
and
. Fine. I think it’s mysterious anyway.)
Well, let’s consider a function with derivative
monotonically decreasing (we could do increasing too, similarly). Let
and
be in the range where
and
exist and
. Then
But that integral is the area under the curve from to
and so
.
Then and
or
.
But that makes sense given the definition of the derivative and the monotonic property of ; with a little thought you could’ve written this down directly. As obvious as it is, it takes a decidedly less obvious form if you specify
,
, which gives
or the formerly mysterious
QED.
I called the original post “Power trip” and this one “Power trip (part 2)”, and there haven’t been any powers here. Sorry. Here:
so
.
That form is… neither obvious nor interesting, though. Unless in which case the left part is
or
which is where we came in.