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
I called the original post “Power trip” and this one “Power trip (part 2)”, and there haven’t been any powers here. Sorry. Here:
That form is… neither obvious nor interesting, though. Unless in which case the left part is
which is where we came in.