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.

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