I stumbled upon a theorem today that I feel like I’ve needed in the past, though I can’t remember any particular applications. I’m writing it up here as a note to my future self should the need reappear.

The theorem gives sufficient conditions to conclude

*f*(*g*(*x*)) ≤ *g*(*f*(*x*))

and uses this to prove, for example, that

arcsin( sinh(*x*/2) ) ≤ sinh( arcsin(*x*)/2 )

on the interval [0, 1].

If you think of any applications, please leave a comment.

Here’s the theorem, found in [1].

Let *f* be continuous with domain 0 ≤ *x* < 1 or 0 ≤ *x* ≤ 1, *f*(0) = 0, *f*(1) > 1 (including the possibility that *f*(1) = +∞); let *g* be continuous with domain the range of *f*, and *g*(1) ≤ 1. Let *f*(*x*)/*x* and *g*(*x*)/*x* be strictly increasing on their domains. Finally let *f*(*x*) ≠ *x* for 0 < *x* < 1. Then *f*(*g*(*x*)) ≤ *g*(*f*(*x*)) for 0 < *x* < 1.

[1] Ralph P. Boas. Inequalities for a Collection. Mathematics Magazine, January 1979, Vol. 52, No. 1, pp. 28–31