Question:
What book(s) contain a proof that e^Pi > Pi^e? I think it might be in Problems in Analysis published by Springer-Verlag but I have not been able to check.

name: Dusty

reason: I would like to compare a published proof to an unpublished proof that I have.

There are dozens, perhaps hundreds of published proofs. New ones (which are probably rediscoveries of old ones) are published every year. If yours is brief and attractive, you should send it to a Journal like MATH MAG or THE COLLEGE MATH JOURNAL. Even if it is not new, anything nice should be brought back from time to time for the next generation.

I know that I've seen it in calculus texts as well as books that provide problems and nice solutions. Three examples were included in A CENTURY OF CALCULUS Part II (1969-1991), ed,. by Tom M. Apostol et al., on pages 445-449. (It is published by the Mathematical Association of America, 1992.) Other references are provided there. These 3 items were taken from The TWO-YEAR COLLEGE MATH JOURNAL VOL 3, no. 2, pp 13-15

(Ivan Niven, Which is larger, e^pi or pi^e?); IBID 6:2, p. 45

(Just and Schaumberger, Two more proofs of a familiar inequality); MATH. MAG. 60:3 p. 65,

(Fouad Nakhli, Proof without words: pi^e < e^pi.)

The "proof without words" shows a drawing of the curve (ln x)/x with its maximum indicated at (e, 1/e). The point (pi, (ln pi)/pi) is to the right and lower. No words are included, but presumably you are supposed to prove that the max occurs at (e, 1/e), which is a simple calculus problem, which can be found in any calculus text.