For which real k do we have cosh x ≤ exp(k x^{2}) for all real x?

**Solution**

Answer: k ≥ 1/2.

cosh x = 1 + x^{2}/2! + x^{4}/4! + ... + x^{2n}/(2n)! + ... and exp(x^{2}/2) = 1 + x^{2}/2 + x^{4}/8 + ... + x^{2n}/(2^{n}n!). But 2^{n}n! = 2n.(2n - 2).(2n - 4) ... 2 < (2n)! for n > 1, so cosh x < e^{x2/2} for all x. Hence the inequality holds for k ≥ 1/2.

cosh x = 1 + x^{2}/2 + o(x), exp(k x^{2}) = 1 + kx^{2} + o(x). So if k < 1/2, then cosh x > exp(k x^{2}) for sufficiently small x. Thus the inequality does not hold for k < 1/2.

© John Scholes

jscholes@kalva.demon.co.uk

10 Dec 1999

Last corrected/updated 21 Nov 02