Agnes Scott College

Sheila Scott Macintyre

A Functional Inequality
Journal of the London Mathematical Society, Vol. 23 (1948), 202-209

Received and read 22 January, 1948


In a recent paper Wright discusses sufficient restrictions on the real function \(f(x)\) and its first N derivatives to ensure that \(f(x) \le \sin(x)\) in the interval \(0 \le x \le \pi/2\). He defines \[a_n = \max_{0 \le x \le \pi/2} |\,f^{(n)}(x)\,|,\] and proves among others the following theorem, where \(f(x)\) is real and \(0 \le x \le \pi/2\).

THEOREM 1. If (i) \(f(x)\) and all its derivatives exist and are continuous, (ii) \(f(0) \le 0, (-1)^{(n-1)/2}f^{(n)}(0) \le 1\) for all odd n, (iii) for some δ > 0 there is a function \(\lambda(x)\) such that, for \(\pi/2-\delta \lt x \lt \pi/2\), \[0 \lt \lambda(x) \lt 1, \quad f(x) \le 1, \quad (-1)^{n/2}f^{(n)}(x) \le \frac{(\pi \lambda(x))^n}{(2x)^n}\] for all even n, and (iv) \[\lim_{n \rightarrow \infty} \frac{\log a_n}{n} \le 0,\] then \(f(x) \le \sin(x)\).

I shall prove a new theorem (Theorem III) of this type by means of a two-point expansion and also prove a theorem (Theorem II) in which some of the inequalities in the hypotheses are omitted and others reversed.

Full article available online at the Journal of the London Mathematical Society (subscription required).