### A Functional Inequality

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

Received and read 22 January, 1948

Introduction

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

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