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) ≤ sin(x) in the interval 0 ≤ x ≤ π/2. He defines
0 ≤ x ≤ π/2
|| f(n)(x) |,|
and proves among others the following theorem, where f(x) is real and 0 ≤ x ≤ π/2.
THEOREM 1. If (i) f(x) and all its derivatives exist and are continuous, (ii) f(0) ≤ 0, (–1)(n-1)/2 f(n)(0) ≤ 1 for all odd n, (iii) for some δ > 0 there is a function λ(x) such that, for π/2–δ < x < π/2,
0 < λ(x) < 1, f(x) ≤ 1, (–1)n/2 f(n)(x) ≤
(π λ(x))n --------- (2x)n
for all even n, and (iv)
log an ------- n ≤ 0,
then f(x) ≤ 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.