Bessel Functions: Monotonicity and Bounds

Monotonicity with respect to the order v of the magnitude of general Bessel functions Cv(x) = aJv(x)+bYv(x) at positive stationary points of associated functions is derived. In particular, the magnitude of Cv at its positive stationary points is strictly decreasing in v for all positive v. It follows that supx∣Jv(x)∣ strictly decreases from 1 to 0 as v increases from 0 to ∞. The magnitude of x1/2Cv(x) at its positive stationary points is strictly increasing in v. It follows that supx∣x1/2Jv(x)∣ equals √2/π for 0 ⩽ v ⩽ 1/2 and strictly increases to ∞ as v increases from 1/2 to ∞. It is shown that v1/3supx∣Jv(x)∣ strictly increases from 0 to b = 0.674885… as v increases from 0 to ∞. Hence for all positive v and real x, | J v ( x ) | < b v - 1 / 3 where b is the best possible such constant. Furthermore, for all positive v and real x, | J v ( x ) | ⩽ c | x | - 1 / 3 where c = 0.7857468704… is the best possible such constant. Additionally, errors in work by Abramowitz and Stegun and by Watson are pointed out.