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He remarks [21 after deriving equation (2) that "The absence of the factor -c/j -x2 from the denominator is remarkable." This paper obtains an inequality for J,(vx) where the factor (1 -x2)1/4 does not appear. The region where it is a better upper bound than (1) is determined. The notation used is exactly that of Watson [21 and most of the derivation is exactly the same as used by Watson to derive (1).

An inequality involving Bessel functions of argument nearly equal to their order