Bessel Functions: Monotonicity and Bounds

Monotonicity with respect to the order v of the magnitude ofgeneral Bessel functions C_v(x) = aJ_v(x)+bY_v(x) at positive stationarypoints of associated functions is derived. In particular, themagnitude of C_v at its positive stationary points is strictlydecreasing in v for all positive v. It follows that sup_x|J_v(x)|strictly decreases from 1 to 0 as v increases from 0 to . Themagnitude of x^1/2C_v(x) at its positive stationary points isstrictly increasing in v. It follows that sup_x|x^1/2J_v(x)| equals2/ for 0 v 1/2 and strictly increases to as v increases from1/2 to . It is shown that v^1/3sup_x|J_v(x)| strictly increases from 0 tob = 0.674885... as v increases from 0 to . Hence for all positivev and real x, where b is the best possible such constant. Furthermore, forall positive v and real x, where c = 0.7857468704... is the best possible such constant. Additionally, errors in work by Abramowitz and Stegun and byWatson are pointed out.