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Bessel functions: monotonicity and bounds. (English) Zbl 0948.33001
The aim of this paper is the derivation of monotonicity properties of Bessel functions, leading to precise bounds which are uniform in order or argument. Monotonicity with respect to the order $\nu$ of the magnitude of general Bessel functions. ${\cal C}_\nu(x)= aJ_\nu(x)+ bY_\nu(x)$ at positive stationary points of associated functions is derived. In particular, the magnitude of ${\cal C}_\nu$ at its positive stationary points is strictly decreasing in $\nu$ for all positive $\nu$. It follows that $\sup_x|J_\nu (x)|$ strictly decreases from 1 to 0 as $\nu$ increases from 0 to $\infty$. The magnitude of $x^{1\over 2}{\cal C}_\nu$ at its positive stationary points is strictly increasing in $\nu$. It follows that $\sup_x|x^{1\over 2}J_\nu (x)|$ equals $\sqrt{2\over \pi}$ for $0\le\nu \le{1 \over 2}$ and strictly increases to $\infty$ as $\nu$ increases from ${1\over 2}$ to $\infty$. It is also shown that $\nu^{1\over 3}\sup|J_\nu(x)|$ strictly increases from 0 to $b=0.674885\dots$ as $\nu$ increases from 0 to $\infty$. Hence for all positive $\nu$ and real $x$, $|J_\nu(x)|<b\nu^{- {1\over 3}}$ where $b$ is the best possible such constant. Additionally, errors in the work by Abramowitz and Stegun by Watson are pointed out.

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
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