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An upper bound for the zeros of the derivative of Bessel functions. (English) Zbl 0880.33003

The aim of this paper is to establish a new upper bound for the positive zeros j νk ' contained in the following theorem: Let j νk ' denote the k-th positive zero of the derivative of Bessel functions (d/dx)J ν (x)=J ν ' (x) for ν>0, and k=1,2,. Then j νk ' <F k (ν), where

F k (ν)=ν+a k (ν+A k 3 /a k 3 ) 1/3 +3 10a k 2 (ν+A k 3 /a k 3 ) -1/3 ,

and

A k =2 3a k 2a k ,a k =2 -1/3 x k '

and x k ' is the k-th positive zero of the derivative A i ' (x) of the Airy function. The bound is sharp for large values of ν and improves known results. A similar inequality holds for the k-th positive zero y νk ' of Y ν ' (x), too, where Y ν (x) denotes the Bessel function of the second kind.

MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1
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