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Inequalities for Bessel functions. (English) Zbl 0596.33011

R. B. Paris, SIAM J. Math. Anal. 15, 203-205 (1984; Zbl 0536.33006) derived upper and lower bounds for the ratio J ν (νx)/[x ν J ν (ν)] by using recurrence relations and Sonine’s integral representation. The present author uses similar techniques to extend such results to general cylinder functions C ν (x,α)=J ν (x)cosα-Y ν (x)sinα and to improve the results of Paris in the case α=0. Some examples of the author’s results run as follows:

C ν (νt,α)/[C ν (ν,α)t ν ]>exp(ν 2 (1-t 2 )/(4ν+4)),ν>0,0<t<1,0α5π/6,

and

J ν (j ν1 ' t)/[J ν (j ν1 ' )t ν ]<exp(j ν1 ' 2 (1-t 2 )/(2ν+4)),ν>0,0<t<1,

where j ν1 ' is the smallest positive zero of J ν ' (x).

Reviewer: M.E.Muldoon
MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1
26D99Inequalities involving real functions