A Unimodal Property of Purely Imaginary Zeros of Bessel and Related Functions

Abstract

We show, among other things, that, for n = 0,1, the negative of the square of a purely imaginary zero of is unimodal on (n — 2, n — 1). One of the important tools in the proof is the Mittag-Leffler partial fractions expansion of .

Bounds for the small real and purely imaginary zeros of Bessel and related functions

“…Prom our bounds it will follow that jii approaches 0 as v -» -2 + . In fact [23], j^ decreases to a minimum and then increases again (to 0) as u increases from -2 to -1. See Figure 1.…”

“…Numerical evidence based on further bounds of this kind indicates that j^i decreases from 0 to -1.60748... as z/ increases from -2 to about -1.697, and then increases to 0 as u increases to -1. (The unimodality of -j^ on (-2,-1) is proved in [23]. See Figure 1.…”

“…Then we have the lower bounds xl > 4(I/ + Q;)(Z/ + 1)3 (i/ + a){v + l)(i/ + 2) 1 / 3 (i/ + 3) 1 / 3 ({a + z/)3 + 6a 2 + 16^2 + 18ai/ + 18a + 381/ + 24) 1 /3' and (a + vf + 4a + 81/ + 8)(i/ + a)(i/ + l)(i/ + 3) (a + z/) 3 + 6a 2 4-16z/ 2 + 18ai/ + 18a + 38z/ + 24' and the upper bounds and (a + v + 2)(i/ + a)(i/ + l)(i/ + 2) Xl "^ (a + J/) 2 + 4a + 81/ + 8 4(j/ + a)(z/ + l)(z/ + 2) 1 / 2 ^i < (a 2 + z/ 2 + 4a + 81/ + 2az/ + 8) 1 /2" R. Spigler[36] showed, in our notation, that (or ■ i/ 2 <*?<(!/-a) 2 , (^>l/2).Remark. In[23, Theorem 4.1], a key part of the proof that -x\ is unimodal on (-1, -a) (where -1/2 < a < 1 and -1 < v < -a) requires the inequality 3^1+^? > 0 > -1<^< -OL.…”

On the Square of the First Zero of the Bessel Function Jv(z)

“…At the point ν = −2 the function j ν,1 is vanishing again. Concerning the local behavior of j ν,1 , R. Piessens [9] has found the following representation Recalling the inequalities [6,(5.11)], [6,(5. In [6], [7] one can find the graph of the function j 2 ν,1 in the interval (−2, 0), indicating the property that j 2 ν,1 is a convex function of ν in that interval. This property was proved for 3 ≤ ν < +∞ by J. T. Lewis and M. E. Muldoon [8].Á.…”

The real zeros of the derivatives of cylinder functions of negative order

“…Reality of zeros. It is well known ( [7], [9]) that the smallest positive zero of J ′ ν (z) approaches 0 as ν ↓ 0 and that it becomes purely imaginary when −1 < ν < 0 returning to the origin when ν ↓ −1. But it becomes real again as ν is further decreased: Suppose that h(ν, z) has a nonreal zero for some ν in the interval in question.…”