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Higher monotonicity properties of certain Sturm-Liouville functions. V†

Published online by Cambridge University Press:  14 February 2012

M. E. Muldoon
M. E. Muldoon
Affiliation:
Department of Mathematics, York University, Downsview, Ontario, Canada
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Synopsis

The principal concern here is with conditions on f or on special solutions of the equation

which ensure that the higher differences of the zeros and related quantities of solutions of (1) are regular in sign. In particular, by choosing f(x)= 2v−2x1/v−2, it is shown that if ⅓ ≦|v|<½, then

where cvk denotes the kth positive zero of a Bessel function of order v and Δµk = Δk+1 − µk. Lorch and Szego [15] conjectured that (2) should hold for the larger range | v | < ½ but the methods used here do not apply to the range | v <| ⅓.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 77 , Issue 1-2 , 1977 , pp. 23 - 37
Copyright
Copyright © Royal Society of Edinburgh 1977

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