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L. Lorch and P. Szeg6 have considered the sign-regularity of the higher differences (with respect to the rank k ) of the sequence {cvk} of positive zeros of the Bessel function ? (x) . Our main purpose here is to extend one of their main results to the higher derivatives with respect to K when cvk is appropriately defined as a function of a continuous variable K rather than the discrete variable k, and the difference operator is replaced by a derivative operator. We also present some inequalities arising from these and other results.

Higher monotonicity properties and inequalities for zeros of Bessel functions