Higher monotonicity properties of normalized Bessel functions

Denote by Jν the Bessel function of the first kind of order ν and μν,k is its kth positive zero. For ν > ½, a theorem of Lorch, Muldoon and Szego states that the sequence $\{\int_{\mu_{\nu, k}}^{\mu_{\nu, k+1}}x^{-\nu}\vert J_{\nu}(x)\vert dx\}$ is decreasing, another theorem of theirs states that the sequence $\{\mu_{\nu, k}^{-2(\nu-1)}J_{\nu-1}^2(\mu_{\nu, k})\}_{k = 1}^{\infty}$ has higher monotonicity properties. In the present paper, we proved that when ν > ½ the sequence $\{\int_{\mu_{\nu, k}}^{\mu_{\nu, k+1}}x^{-\nu+1}\vert J_{\nu}(x)\vert dx\}$ has higher monotonicity properties and the properties imply those of the sequence of the local maxima of the function x-ν+1|Jν-1(x)|, x ∈ (0, ∞), i.e. the sequence $\{\mu_{\nu, k}^{-(\nu-1)}\vert J_{\nu-1}(\mu_{\nu, k})\vert \}_{k = 1}^{\infty}$ has higher monotonicity properties.