A family of logarithmically concave functions defined by an integral over the modified Bessel function of order \(1\) (Q1910824)

This paper deals with questions regarding logarithmic concavity of the function \[ c(\tau, z):= \int^\beta_0 zI_1 (\cos (s) z)ds \] (\(\beta= \text{arcsin} (\tau))\), where \(I_1\) stands for the modified Bessel function of order one. It is shown that for fixed \(\tau\in (0,1)\) the function \(c(\tau, z)\) is strictly logarithmic-concave on \(\mathbb{R}^*= \mathbb{R} \setminus \{0\}\) (see Theorem 7). Also the logarithmic-concavity (on \(\mathbb{R}^*\)) of \(zI_1 (z)\) is established.