Inequalities and monotonicity results for zeros of modified Bessel functions of purely imaginary order

Let ${x_k}\left ( v \right )$ and ${x’_k}\left ( v \right )$ denote the $k$ th positive zeros, in decreasing order, of the modified Bessel function ${K_{iv}}\left ( x \right )$ of purely imaginary order and of its derivative ${K’_{iv}}\left ( x \right ) = {\textstyle {d \over {dx}}}{K_{iv}}\left ( x \right )$, respectively. We show that for $k = 1,2,...$ and $0 < v < \infty ,{x_k}\left ( v \right ) < {x’_k}\left ( v \right ) < v$ and ${x_k}\left ( v \right )/{x_{k + 1}}\left ( v \right )$ and ${x_k}\left ( v \right ) - {x_{k + 1}}\left ( v \right )$ decrease as $k$ increases. Some related results are mentioned for the zeros of ${K’_{iv}}\left ( x \right )$ and the chain of inequalities $\left | {{K_{iv}}\left ( {{{x’}_1}\left ( v \right )} \right )} \right | > \left | {{K_{iv}}\left ( {{{x’}_2}\left ( v \right )} \right )} \right | > \cdot \cdot \cdot > \left | {{K_{iv}}\left ( {{{x’}_n}\left ( v \right )} \right )} \right | > \left | {{K_{iv}}\left ( {{{x’}_{n + 1}}\left ( v \right )} \right )} \right | > \cdot \cdot \cdot$ is established.