On the log-concavity of the composite Bessel function $x^{\alpha }{J}_{\nu }\left(\beta x^{\gamma }\right)$

For a twice differentiable function $f:(a,b)\to \mathbb{R}$ define $v\left(f\right)=f^{\mathrm{\prime }}{f}^{\mathrm{\prime }}-f^{\mathrm{\prime }\mathrm{\prime }}f\mathrm{.}$ It is well known that the positivity of $v\left(f\right)$ implies that the function $\mid f\mid$ is strictly log-concave on each subinterval which does not contain zeros of $f\mathrm{.}$ In this paper we provide criteria for the positivity of $v\left(F\right)$ for the composite Bessel function $F\left(x\right)=J_{\alpha ,\beta ,\gamma ,\nu }\left(x\right):=x^{\alpha }{J}_{\nu }\left(\beta x^{\gamma }\right)$ for positive numbers $\beta$ and $\gamma$ and real numbers $\alpha$ and $\nu \mathrm{.}$