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

Abstract

For a twice differentiable function $f:\left(a,b\right)\to \mathbb{R}$ define $v\left(f\right)=f^{\prime }{f}^{\prime }-f^{\prime \prime }f.$ It is well known that the positivity of $v\left(f\right)$ implies that the function $\left|f\right|$ is strictly log-concave on each subinterval which does not contain zeros of $f.$ 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 $\nu .$

20 pages, 5 figures