Let $\gamma>0$ be a real number and $\Phi(r)=\frac{1}{r}-\frac{\pi}{4}\left(H_0(r/2)-Y_0(r/2)\right)$ defined on $[0,\infty)$, where $H_0$ is the Struve function of order zero and $Y_0$ is the Bessel function of second kind of order zero.
My goal is to find $\gamma_c$ such that for all $\gamma\geq\gamma_c$ the function $\Phi(r)-\frac{\gamma}{r^2}\leq0$ for all $r\in[0,\infty)$. Plots suggest that the critical $\gamma_c$ is about $0.5$. The root in this regime is about $r_0\approx 0.5$.
1/r - (StruveH[0, r/2] - BesselK[0, r/2]) Pi/4
, or something else?Y0
typically is used to represent a Bessel function, not a modified Bessel function. $\endgroup$