Determine parameter from which on there is no more root for a given function

Question

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 $r_0\approx 0.5$.

Answer 1

Defining

f = 1/r - (StruveH[0, r/2] - BesselY[0, r/2]) Pi/4

the maximum value that f r^2 assumes is

NMaximize[{f r^2, r > 0}, r]
(* {0.498508, {r -> 2.58341}} *)

Thus, the critical value of γ is 0.498508. The following illustrates this maximum.

Plot[r^2 f, {r, 0, 10}, AxesLabel -> {r, "f"}]