Consider a function from the Bessel family, for concreteness say $f(x) := J_0(x)$, depicted in blue below (the question can be asked for any order of the first or second kind):

I'm interested in the orange tangent curve to the function. That is, an analytic, decreasing, convex (possibly even completely monotonic) function $g(x)$ that always upper bounds $f(x)$ and meets the function (with a matching derivative) once for each oscillation (the intersection points converging to the local maxima of $f$ as $x$ grows).

We know from the asymptotics of the Bessel function that $g(x)$ asymptotically behaves like $\sqrt{2/(\pi x)}$. But is there an explicit exact expression known in the literature, in any form (in terms of special functions, as a series expansion, integral form, etc)?

In general, what are the known techniques to approach questions of this kind (deriving tangent curves to oscillating functions, or even proving that they exist)?