My question is tricky to explain so please bear with me...
I have the following:
$\zeta \leq f(x,a,b,c) = ax^{4/3} + be^{-cx/2}$,
where a, b, c, x are all > 0. I would like to minimize the RHS with respect to x to produce something like:
$\zeta \leq g(a,b,c)$
However, I would like this g to be "neat" (to produce a nice physics result!), and as far as I can tell minimising f with respect to x gives a complicated solution involving the product log function. So, my idea is to first find a polynomial upper bound of f:
$f(x,a,b,c) \leq h(x,a,b,c)$
where h is a polynomial. Then I can minimise h to find a "neat" solution to finally get:
$\zeta \leq k(a,b,c) = \min_x [h(x,a,b,c)].$
I know that I can upper bound the exponential, using the result in this post: upper bound of exponential function. But to have a reasonably tight upper bound I would need a number of polynomial terms, which again won't be too "neat".
So, my question is: Does anyone have an idea to solve this problem - i.e. find a neat upper bound for $\zeta$, minimizing over x?
Thanks a lot!
Write $x:={2\over c} t$ $\ (t\geq0)$, and put $${a\over b}\left({2\over c}\right)^{4/3}=:p>0\ .$$ Then we have to minimize the function$$z(t):=p\,t^{4/3}+e^{-t}\qquad(t\geq0)\ .$$ Determining the unique critical point leads to a transcendental equation; hence we have to manoeuvre with trade offs, etcetera. For this work one should know the order of magnitude of $p$. Since your problem comes from physics it may be the case that $p$ is not "any number between $0$ and $\infty$".