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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!

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  • $\begingroup$ Presumably you have $a,x \ge 0$? $\endgroup$
    – copper.hat
    Commented Aug 25, 2017 at 18:35
  • $\begingroup$ a, b, c, x are all > 0. Thanks for this comment $\endgroup$
    – Paul K
    Commented Aug 28, 2017 at 7:36

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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 $\infty$".

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  • $\begingroup$ Thanks for this suggestion. The transcendental equation is actually what I want to avoid because the solution will generally not be neat (by neat I mean e.g. polynomial, logarithmic, exponential, trignometric, etc). But not something like the product log function. This is why I'd like to first find an upper bound for this function z(t) in your answer, and then minimise it. E.g. I think we can upper bound z(t) by p*t^2+1-t+t^2/2, then minimise this quadratic. But this upper bound isn't very tight so it's not ideal! To answer your question: unfortunately p can be anything from 0 to infinity! $\endgroup$
    – Paul K
    Commented Aug 29, 2017 at 7:59

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