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Consider the following minimization problem: $$ \text{minimize} \quad \frac{( a + b )^2}{y - x} +\frac{a^2}{x} +\frac{b^2}{1 - y} \quad \text{in} \quad 0 < x < y < 1, $$ where $a$ and $b$ are fixed positive numbers. Note how the minimization requirement forces 1) $x$ and $y$ to stay away from each other, 2) $x$ to stay away from $0$ and 3) $y$ to stay away from $1$.

This problem is quite easy to solve directly by taking derivatives and such, but in view of some possible generalizations which are of interest for me, I wonder whether there is a simple trick to find the answer. A trick which involves as little computations as possible.

I know my question is a bit vague, but I'd appreciate any ideas. I'd also like to hear about different possible approaches to similar minimization problems.

tsnao
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    Yes, use the weighted version of the inequality between the harmonic mean and the arithmetic mean The conditions on the variables makes each summand non-negative. Note how the weighted sum of the reciprocals of the summands is $(a+b)^2\frac{y-x}{(a+b)^2}+a^2\frac{x}{a^2}+b^2\frac{1-y}{b^2}=1$. – plop Jan 09 '23 at 11:32
  • Could you give an idea about the motivation for this expression ? – Jean Marie Jan 09 '23 at 11:34
  • @owl, very nice approach! – tsnao Jan 09 '23 at 11:34
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    @JeanMarie, hm... My own motivation comes from a certain high excursion problem for Gaussian processes. I do not know how to state it in the space of a comment :) – tsnao Jan 09 '23 at 11:36
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    Hmm. The Wikipedia doesn't mention the weighted version. Well, you can find them in other places, like here. or in this paper, the very first inequality, Bergstrom inequality, which is just a weighted HM-AM inequality re-arranged, can be applied directly to your problem. – plop Jan 09 '23 at 11:42

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