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I am working to prove the following inequality, however I have not reached to any definite solutions. I would be so thankful if anyone could help me prove this inequality or at least give me some hints and ideas to prove this inequality:

if $a , b$ are positive numbers such that $a + b = 1$ prove that for all $x$

$ae^{\frac{x}{a}} + be^{-\frac{x}{b}} \le e^{\frac{x^2}{8a^2b^2}}$

one of my friends told me to use the Taylor series of the two exponents on the left side of the inequality but I did not reach to the proper solution. Moreover, I chose to use the weighted arithmetic mean of the two exponents on the left side of the inequality but I did not reach to the solution either!!!

Please help me

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    $\begingroup$ You already asked over at Mathematics StackExchange (twice). Give it a week, and please: don't cross-post without linking, at every SE site where you ask, to every other SE site where you ask. $\endgroup$
    – Todd Trimble
    Commented Oct 25, 2016 at 15:14
  • $\begingroup$ @ToddTrimble sure thanks a lot, the answer to this questions is very vital for me therefore I thought it might be helpful to post the same question in different places. Anyway, from now on I would definitely give a link of the previous places where I posted this question. $\endgroup$
    – alfred noble
    Commented Oct 25, 2016 at 15:46
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    $\begingroup$ If anyone wants to have a crack at this, here's the MSE link: math.stackexchange.com/questions/1978987/… I see Fedor Petrov has given an answer, but it seems not to be easy... $\endgroup$
    – Todd Trimble
    Commented Oct 25, 2016 at 18:21
  • $\begingroup$ This is a well-known problem, also posed on dxdy forum. Why to ban it immediately for no reason at all? Let start a company to ban a band of unprofessional banners! $\endgroup$
    – Sergei
    Commented Oct 25, 2016 at 18:42
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    $\begingroup$ @Sergei Could you give the link to where it is posted at dxdy? Is it a contest problem? $\endgroup$
    – Todd Trimble
    Commented Oct 25, 2016 at 20:35

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