Let x, y, p, q be positive numbers with 1/p + 1/q = 1. Prove that xy ≤ x^p/ p + y^q/

[Riya]

Let x, y, p, q be positive numbers with 1/p + 1/q = 1. Prove that xy ≤ x^p/ p + y^q/

Let x, y, p, q be positive numbers with 1/p + 1/q = 1. Prove that xy ≤ x^p/ p + y^q/q

 

[stapel]

Let x, y, p, q be positive numbers with 1/p + 1/q = 1. Prove that xy ≤ x^p/ p + y^q/q

Please reply with a clear listing of your thoughts and efforts so far, including any results (theorems, rules, etc) from class or your textbook which you think may be applicable. Thank you!

 

[Deleted member 4993]

Young's inequality:

The claim is certainly true if a = 0 or b = 0. Therefore, assume a > 0 and b > 0 in the following. Put t = 1/p, and (1 − t) = 1/q.
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\(\displaystyle \log\left(ta^{\,p}\, +\, (1\, -\, t)\, b^q\right)\, \geq\, t\, \log\left(a^{\,p}\right)\, +\, (1\, -\, t)\, \log\left(b^q\right)\, =\, \log(a)\, +\, \log(b)\, =\, \log(ab)\)

with the equality holding if and only if a^p = b^q. Young's inequality follows by exponentiating
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Where did a & b come from?

 

[Steven G]

Where did a & b come from?

I thought that I was missing something when I saw the a and b. Thanks for letting me know that I am sane at least for another day.