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## Proof: (related to "Upper Bound for the Product of General Powers")

• By hypothesis, $x,y$ are positive numbers, and $p,q\in(1,\infty)$ with $\frac 1p+\frac 1q=1$.
• The second derivative of the natural logarithm is negative $\log^{\prime\prime}(x)=-\frac 1{x^2} < 0$ for all positive numbers $x > 0.$
• Thus, the natural logarithm fulfills the test for concaveness.
• Since by definition $\frac 1p,\frac 1q\in(0,1),$ $\frac 1p+\frac 1q=1,$ we get by definition of concave the following inequality $$\log\left(\frac 1px+\frac 1qy\right)\ge \frac 1p\log(x)+\frac 1q\log(y).$$
• Taking the exponential function on both sides of the inequation1 yields
$$\frac xp+\frac yq\ge x^{1/p}\cdot y^{1/q}.$$

1 Note that the exponential function is the inverse function to the logarithm.

q.e.d

| | | | created: 2020-02-07 17:29:18 | modified: 2020-02-07 17:40:02 | by: bookofproofs | references: [581]

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