The inequality $xy \leq \frac{x^p}{p}+\frac{y^q}{q}$

Question

I struggled to show that this is true when $x,y >0$, $p>1$ and $q=\frac{p}{p-1}$. But, I managed to show that if we assume that $x \geq 1$ the assertion is possibly false only for a finite number of $p$ by taking the limit as $p \rightarrow \infty$.

I have a feeling that this inequality involves convexity somehow but I struggled to use that property.

Answer 1

Use concavity of the logarithm: $$\log\Big(\frac 1p x^p + \frac 1q y^q\Big) \ge \frac 1p \log x^p + \frac 1q \log y^q = \log x + \log y = \log xy.$$ To conclude, exponentiate both sides.