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Lemma: Upper Bound for the Product of General Powers

Let $p,q\in(1,\infty)$ with $\frac 1p+\frac 1q=1$. Then the product of general powers $x^{1/p}y^{1/q}$ (for all positive numbers $x,y$) has the following upper bound:

$$x^{1/p}y^{1/q}\le \frac xp+\frac yq.$$

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983