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# Tag Archives: Inequality

## Double inequality

In a triangle prove that

## Vector inequality

Let be endowed with the usual product and the usual norm. If then we define . Prove that

Solution ( Robert Tauraso )

We will show the more general inequality

where . Taking we get the requested inequality. If are linearly dependent then the inequality holds. We assume now that and are linearly independent. Then where and . Moreover,

and by solving the linear system we find

Hence,

## Double inequality

Prove the following double inequality, where the sum and product are cyclic over the angles of a triangle

## Gamma inequality

Let be three positive real numbers such that . Prove that:

where is Euler’s Gamma function.

Solution

We can rewrite the inequality as

since is convex.

## Integral inequality

Let be a continuous function such that

Prove that .

Solution

We have successively:

Since is continuous  equality holds only if for some constant . From the data we get .