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Tag Archives: Inequality
In a triangle prove that
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
Prove the following double inequality, where the sum and product are cyclic over the angles of a triangle
Let be three positive real numbers such that . Prove that:
where is Euler’s Gamma function.
We can rewrite the inequality as
since is convex.
Let be a continuous function such that
Prove that .
We have successively:
Since is continuous equality holds only if for some constant . From the data we get .