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# Tag Archives: Inequality

## Root inequality

Let such that . Prove that Solution

It follows from Cauchy – Schwartz that ## Double inequality involving matrix

Prove that Solution

Fix . Apply the matrix on thus: Since is exactly the – th column of the previous equality can be rerwritten as Since this holds for all we get and the left inequality follows.

For a random unit vector the coordinate of the vector is . It follows from Cauchy – Schwartz that Summing over all ‘s till we find we conclude that, for every unit vector , it holds that is less than the right hand side. Taking supremum with respect to all the right hand side inequality follows.

## Homogeneity of inequality

Let . Prove that: Solution

Due to homogeneity we may assume . Thus there exist positive such that Hence, ## Trigonometric inequality on a triangle

Let be a triangle. Prove that Solution

The function is convex, thus: The result follows. In fact, something a bit stronger holds. Let denote the inradius , the circumradius and the semiperimeter. Then, Indeed, in view of the known identities

(1) (2) (3) ## Inequality involving area of a triangle

Show that in any triangle with area the following holds: Solution

Let is the semiperimeter , the circumradius and the inradius. From the law of sines we find

(1) as well as

(2) Now, Substitute the preceding equalities into the last inequality  and simplifying we obtain From the last inequality becomes which is true by the rearrangement inequality.

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