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# Tag Archives: Inequality

## 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.

## Trigonometric inequality on an acute triangle

Prove that in any acute triangle the following inequality holds:

**Solution**