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Tag Archives: Inequality

Root inequality

Let be positive real numbers such that . Prove that Solution

Due to the AM – GM we have that

(1) and

(2) Thus, Inequality of a triangle

Let be a triangle and denote the lengths of the sides , and respectively. If then prove that Solution

Applying Cauchy’s inequality to the vectors and we get that due to the well known fact

(1) It suffices to prove that . Let be two positive real numbers. Then,

(2) On the other hand if then and

(3) Then it follows that 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, Who is Tolaso?

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