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# Tag Archives: Inequality

## Integral and inequality

Let be a continuous function such that

(1)

Prove that

**Solution**

We note that

On the other hand by Cauchy – Schwartz we have that

The result follows.

## Inequality of a concave function

Let be a concave function. Prove that

**Solution**

Since is concave , it holds that

By setting and we get that . Thus,

and the exercise is complete.

## Root inequality

Let be positive real numbers satisfying the following equality

Prove that

**Solution**

We begin by stating a lemma:

**Lemma:** Let be positive real numbers, then:

Now, making use of the lemma we have that:

Making use of the Cauchy – Schwartz inequality we have that

The inequality now follows.

## A logarithmic inequality

Let be positive numbers other than such that . Prove that

**Solution**

The Engels form of the Cauchy – Schwartz inequality gives us:

and the inequality is proven.

## Root inequality

Let be three positive real numbers such that . Prove that

**Solution**

By AM – GM we have,

However,

Hence and the exercise is complete.