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# Tag Archives: Inequality

## Logarithmic mean inequality

Let such that . Prove that

Solution

We are invoking the Hermite – Hadamard Inequality for the convex function . Thus,

## Complex inequality

Prove that forall and it holds that

Solution

Well,

Done!

## Value of parameter

This is a very classic exercise and can be dealt with various ways. We know the result in advance. Why? Because it is the Taylor Polynomial of the exponential function. Let us see however how we gonna deal with it with High School Methods.

Find the positive real number such that

Solution

Define the function

and note that forall . Clearly , is differentiable and its derivative is given by

It follows that . Suppose that . Then the monotony of as well as the sign of is seen at the following table.

It follows then that . This is an obscurity due to the fact that . Similarly, if we suppose that . Hence

For we easily see that the given inequality holds.

## 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.

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