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# Tag Archives: Inequality

## Gamma inequality

Let be three positive real numbers such that . Prove that:

where is Euler’s Gamma function.

Solution

We can rewrite the inequality as

since is convex.

## Integral inequality

Let be a continuous function such that

Prove that .

Solution

We have successively:

Since is continuous  equality holds only if for some constant . From the data we get .

## “Upper” bound

Let be a function satisfying

for all positive real numbers and . Prove that

Solution

For starters, let us assume that . Dividing the interval into subintervals each of length so that . Thus,

The inequality implies that

Hence,

The limit exists and equals to . Hence , the inequality is proved for .

Now, assume that . Dividing the interval into subintervals each of length so that . Thus,

The inequality implies that

Hence,

The limit exists and equals to . Hence , the inequality is also proved for . This completes the proof!

## Upper bound of max product

Let be the roots of the polynomial

Prove that:

Solution

Suppose the roots of polynomial are where

Let . Then, the are the zeros of in the disk where is chosen such that for .

Jensen’s inequality implies that

Applying Cauchy – Schwartz yields,

Therefore,

Letting and we get the result.

## Offset logarithmic integral inequality

Prove that

Solution

We have successively: