## Inequality with roots

Let be positive real numbers. Prove that

Solution

We apply the AM – GM inequality, thus:

Hence it suffices to prove that which holds because it is equivalent to .

## On permutation

Let be vectors of – dimensional Euclidean space such that . Prove that there exists a permutation of the integers such that

for each .

Solution

We define inductively. Set .Assume is defined for and also

(1)

Note that is true for . We choose in a way that is fulfilled for instead of . Set and . Assume that for all .  Then and in view of ones gets which is impossible. Hence , there is such that

(2)

Put . Then using and we have

which verifies for . Thus we define for every . Finally from we get

## Homomorphism and inequality

Let be a group and be a homomorphism. Prove that

Solution

In general it holds that  ( first isomorphism theorem ) . Taking that for granted we also have

and the inequality is equivelant to  which is obviously true.

The exercise can also be found at mathematica.gr .

## On linear operators

Let and suppose that , are linear operators from into satisfying

(1)

1. Show that for all one has

2. Show that there exists such that .

Solution

1. Using the assumptions we have

2. Consider the linear operator acting over all matrices . It may have at most different eigenvalues. Assuming that for every we get that has infinitely many different eigenvalues in view of (i). This is a contradiction.

## On the sum of inverse binomial

In this post we are discussing the sum

In Staver was the first to study the sum . He observed that . He was , then , able to extract the recursive relation

(1)

He then proved a great result which is well known in literature

(2)

Later, in Rocket combining the identity

(3)

along with induction he was able to give another proof of . In Surin provided another proof using the well known integral representation of the binomial coefficient,

(4)

Since then the cases and have been studied extensively. However, Mansour generalising the idea of Sury provided a theorem which states the following:

Theorem [Mansour]: Let be non negative integers and be given by

where are two functions defined on . Let , be two sequences and , be their corresponding generating functions. Then,

Proof: The proof is a standard generating type one and is left to the reader.

Using the above theorem along with equation we can generate wonderful stuff. For example:

Example 1: Pick and . Then

which , after a bit of transformations gives the general result

Fabulous, isn’t it? If we set we get equation . Of course there are other applications of the above theorem. We can establish a similar equality for the sum . Another relation that can be established by the above theorem is the following:

(5)

We are not gonna go into a deep analysis but the following equalities also hold:

(6)

and

(7)