Inequalities and...inequalities

Inequality 53 (Vo Quoc Ba Can)

Basdekis George — Sun, 16 Oct 2011 15:05:30 +0000

Problem:

Let 0' title='\displaystyle a,b,c>0' class='latex' /> such that and . Find the minimum and the maximum value of the expression

Solution:

From the Cauchy – Schwarz Inequality we have that

or due to the hypothesis

From this last Inequality, we acquire

Let us now divide with . Then we get that

Solving this Inequality, gives us

Thus, and , Q.E.D.

Inequality 52 (Unknown Author)

Basdekis George — Sun, 16 Oct 2011 14:52:15 +0000

Problem:

Let be positive real numbers with sum equal to . Find the minimum value of the expression

Solution:

From the definition of , we have that

Solving the 1st Inequality we get that

Solving now the 2nd Inequality we have that

The last relation reduces to the

So, we see now that for .

Summing up these Inequalities we acquire

or, due to the hypothesis we now have that

If we solve this last Inequality it gives us the result , which is also and the value we are searching, Q.E.D.

Inequality 51 (George Basdekis)

Basdekis George — Sat, 15 Oct 2011 16:40:51 +0000

Problem:

If 0' title='\displaystyle a,b,c>0' class='latex' /> then prove that

Solution:

From the Cauchy – Schwarz Inequality we have that

Expanding the RHS, we get that

from the obvious .

Moreover, notice that the following beautiful Inequality holds, that is

Thus, we have that

We have proved our Inequality, because , or

, Q.E.D.

Inequality 50 (Vasile Cirtoaje)

Basdekis George — Sat, 15 Oct 2011 16:28:24 +0000

Problem:

For all 0' title='\displaystyle a,b,c>0' class='latex' /> prove that

Solution:

It holds that . Thus, we only need to prove that

Observe that

From the above result we have to prove now that

Using the AM – GM Inequality we get that

So, it is enough to prove that

The last Inequality can be reduced to the .

So, it is enough to check the Inequality

This one can be prove by the following way. Set and , with and remake the Inequality to the form

Then, the Inequality substitutes to

But this one holds because we have that

due to the AM – GM Inequality and the hypothesis , Q.E.D.

P.S The following nice Inequality also holds:

This Inequality belongs to Thomas Mildorf and the proof for this Inequality is the same as the above.

Inequality 49 (Unknown Author)

Basdekis George — Mon, 03 Oct 2011 16:10:38 +0000

Problem:

Let be positive real numbers such that , and also let be a natural number. Prove that

Solution:

Observe that . Thus, from the AM – GM Inequality we get that

From the above Inequality, we are now capable of building our problem. Specifically, we have that

We must see now that

From the above result we can apply once again the AM – GM Inequality and acquire

Summing up the Inequalities together we get that

Moreover, we have that

which reduces to the , Q.E.D.

Inequality 48 (George Basdekis)

Basdekis George — Sun, 02 Oct 2011 14:52:01 +0000

Problem:

If such that and , the prove that

Solution:

Using the Cauchy – Schwarz Inequality we get that

From the above Inequality we can see that . Similarly we acquire .

Observe now that

Using once again the Cauchy – Schwarz Inequality we get that

Using the hypothesis, we can calculate the last fraction. Thus we have that

Returning to previous Inequality, we now can see that

, Q.E.D.

Inequality 47(Christos Patilas)

Basdekis George — Thu, 13 May 2010 12:10:27 +0000

Problem:

If for are positive real numbers then prove that

Solution(An Idea by Vo Quoc Ba Can):

We only need to prove that

for all 0' title='\displaystyle a>0' class='latex' />.

So, using the AM-GM Inequality we have that

It follows that

Therefore it suffices to prove that

which is obviously true from the AM-GM Inequality, Q.E.D.

Inequality 46(George Basdekis)

Basdekis George — Sun, 11 Apr 2010 11:38:48 +0000

Problem:

Let be positive real numbers. Prove that .

Inequality 45(Vo Quoc Ba Can)

Basdekis George — Wed, 24 Mar 2010 13:26:10 +0000

Problem:

Let be positive real numbers such that . Prove that .

Inequality 44(Unknown Author)

Basdekis George — Wed, 24 Mar 2010 13:13:09 +0000

Problem:

Let be positive real numbers. Prove that

Solution (An idea by Silouanos Brazitikos):

From the above inequality is it enough to show that

But from Schur’s Inequality we know that

So it is enough to check that

From Cauchy-Schwarz Inequality we know that

Therefore we only need to prove that

which is obviously true from AM-GM Inequality, Q.E.D.