If are non-negative real numbers with sum equal to then find the maximum value of the expression
First we may, without loss of generality, assume that Now, we shall prove that
It holds that
Therefore, we only need to prove that
For we get
or equivalently the above Inequality reduces to the But this one holds due to the AM-GM Inequality since we have
Now, for on the other side, we only need to show that
But, again, from the AM-GM Inequality we acquire
So, finally, we are left to find the maximum value of the expression
We have according to the AM-GM Inequality
Thus, the maximum value of the expression is equal to and it is obtained if and only if Q.E.D.
Let be positive real numbers. Prove the Inequality
First, we observe that from the AM-GM Inequality we have that
and now taking square roots for both sides, we see that
Therefore, we acquire
Now, from the Cauchy-Schwarz Inequality we have for the right hand side
Moreover, once again, from the AM-GM Inequality we have for the denominators of the above sum that
and thus, it will hold that
Finally, we have proved that
The Inequality we have proved is
which also is the Inequality we are given to prove. The proof is completed Q.E.D.
If are positive real numbers then prove that
The Inequality is obvious for . So, let’s consider the case where . We will now divide the Inequality into two cases.
- Case 1st: Then it holds that
therefore, according to Bernulli’s Inequality it will hold that
- Case 2nd: Then it is
therefore, once again from Bernulli’s Inequality we get
and the proof is completed Q.E.D.
Let and be real numbers. Prove the Inequality
Without loss of generality, let us assume that . Then according to the AM-GM Inequality we have that
Therefore, it holds that
It remains now to prove that
which reduces to the obvious, after expansions, Inequality Equality holds if and only if or any cyclic permutations of the previous equality Q.E.D.
Old loves cannot and definitely must not be forgotten so I decided to make my blog here active again. I will try to post Inequalities from around the world. Happy sharing Inequalities.
Let such that and . Find the minimum and the maximum value of the expression
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.
Let be positive real numbers with sum equal to . Find the minimum value of the expression
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.