Mathematical inequality
In mathematics, the following inequality is known as Titu's lemma, Bergström's inequality, Engel's form or Sedrakyan's inequality, respectively, referring to the article About the applications of one useful inequality of Nairi Sedrakyan published in 1997,[1] to the book Problem-solving strategies of Arthur Engel published in 1998 and to the book Mathematical Olympiad Treasures of Titu Andreescu published in 2003.[2][3]
It is a direct consequence of Cauchy–Bunyakovsky–Schwarz inequality. Nevertheless, in his article (1997) Sedrakyan has noticed that written in this form this inequality can be used as a proof technique and it has very useful new applications. In the book Algebraic Inequalities (Sedrakyan) several generalizations of this inequality are provided.[4]
Statement of the inequality
For any real numbers
and positive reals
we have
(Nairi Sedrakyan (1997), Arthur Engel (1998), Titu Andreescu (2003))
Probabilistic statement
Similarly to the Cauchy–Schwarz inequality, one can generalize Sedrakyan's inequality to random variables.
In this formulation let
be a real random variable, and let
be a positive random variable. X and Y need not be independent, but we assume
and
are both defined.
Then
Direct applications
Example 1. Nesbitt's inequality.
For positive real numbers
Example 2. International Mathematical Olympiad (IMO) 1995.
For positive real numbers
, where
we have that
Example 3.
For positive real numbers
we have that
Example 4.
For positive real numbers
we have that
Proofs
Example 1.
Proof: Use
and
to conclude:
Example 2.
We have that
Example 3.
We have
so that
Example 4.
We have that
References