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Titu's lemma

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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

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For any real numbers and positive reals we have (Nairi Sedrakyan (1997), Arthur Engel (1998), Titu Andreescu (2003))

Probabilistic statement

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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

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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

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Example 1.

Proof: Use and to conclude:

Example 2.

We have that

Example 3.

We have so that

Example 4.

We have that

References

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  1. ^ Sedrakyan, Nairi (1997). "About the applications of one useful inequality". Kvant Journal. pp. 42–44, 97(2), Moscow.
  2. ^ Sedrakyan, Nairi (1997). A useful inequality. Springer International publishing. p. 107. ISBN 9783319778365.
  3. ^ "Statement of the inequality". Brilliant Math & Science. 2018.
  4. ^ Sedrakyan, Nairi (2018). Algebraic inequalities. Springer International publishing. pp. 107–109.