Sedrakyan's inequality
The following inequality is known as Sedrakyan's inequality, Bergström's inequality, Engel's form or Titu's lemma, respectively, referring to the article “"About the applications of one useful inequality"” of Nairi Sedrakyan published in 1997, to the book "Problem-solving strategies" of Arthur Engel (mathematician) published in 1998 and to the book "Mathematical Olympiad Treasures" of Titu Andreescu published in 2003. 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 mathematical proof technique and  it has very useful new applications. In the book "Algebraic Inequalities" (Sedrakyan) are provided several generalizations of this inequality. Statement of the inequality (Nairi Sedrakyan (1997), Arthur Engel (mathematician) (1998), Titu Andreescu (2003)). For any reals formula_1 and positive reals formula_2 we have formula_3 Direct applications. Example 1. Nesbitt's inequality. For positive real numbers formula_4 formula_5 Example 2. International Mathematical Olympiad (IMO) 1995. For positive real numbers formula_6, where formula_7 we have that formula_8 Example 3. For positive real numbers formula_9 we have that formula_10 Example 4. For positive real numbers formula_6 we have that formula_12 Proofs. Example 1 . Proof: Use formula_13 formula_14 and formula_15 to conclude: formula_16 Example 2. We have that formula_17 Example 3. We have formula_18 so that formula_19 Example 4. We have that formula_20