Titu's Lemma

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, 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.
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.

Statement of the inequality

For any real numbers $a_1, a_2, a_3, \ldots, a_n$ and positive reals $b_1, b_2, b_3,\ldots, b_n,$ we have $\frac + \frac + \cdots + \frac \geq \frac.$ ( 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 $X$ be a real random variable, and let $Y$ be a positive random variable. ''X'' and ''Y'' need not be independent, but we assume E /math> are both defined.
Then
\operatorname ^2/Y\ge \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, /math> and E /math> are both defined.
Then
\operatorname ^2/Y\ge \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, 2 / \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, /math> and E /math> are both defined.
Then
\operatorname ^2/Y\ge \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, /math> and E /math> are both defined.
Then
\operatorname ^2/Y\ge \operatorname X, 2 / \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, 2 / \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, 2 / \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, 2 / \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, /math> and E /math> are both defined.
Then
\operatorname ^2/Y\ge \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, /math> and E /math> are both defined.
Then
\operatorname ^2/Y\ge \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, 2 / \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, /math> and E /math> are both defined.
Then
\operatorname ^2/Y\ge \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, /math> and E /math> are both defined.
Then
\operatorname ^2/Y\ge \operatorname X, 2 / \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, 2 / \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, 2 / \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities>X, 2 / \operatorname \ge \operatorname 2 / \operatorname

Direct applications

Example 1. Nesbitt's inequality.

For positive real numbers $a, b, c:$
$\frac + \frac + \frac \geq \frac.$

Example 2. International Mathematical Olympiad (IMO) 1995.

For positive real numbers $a,b,c$, where $abc=1$ we have that $\frac+\frac+\frac \geq \frac.$

Example 3.

For positive real numbers $a,b$ we have that $8(a^4+b^4) \geq (a+b)^4.$

Example 4.

For positive real numbers $a,b,c$ we have that $\frac+\frac+\frac \geq \frac.$

Proofs

Example 1.

Proof: Use $n = 3,$ $\left(a_1, a_2, a_3\right) := (a, b, c),$ and $\left(b_1, b_2, b_3\right) := (a(b + c), b(c + a), c(a + b))$ to conclude: $\frac + \frac + \frac \geq \frac = \frac = \frac + 1 \geq \frac (1) + 1 = \frac. \blacksquare$

Example 2.

We have that $\frac + \frac + \frac \geq \frac = \frac \geq \frac = \frac.$

Example 3.

We have $\frac + \frac \geq \frac$ so that $a^4 + b^4 = \frac + \frac \geq \frac \geq \frac = \frac.$

Example 4.

We have that $\frac + \frac + \frac \geq \frac = \frac.$

References

{{DEFAULTSORT:Sedrakyan's inequality
Inequalities (mathematics)
Linear algebra
Operator theory
Articles containing proofs
Mathematical analysis
Probabilistic inequalities