Difference between revisions of "Cauchy-Schwarz Inequality"

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The '''Cauchy-Schwarz Inequality''' (which is known by other names, including Cauchy's Inequality) states that, for two sets of real numbers <math>a_1,a_2,\ldots,a_n</math> and <math>b_1,b_2,\ldots,b_n</math>, the following inequality is always true:
+
The '''Cauchy-Schwarz Inequality''' (which is known by other names, including '''Cauchy's Inequality''', '''Schwarz's Inequality''', and the '''Cauchy-Bunyakovsky-Schwarz Inequality''') is a well-known [[inequality]] with many elegant applications. It has an elementary form, a complex form, and a general form.
  
<math>\displaystyle({a_1}^2+{a_2}^2+...+{a_n}^2)({b_1}^2+{b_2}^2+...+{b_n}^2)\geq(a_1b_1+a_2b_2+...+a_nb_n)^2</math>
+
[https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy|Augustin Louis Cauchy] wrote the first paper about the elementary form in 1821. The general form was discovered by [https://en.wikipedia.org/wiki/Viktor_Bunyakovsky|Viktor Bunyakovsky] in 1849 and independently by [https://en.wikipedia.org/wiki/Hermann_Amandus_Schwarz|Hermann Schwarz] in 1888.
  
Equality holds if and only if <math>\frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}</math>.
+
== Elementary Form ==
  
There are many ways to prove this; one of the more well-known is to consider the equation
+
For any real numbers <math> a_1, \ldots, a_n </math> and <math> b_1, \ldots, b_n </math>,
<math>(a_1x+b_1)^2+(a_2x+b_2)^2+...+(a_nx+b_n)^2=0</math>.  Expanding, we find the equation to be of the form <math>Ax^2+Bx+C</math>, where <math>A=\sum_{i=1}^n a_i^2</math>, <math>B=2\sum_{j=1}^n a_jb_j</math>, and <math>C=\sum_{k=1}^n b_k^2.</math>.  By the [[Trivial inequality | Trivial Inequality]], we know that the left-hand-side of the original equation is always at least 0, so either both roots are [[Complex Numbers]], or there is a double root at <math>x=0</math>.  Either way, the [[discriminant]] of the equation is nonpositive.  Taking the [[discriminant]], <math>B^2-4AC \leq 0</math> and substituting the above values of A, B, and C leaves us with the '''Cauchy-Schwarz Inequality''', <math>(a_1b_1+a_2b_2+...+a_nb_n)^2 \leq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2)</math>,
+
<cmath>
or, in the more compact [[sigma notation]],
+
\left( \sum_{i=1}^{n}a_ib_i \right)^2 \le \left(\sum_{i=1}^{n}a_i^2 \right) \left(\sum_{i=1}^{n}b_i^2 \right)
<math>\left(\sum a_ib_i\right)</math> <math>\leq \left(\sum a_i^2\right)\left(\sum b_i^2\right)</math>  
+
</cmath>
+
with equality when there exists a nonzero constant <math>\mu</math> such that for all <math> 1 \le i \le n </math>, <math>\mu a_i = b_i </math>.
Note that this also gives us the equality case; equality holds if and only if the discriminant is equal to 0, which is true if and only if the equation has 0 as a double root, which is true if and only if <math>\frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}</math>.
 
  
This inequality is used very frequently to solve Olympiad-level Inequality problems, such as those on the [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]].
+
=== Discussion ===
 +
 
 +
Consider the vectors <math> \mathbf{a} = \langle a_1, \ldots a_n \rangle </math> and <math> {} \mathbf{b} = \langle b_1, \ldots b_n \rangle </math>.  If <math>\theta </math> is the [[angle]] formed by <math> \mathbf{a} </math> and <math> \mathbf{b} </math>, then the left-hand side of the inequality is equal to the square of the [[dot product]] of <math>\mathbf{a} </math> and <math> \mathbf{b} </math>, or <math>(\mathbf{a} \cdot \mathbf{b})^2 = a^2 b^2 (\cos\theta) ^2</math>  .The right hand side of the inequality is equal to <math> \left( ||\mathbf{a}|| * ||\mathbf{b}|| \right)^2  =  a^2b^2</math>.  The inequality then follows from <math> |\cos\theta | \le 1 </math>, with equality when one of <math> \mathbf{a,b} </math> is a multiple of the other, as desired.
 +
 
 +
=== Complex Form ===
 +
 
 +
The inequality sometimes appears in the following form.
 +
 
 +
Let <math> a_1, \ldots, a_n </math> and <math> b_1, \ldots, b_n </math> be [[complex numbers]].  Then
 +
<cmath>
 +
\left| \sum_{i=1}^na_ib_i \right|^2 \le \left(\sum_{i=1}^{n}|a_i^2| \right) \left( \sum_{i=1}^n |b_i^2| \right)
 +
</cmath>
 +
This appears to be more powerful, but it follows from
 +
<cmath>
 +
\left| \sum_{i=1}^n a_ib_i \right| ^2 \le \left( \sum_{i=1}^n |a_i| \cdot |b_i| \right)^2 \le \left(\sum_{i=1}^n |a_i^2| \right) \left( \sum_{i=1}^n |b_i^2| \right)
 +
</cmath>
 +
 
 +
== Upper Bound on (Σa)(Σb) ==
 +
 
 +
Let <math>a_1, a_2, \ldots, a_n</math> and <math>b_1, b_2, \ldots, b_n</math> be two sequences of positive real numbers with
 +
<cmath>
 +
0 < m \le \frac{a_i}{b_i} \le M
 +
</cmath>
 +
for <math>1 \le i \le n</math>. Then
 +
<cmath>
 +
\left(\sum_{i=1}^{n}a_i^2 \right) \left(\sum_{i=1}^{n}b_i^2 \right) \le \frac{(M+m)^2}{4Mm} \left( \sum_{i=1}^{n}a_ib_i \right)^2,
 +
</cmath>
 +
with equality if and only if, for some ordering of the pairs <math>(a_i,b_i) \mapsto (a_{\sigma(i)},b_{\sigma(i)})</math>, some <math>0 \le j \le n</math> exists such that <math>a_{\sigma(i)}=mb_{\sigma(i)}</math> for <math>1 \le \sigma(i) \le j</math> and <math>a_{\sigma(i)}=Mb_{\sigma(i)}</math> for <math>j+1 \le \sigma(i) \le n</math>, and
 +
<cmath>
 +
m\sum_{\sigma(i)=1}^{j}b_{\sigma(i)}^2 = M\sum_{\sigma(i)=j+1}^{n}b_{\sigma(i)}^2.
 +
</cmath>
 +
If we restrict that <math>m_1 \le a_i \le M_1</math> and <math>m_2 \le b_i \le M_2</math> for all <math>i</math>, then it's clear that for <math>a_i/b_i</math> to be <math>m=m_1/M_2</math> or <math>M=M_1/m_2</math> for all <math>i</math>, then <math>a_i=m_1 \Longleftrightarrow b_i=M_2</math> and <math>a_i=M_1 \Longleftrightarrow b_i=m_2</math>, so
 +
<cmath>
 +
m\sum_{\sigma(i)=1}^{j}b_{\sigma(i)}^2 = M\sum_{\sigma(i)=j+1}^{n}b_{\sigma(i)}^2
 +
</cmath>
 +
is equivalent to
 +
<cmath>
 +
\begin{align*}
 +
m(jM_2^2) = M((n-j)m_2^2) &\Longleftrightarrow m_1M_2j = M_1m_2(n-j)\\
 +
&\Longleftrightarrow j = \left(\frac{M_1m_2}{M_1m_2+m_1M_2}\right) n.
 +
\end{align*}
 +
</cmath>
 +
(When this is not an integer, the maximum occurs when <math>j</math> is either the ceiling or floor of the right-hand side.) In the special case that <math>a_ib_i = k > 0</math> is constant for all <math>i</math>, we have <math>M_1=1/m_2</math> and <math>m_1=1/M_2</math>, so here <math>j</math> must be <math>n/2</math>.
 +
 
 +
=== Proof ===
 +
 
 +
Note that for all <math>i</math>, we have
 +
<cmath>
 +
0 \le \left(\frac{a_i}{b_i}-m\right)\left(M-\frac{a_i}{b_i}\right) = \frac{1}{b_i^2}(a_ib_iM-a_i^2-b_i^2Mm+a_ib_im)
 +
</cmath>
 +
or
 +
<cmath>
 +
(M+m)a_ib_i \ge a_i^2+(Mm)b_i^2,
 +
</cmath>
 +
with equality if and only if <math>a_i=mb_i</math> or <math>a_i=Mb_i</math>. Summing up these inequalities over <math>1 \le i \le n</math>, we obtain from AM-GM that
 +
<cmath>
 +
\begin{align*}
 +
(M+m)\sum_{i=1}^{n}a_ib_i &\ge \sum_{i=1}^{n}a_i^2 + (Mm)\sum_{i=1}^{n}b_i^2\\
 +
&\ge 2\sqrt{Mm \left(\sum_{i=1}^{n}a_i^2 \right) \left(\sum_{i=1}^{n}b_i^2 \right)},
 +
\end{align*}
 +
</cmath>
 +
and squaring gives us the desired bound. For equality to occur, we must have <math>a_i=mb_i</math> or <math>a_i=Mb_i</math> for all <math>i</math>. If, without loss of generality, <math>a_i=mb_i</math> for <math>1 \le i \le j</math> and <math>a_i=Mb_i</math> for <math>j+1 \le i \le n</math> for some <math>0 \le j \le n</math>, then for the AM-GM to reach equality we must have (assume <math>M>m</math> since <math>M=m</math> is trivial)
 +
<cmath>
 +
\begin{align*}
 +
\sum_{i=1}^{n}a_i^2 &= Mm\sum_{i=1}^{n}b_i^2\\
 +
m^2\sum_{i=1}^{j}b_i^2 + M^2\sum_{i=j+1}^{n}b_i^2 &= Mm\sum_{i=1}^{j}b_i^2 + Mm\sum_{i=j+1}^{n}b_i^2\\
 +
(m-M)m\sum_{i=1}^{j}b_i^2 &= (m-M)M\sum_{i=j+1}^{n}b_i^2\\
 +
m\sum_{i=1}^{j}b_i^2 &= M\sum_{i=j+1}^{n}b_i^2.
 +
\end{align*}
 +
</cmath>
 +
 
 +
== General Form ==
 +
 
 +
Let <math>V </math> be a [[vector space]], and let <math> \langle \cdot, \cdot \rangle : V \times V \to \mathbb{R} </math> be an [[inner product]].  Then for any <math> \mathbf{a,b} \in V </math>,
 +
<cmath>
 +
\langle \mathbf{a,b} \rangle^2 \le \langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle ,
 +
</cmath>
 +
with equality if and only if there exist constants <math>\mu, \lambda </math> not both zero such that <math> \mu\mathbf{a} = \lambda\mathbf{b} </math>.
 +
 
 +
=== Proof 1 ===
 +
 
 +
Consider the polynomial of <math> t </math>
 +
<cmath>
 +
\langle t\mathbf{a + b}, t\mathbf{a + b} \rangle = t^2\langle \mathbf{a,a} \rangle + 2t\langle \mathbf{a,b} \rangle + \langle \mathbf{b,b} \rangle .
 +
</cmath>
 +
This must always be greater than or equal to zero, so it must have a non-positive discriminant, i.e., <math> \langle \mathbf{a,b} \rangle^2 </math> must be less than or equal to <math> \langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle </math>, with equality when <math> \mathbf{a = 0} </math> or when there exists some scalar <math>-t </math> such that <math> -t\mathbf{a} = \mathbf{b} </math>, as desired.
 +
 
 +
=== Proof 2 ===
 +
 
 +
We consider
 +
<cmath>
 +
\langle \mathbf{a-b, a-b} \rangle = \langle \mathbf{a,a} \rangle + \langle \mathbf{b,b} \rangle - 2 \langle \mathbf{a,b} \rangle .
 +
</cmath>
 +
Since this is always greater than or equal to zero, we have
 +
<cmath>
 +
\langle \mathbf{a,b} \rangle \le \frac{1}{2} \langle \mathbf{a,a} \rangle + \frac{1}{2} \langle \mathbf{b,b} \rangle .
 +
</cmath>
 +
Now, if either <math> \mathbf{a} </math> or <math> \mathbf{b} </math> is equal to <math> \mathbf{0} </math>, then <math> \langle \mathbf{a,b} \rangle^2 = \langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle = 0 </math>.  Otherwise, we may [[normalize]] so that <math> \langle \mathbf {a,a} \rangle = \langle \mathbf{b,b} \rangle = 1 </math>, and we have
 +
<cmath>
 +
\langle \mathbf{a,b} \rangle \le 1 = \langle \mathbf{a,a} \rangle^{1/2} \langle \mathbf{b,b} \rangle^{1/2} ,
 +
</cmath>
 +
with equality when <math>\mathbf{a} </math> and <math> \mathbf{b} </math> may be scaled to each other, as desired.
 +
 
 +
=== Examples ===
 +
 
 +
The elementary form of the Cauchy-Schwarz inequality is a special case of the general form, as is the '''Cauchy-Schwarz Inequality for Integrals''': for integrable functions <math> f,g : [a,b] \mapsto \mathbb{R} </math>,
 +
<cmath>
 +
\biggl( \int_{a}^b f(x)g(x)dx \biggr)^2 \le \int_{a}^b \bigl[ f(x) \bigr]^2dx \cdot \int_a^b \bigl[ g(x) \bigr]^2 dx
 +
</cmath>
 +
with equality when there exist constants <math> \mu, \lambda </math> not both equal to zero such that for <math> t \in [a,b] </math>,
 +
<cmath>
 +
\mu \int_a^t f(x)dx = \lambda \int_a^t g(x)dx .
 +
</cmath>
 +
 
 +
==Problems==
 +
 
 +
===Introductory===
 +
 
 +
*Consider the function <math>f(x)=\frac{(x+k)^2}{x^2+1},x\in (-\infty,\infty)</math>, where <math>k</math> is a positive integer. Show that <math>f(x)\le k^2+1</math>. ([[User:Temperal/The_Problem_Solver's Resource Competition|Source]])
 +
* [http://www.mathlinks.ro/Forum/viewtopic.php?t=78687 (APMO 1991 #3)] Let <math>a_1</math>, <math>a_2</math>, <math>\cdots</math>, <math>a_n</math>, <math>b_1</math>, <math>b_2</math>, <math>\cdots</math>, <math>b_n</math> be positive real numbers such that <math>a_1 + a_2 + \cdots + a_n = b_1 + b_2 + \cdots + b_n</math>.  Show that
 +
<cmath>\frac {a_1^2}{a_1 + b_1} + \frac {a_2^2}{a_2 + b_2} + \cdots + \frac {a_n^2}{a_n + b_n} \geq \frac {a_1 + a_2 + \cdots + a_n}{2}</cmath>
 +
 
 +
===Intermediate===
 +
 
 +
*Let <math>ABC</math> be a triangle such that
 +
<cmath>
 +
\left( \cot \frac{A}{2} \right)^2 + \left( 2 \cot \frac{B}{2} \right)^2 + \left( 3 \cot \frac{C}{2} \right)^2 = \left( \frac{6s}{7r} \right)^2 ,
 +
</cmath>
 +
where <math>s</math> and <math>r</math> denote its [[semiperimeter]] and [[inradius]], respectively.  Prove that triangle <math>ABC </math> is similar to a triangle <math>T </math> whose side lengths are all positive integers with no common divisor and determine those integers.
 +
([[2002 USAMO Problems/Problem 2|Source]])
 +
 
 +
===Olympiad===
 +
 
 +
*<math>P</math> is a point inside a given triangle <math>ABC</math>.  <math>D, E, F</math> are the feet of the perpendiculars from <math>P</math> to the lines <math>BC, CA, AB</math>, respectively.  Find all <math>P</math> for which
 +
<cmath>
 +
\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}
 +
</cmath>
 +
is least.
 +
 
 +
([[1981 IMO Problems/Problem 1|Source]])
 +
 
 +
== Other Resources ==
 +
* [http://en.wikipedia.org/wiki/Cauchy-Schwarz_inequality Wikipedia entry]
 +
 
 +
===Books===
 +
* [http://www.amazon.com/exec/obidos/ASIN/052154677X/artofproblems-20 The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities] by J. Michael Steele.
 +
* [http://www.amazon.com/exec/obidos/ASIN/0387982191/artofproblems-20 Problem Solving Strategies] by Arthur Engel contains significant material on inequalities.
 +
 
 +
 
 +
[[Category:Algebra]]
 +
[[Category:Inequality]]
 +
[[Category:Theorems]]

Latest revision as of 02:12, 19 August 2018

The Cauchy-Schwarz Inequality (which is known by other names, including Cauchy's Inequality, Schwarz's Inequality, and the Cauchy-Bunyakovsky-Schwarz Inequality) is a well-known inequality with many elegant applications. It has an elementary form, a complex form, and a general form.

Louis Cauchy wrote the first paper about the elementary form in 1821. The general form was discovered by Bunyakovsky in 1849 and independently by Schwarz in 1888.

Elementary Form

For any real numbers $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$, \[\left( \sum_{i=1}^{n}a_ib_i \right)^2 \le \left(\sum_{i=1}^{n}a_i^2 \right) \left(\sum_{i=1}^{n}b_i^2 \right)\] with equality when there exists a nonzero constant $\mu$ such that for all $1 \le i \le n$, $\mu a_i = b_i$.

Discussion

Consider the vectors $\mathbf{a} = \langle a_1, \ldots a_n \rangle$ and ${} \mathbf{b} = \langle b_1, \ldots b_n \rangle$. If $\theta$ is the angle formed by $\mathbf{a}$ and $\mathbf{b}$, then the left-hand side of the inequality is equal to the square of the dot product of $\mathbf{a}$ and $\mathbf{b}$, or $(\mathbf{a} \cdot \mathbf{b})^2 = a^2 b^2 (\cos\theta) ^2$ .The right hand side of the inequality is equal to $\left( ||\mathbf{a}|| * ||\mathbf{b}|| \right)^2  =  a^2b^2$. The inequality then follows from $|\cos\theta | \le 1$, with equality when one of $\mathbf{a,b}$ is a multiple of the other, as desired.

Complex Form

The inequality sometimes appears in the following form.

Let $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ be complex numbers. Then \[\left| \sum_{i=1}^na_ib_i \right|^2 \le \left(\sum_{i=1}^{n}|a_i^2| \right) \left( \sum_{i=1}^n |b_i^2| \right)\] This appears to be more powerful, but it follows from \[\left| \sum_{i=1}^n a_ib_i \right| ^2 \le \left( \sum_{i=1}^n |a_i| \cdot |b_i| \right)^2 \le \left(\sum_{i=1}^n |a_i^2| \right) \left( \sum_{i=1}^n |b_i^2| \right)\]

Upper Bound on (Σa)(Σb)

Let $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ be two sequences of positive real numbers with \[0 < m \le \frac{a_i}{b_i} \le M\] for $1 \le i \le n$. Then \[\left(\sum_{i=1}^{n}a_i^2 \right) \left(\sum_{i=1}^{n}b_i^2 \right) \le \frac{(M+m)^2}{4Mm} \left( \sum_{i=1}^{n}a_ib_i \right)^2,\] with equality if and only if, for some ordering of the pairs $(a_i,b_i) \mapsto (a_{\sigma(i)},b_{\sigma(i)})$, some $0 \le j \le n$ exists such that $a_{\sigma(i)}=mb_{\sigma(i)}$ for $1 \le \sigma(i) \le j$ and $a_{\sigma(i)}=Mb_{\sigma(i)}$ for $j+1 \le \sigma(i) \le n$, and \[m\sum_{\sigma(i)=1}^{j}b_{\sigma(i)}^2 = M\sum_{\sigma(i)=j+1}^{n}b_{\sigma(i)}^2.\] If we restrict that $m_1 \le a_i \le M_1$ and $m_2 \le b_i \le M_2$ for all $i$, then it's clear that for $a_i/b_i$ to be $m=m_1/M_2$ or $M=M_1/m_2$ for all $i$, then $a_i=m_1 \Longleftrightarrow b_i=M_2$ and $a_i=M_1 \Longleftrightarrow b_i=m_2$, so \[m\sum_{\sigma(i)=1}^{j}b_{\sigma(i)}^2 = M\sum_{\sigma(i)=j+1}^{n}b_{\sigma(i)}^2\] is equivalent to \begin{align*} m(jM_2^2) = M((n-j)m_2^2) &\Longleftrightarrow m_1M_2j = M_1m_2(n-j)\\ &\Longleftrightarrow j = \left(\frac{M_1m_2}{M_1m_2+m_1M_2}\right) n. \end{align*} (When this is not an integer, the maximum occurs when $j$ is either the ceiling or floor of the right-hand side.) In the special case that $a_ib_i = k > 0$ is constant for all $i$, we have $M_1=1/m_2$ and $m_1=1/M_2$, so here $j$ must be $n/2$.

Proof

Note that for all $i$, we have \[0 \le \left(\frac{a_i}{b_i}-m\right)\left(M-\frac{a_i}{b_i}\right) = \frac{1}{b_i^2}(a_ib_iM-a_i^2-b_i^2Mm+a_ib_im)\] or \[(M+m)a_ib_i \ge a_i^2+(Mm)b_i^2,\] with equality if and only if $a_i=mb_i$ or $a_i=Mb_i$. Summing up these inequalities over $1 \le i \le n$, we obtain from AM-GM that \begin{align*} (M+m)\sum_{i=1}^{n}a_ib_i &\ge \sum_{i=1}^{n}a_i^2 + (Mm)\sum_{i=1}^{n}b_i^2\\ &\ge 2\sqrt{Mm \left(\sum_{i=1}^{n}a_i^2 \right) \left(\sum_{i=1}^{n}b_i^2 \right)}, \end{align*} and squaring gives us the desired bound. For equality to occur, we must have $a_i=mb_i$ or $a_i=Mb_i$ for all $i$. If, without loss of generality, $a_i=mb_i$ for $1 \le i \le j$ and $a_i=Mb_i$ for $j+1 \le i \le n$ for some $0 \le j \le n$, then for the AM-GM to reach equality we must have (assume $M>m$ since $M=m$ is trivial) \begin{align*} \sum_{i=1}^{n}a_i^2 &= Mm\sum_{i=1}^{n}b_i^2\\ m^2\sum_{i=1}^{j}b_i^2 + M^2\sum_{i=j+1}^{n}b_i^2 &= Mm\sum_{i=1}^{j}b_i^2 + Mm\sum_{i=j+1}^{n}b_i^2\\ (m-M)m\sum_{i=1}^{j}b_i^2 &= (m-M)M\sum_{i=j+1}^{n}b_i^2\\ m\sum_{i=1}^{j}b_i^2 &= M\sum_{i=j+1}^{n}b_i^2. \end{align*}

General Form

Let $V$ be a vector space, and let $\langle \cdot, \cdot \rangle : V \times V \to \mathbb{R}$ be an inner product. Then for any $\mathbf{a,b} \in V$, \[\langle \mathbf{a,b} \rangle^2 \le \langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle ,\] with equality if and only if there exist constants $\mu, \lambda$ not both zero such that $\mu\mathbf{a} = \lambda\mathbf{b}$.

Proof 1

Consider the polynomial of $t$ \[\langle t\mathbf{a + b}, t\mathbf{a + b} \rangle = t^2\langle \mathbf{a,a} \rangle + 2t\langle \mathbf{a,b} \rangle + \langle \mathbf{b,b} \rangle .\] This must always be greater than or equal to zero, so it must have a non-positive discriminant, i.e., $\langle \mathbf{a,b} \rangle^2$ must be less than or equal to $\langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle$, with equality when $\mathbf{a = 0}$ or when there exists some scalar $-t$ such that $-t\mathbf{a} = \mathbf{b}$, as desired.

Proof 2

We consider \[\langle \mathbf{a-b, a-b} \rangle = \langle \mathbf{a,a} \rangle + \langle \mathbf{b,b} \rangle - 2 \langle \mathbf{a,b} \rangle .\] Since this is always greater than or equal to zero, we have \[\langle \mathbf{a,b} \rangle \le \frac{1}{2} \langle \mathbf{a,a} \rangle + \frac{1}{2} \langle \mathbf{b,b} \rangle .\] Now, if either $\mathbf{a}$ or $\mathbf{b}$ is equal to $\mathbf{0}$, then $\langle \mathbf{a,b} \rangle^2 = \langle \mathbf{a,a} \rangle \langle \mathbf{b,b} \rangle = 0$. Otherwise, we may normalize so that $\langle \mathbf {a,a} \rangle = \langle \mathbf{b,b} \rangle = 1$, and we have \[\langle \mathbf{a,b} \rangle \le 1 = \langle \mathbf{a,a} \rangle^{1/2} \langle \mathbf{b,b} \rangle^{1/2} ,\] with equality when $\mathbf{a}$ and $\mathbf{b}$ may be scaled to each other, as desired.

Examples

The elementary form of the Cauchy-Schwarz inequality is a special case of the general form, as is the Cauchy-Schwarz Inequality for Integrals: for integrable functions $f,g : [a,b] \mapsto \mathbb{R}$, \[\biggl( \int_{a}^b f(x)g(x)dx \biggr)^2 \le \int_{a}^b \bigl[ f(x) \bigr]^2dx \cdot \int_a^b \bigl[ g(x) \bigr]^2 dx\] with equality when there exist constants $\mu, \lambda$ not both equal to zero such that for $t \in [a,b]$, \[\mu \int_a^t f(x)dx = \lambda \int_a^t g(x)dx .\]

Problems

Introductory

  • Consider the function $f(x)=\frac{(x+k)^2}{x^2+1},x\in (-\infty,\infty)$, where $k$ is a positive integer. Show that $f(x)\le k^2+1$. (Source)
  • (APMO 1991 #3) Let $a_1$, $a_2$, $\cdots$, $a_n$, $b_1$, $b_2$, $\cdots$, $b_n$ be positive real numbers such that $a_1 + a_2 + \cdots + a_n = b_1 + b_2 + \cdots + b_n$. Show that

\[\frac {a_1^2}{a_1 + b_1} + \frac {a_2^2}{a_2 + b_2} + \cdots + \frac {a_n^2}{a_n + b_n} \geq \frac {a_1 + a_2 + \cdots + a_n}{2}\]

Intermediate

  • Let $ABC$ be a triangle such that

\[\left( \cot \frac{A}{2} \right)^2 + \left( 2 \cot \frac{B}{2} \right)^2 + \left( 3 \cot \frac{C}{2} \right)^2 = \left( \frac{6s}{7r} \right)^2 ,\] where $s$ and $r$ denote its semiperimeter and inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisor and determine those integers. (Source)

Olympiad

  • $P$ is a point inside a given triangle $ABC$. $D, E, F$ are the feet of the perpendiculars from $P$ to the lines $BC, CA, AB$, respectively. Find all $P$ for which

\[\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}\] is least.

(Source)

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