Titu's Lemma
Titu's lemma (also known as T2 Lemma, Engel's form, or Sedrakyan's inequality) states that for positive reals \( a_1, a_2, \ldots, a_n \) and \( b_1, b_2, \ldots, b_n \), we have
\[ \frac{ a_1^2 } { b_1 } + \frac{ a_2 ^2 } { b_2 } + \cdots + \frac{ a_n ^2 } { b_n } \geq \frac{ (a_1 + a_2 + \cdots+ a_n ) ^2 } { b_1 + b_2 + \cdots+ b_n }. \]
It is a direct consequence of Cauchy-Schwarz inequality. This form is especially helpful when the inequality involves fractions where the numerator is a perfect square.
It is obtained by applying the substitution \(a_i= \frac{x_i}{ \sqrt{y_i} }\) and \( b_i = \sqrt{y_i} \) into the Cauchy-Schwarz inequality. Equality holds if and only if \( a_i = k b_i\) for a non-zero real constant \(k\).
It then becomes
\[\begin{align} \left( \frac{ x_1^2 }{ y_1 } + \frac{ x_2^2 }{ y_2 } + \cdots + \frac{ x_n^2 }{ y_n } \right) (y_1 + y_2 + \cdots + y_n ) &\geq ( x_1 + x_2 + \cdots + x_n )^2 \\ \frac{ x_1^2 }{ y_1 } + \frac{ x_2^2 }{ y_2 } + \cdots + \frac{ x_n^2 }{ y_n } &\geq \frac{ ( x_1 + x_2 + \cdots + x_n )^2 }{ (y_1 + y_2 + \cdots + y_n ) }. \end{align}\]
Statement
For positive reals \( a_1, a_2, \ldots, a_n \) and \( b_1, b_2, \ldots, b_n \), we have
\[ \frac{ a_1^2 } { b_1 } + \frac{ a_2 ^2 } { b_2 } + \cdots + \frac{ a_n ^2 } { b_n } \geq \frac{ (a_1 + a_2 + \cdots+ a_n ) ^2 } { b_1 + b_2 + \cdots+ b_n }. \]
It is obtained by applying the substitution \(a_i= \frac{x_i}{ \sqrt{y_i} }\) and \( b_i = \sqrt{y_i} \) into the Cauchy-Schwarz inequality.
It then becomes
\[\begin{align} \left( \frac{ x_1^2 }{ y_1 } + \frac{ x_2^2 }{ y_2 } + \cdots + \frac{ x_n^2 }{ y_n } \right) (y_1 + y_2 + \cdots + y_n ) &\geq ( x_1 + x_2 + \cdots + x_n )^2 \\ \frac{ x_1^2 }{ y_1 } + \frac{ x_2^2 }{ y_2 } + \cdots + \frac{ x_n^2 }{ y_n } &\geq \frac{ ( x_1 + x_2 + \cdots + x_n )^2 }{ (y_1 + y_2 + \cdots + y_n ) }. \ _\square \end{align}\]
Direct Applications
Prove that for positive reals \(a,b,c\) summing up to \(1\), we have
\[\dfrac{1}{a+b}+\dfrac{16}{c}+\dfrac{81}{a+b+c}\ge 98.\]
The hypothesis implies \(a+b+c=1\). We have square terms in the numerator, so
\[\begin{eqnarray} \dfrac{1}{a+b}+\dfrac{16}{c}+\dfrac{81}{a+b+c} &=& \dfrac{1}{a+b}+\dfrac{4^2}{c}+\dfrac{9^2}{a+b+c} \\ \\ &\ge &\dfrac{(1+4+9)^2}{(a+b)+c+(a+b+c)} \\ \\ & = & \dfrac{(14)^2}{2(a+b+c)} = \dfrac{196}{2} = 98, \end{eqnarray}\]
which is exactly what we wanted. \(_\square\)
Note that the equality condition \( a+b = k, c = 16k, a+b+c = 81k \) cannot be satisfied.
In fact, we have the stronger statement:
\[ \dfrac{1}{a+b}+\dfrac{16}{c}+\dfrac{81}{a+b+c}\ge 106. \]
Can you prove this using Titu's lemma?
Over all triplets of positive reals \(x,y,z\) satisfying \(x+y+z=3\), find the minimum value of
\[f(x,y,z)=\dfrac{yz+4zx+9xy}{xyz}.\]
Dividing out simplifies \(f(x,y,z)\) into \(\frac{1}{x}+\frac{4}{y}+\frac{9}{z}\). Now we have square terms on the numerator. We get
\[f(x,y,z)=\dfrac{1^2}{x}+\dfrac{2^2}{y}+\dfrac{3^2}{z} \ge \dfrac{(1+2+3)^2}{x+y+z} = \dfrac{36}{3}=12.\]
Equality holds when
\[\dfrac 1 x = \dfrac 2 y = \dfrac 3 z~\Longleftrightarrow ~ y=2x, ~z=\dfrac{3y}{2}=3x,\]
and solving for \(x\) using the constraint \(x+y+z=3\), we have equality at \(x=\frac{1}{2}\), \(y=1\), \(z=\frac{3}{2}\). We check that equality indeed occurs.
Therefore, the minimum value is 12. \(_\square\)
Let \(x,y,z\in\mathbb{R^{+}}\). Prove that \[\frac{2}{x+y}+\frac{2}{y+z}+\frac{2}{z+x}≥\frac{9}{x+y+z}.\]
Observe that
\[\frac{2}{x+y}+\frac{2}{y+z}+\frac{2}{z+x}=\frac{\big(\sqrt{2}\big)^{2}}{x+y}+\frac{\big(\sqrt{2}\big)^{2}}{y+z}+\frac{\big(\sqrt{2}\big)^{2}}{z+x}.\]
Then,
\[\frac{\big(\sqrt{2}\big)^{2}}{x+y}+\frac{\big(\sqrt{2}\big)^{2}}{y+z}+\frac{\big(\sqrt{2}\big)^{2}}{z+x}≥\frac{\big(3\sqrt{2}\big)^{2}}{2(x+y+z)}=\frac{9}{x+y+z}. \ _\square\]
Let \(a,b\in\mathbb{R^{+}}\). Prove that
\[8\big(a^{4}+b^{4}\big)\geq(a+b)^{4}.\]
We have
\[a^{4}+b^{4}=\dfrac{a^{4}}{1}+\dfrac{b^{4}}{1}\geq\dfrac{\left(a^{2}+b^{2}\right)^{2}}{2}\geq\dfrac{\Big(\frac{(a+b)^{2}}{2}\Big)^{2}}{2}=\dfrac{(a+b)^{4}}{8}. \ _\square\]
For \(a,b,c\in\mathbb{R^{+}}\), prove that
\[\frac{a^{2}+b^{2}}{a+b}+\frac{b^{2}+c^{2}}{b+c}+\frac{c^{2}+a^{2}}{c+a}≥a+b+c.\]
Write the left side of the given inequality as
\[\dfrac{a^{2}}{a+b}+\dfrac{b^{2}}{a+b}+\dfrac{b^{2}}{b+c}+\dfrac{a^{2}}{c+a}+\dfrac{c^{2}}{a+c}+\dfrac{c^{2}}{c+b},\]
which is \(≥\frac{(2a+2b+2c)^{2}}{4(a+b+c)},\) so we are done. \(_\square\)
Now you can try to prove the Nesbitt's inequality.
Let \( a, b, c \in \mathbb{R}^+ \). Show that
\[ \frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{b + a} \ge \frac{3}{2}.\]
The numerator currently isn't a perfect square. Let's make it a perfect square:\[ \frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{b + a} = \frac{ a^2 } { ab+ac} + \frac{ b^2 } { ab + bc } + \frac{ c^2 } { bc + ac } . \]
We can then apply Titu's lemma to obtain
\[ \frac{ a^2 } { ab+ac} + \frac{ b^2 } { ab + bc } + \frac{ c^2 } { bc + ac } \geq \frac{ (a+b+c) ^2 } { 2 ( ab+bc+ca) } . \]
Now,
\[ \begin{array} { c r c l } & \frac{ (a+b+c) ^2 } { 2 ( ab+bc+ca) } & \geq & \frac{3}{2} \\ \Leftrightarrow & (a+b+c)^2 & \geq & 3 (ab+bc+ca) \\ \Leftrightarrow & a^2 + b^2 + c^2 & \geq & ab + bc + ca \\ \Leftrightarrow & (a-b)^2+(b-c)^2+(c-a)^2 & \geq & 0. \end{array}\]
Note that if we tried to apply Titu's lemma directly, we end up with
\[ \frac{a}{b + c} + \frac{b}{a + c} + \frac{c}{b + a} \geq \frac{ \big( \sqrt{a} + \sqrt{b} + \sqrt{c} \big)^2 } { 2 ( a + b + c ) }. \]
Then, the right-hand side actually has a maximum value of \( \frac{3}{2} \), so we cannot proceed any further. \(_\square\)
Another approach is to apply Titu's lemma to show that:
\[ \frac{1}{ b+c} + \frac{1}{ c+a} + \frac{1}{a+b} \geq \frac{ (1 + 1 + 1)^2 } { 2(a+b+c) }. \]
Hence,
\[ \frac{a+b+c}{ b+c} + \frac{a+b+c}{ c+a} + \frac{a+b+c}{a+b} \geq \frac{9}{2}. \]
Hence,
\[ \frac{a}{ b+c} + \frac{b}{ c+a} + \frac{c}{a+b} \geq \frac{9}{2} - 3 = \frac{3}{2} . \]
Generalized Form
Here is the generalized form:
For any positive integer \(m\), we have \[\frac{x_1^m}{a_1^{m-1}}+\frac{x_2^m}{a_2^{m-1}}+\cdots+\frac{x_n^m}{a_n^{m-1}}\ge\frac{(x_1+x_2+\cdots+x_n)^m}{(a_1+a_2+\cdots+a_n)^{m-1}}.\]
To learn how we reached this generalized form, follow the process below.
Let \(\displaystyle \bigg(\sum_{i=1}^n\sqrt[\frac{m}{m-1}]{a_i}^{\frac{m}{m-1}}\bigg)^{\frac{1}{\ \frac{m}{m-1}\ }}\left(\sum_{i=1}^n\dfrac{x_i^m}{\left(\frac{a_i}{\sqrt[m]{a_i}}\right)^m}\right)^{\frac{1}{m}} \ge \sum_{i=1}^n x_i,\) which is true by Hölder's inequality.
Then by simplifying, we get
\[\begin{split}\displaystyle \bigg(\sum_{i=1}^n a_i\bigg)^{\frac{m-1}{m}}\bigg(\sum_{i=1}^n\dfrac{x_i^m}{a_i^{m-1}}\bigg)^{\frac{1}{m}}&\ge \sum_{i=1}^n x_i \\\bigg(\sum_{i=1}^n a_i\bigg)^{m-1}\bigg(\sum_{i=1}^n\dfrac{x_i^m}{a_i^{m-1}}\bigg)&\ge \bigg(\sum_{i=1}^n x_i\bigg)^m \\ \bigg(\sum_{i=1}^n\dfrac{x_i^m}{a_i^{m-1}}\bigg)&\ge \dfrac{\bigg(\sum_{i=1}^n x_i\bigg)^m}{\bigg(\sum_{i=1}^n a_i\bigg)^{m-1}}.\end{split}\]
Thus we conclude that \({\frac{x_1^m}{a_1^{m-1}}+\frac{x_2^m}{a_2^{m-1}}+\cdots+\frac{x_n^m}{a_n^{m-1}}\ge\frac{(x_1+x_2+\cdots+x_n)^m}{(a_1+a_2+\cdots+a_n)^{m-1}}}\) for \( m \ge 2.\)
When we put \(m=2\) in the above inequality, we get
\[\frac{x_1^2}{a_1}+\frac{x_2^2}{a_2}+\cdots+\frac{x_n^2}{a_n}\ge\frac{(x_1+x_2+\cdots+x_n)^2}{(a_1+a_2+\cdots+a_n)}. \]
Problem Solving
We start with some examples followed by several problems to try.
[IMO/1995] Given \(a,b,c\in\mathbb{R^{+}} \text{ and } abc=1\), prove that
\[\frac{1}{a^{3}(b+c)}+\frac{1}{b^{3}(a+c)}+\frac{1}{c^{3}(a+b)}≥\frac{3}{2}.\]
We write the left side of the given inequality as \(\dfrac{\frac{1}{a^{2}}}{ab+ac}+\dfrac{\frac{1}{b^{2}}}{ba+bc}+\dfrac{\frac{1}{c^{2}}}{ca+cb}\).
This is \(\ge \dfrac{\big(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\big)^{2}}{2(ab+bc+ca)}\).
Since \(abc=1\), it follows that \(\big(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\big)^{2}=(ab+bc+ca)^{2}\).
Hence, the left-hand side of the given inequality is \(≥\dfrac{(ab+bc+ca)}{2}≥3\dfrac{\sqrt[3]{(abc)^{2}}}{2}=\dfrac{3}{2}\). \(_\square\)
For \(a,b,c\in\mathbb{R^{+}}\), prove that
\[\frac{a}{2a+b}+\frac{b}{2b+c}+\frac{c}{2c+a}≤1.\]
Let's modify the inequality that we have to prove as follows:
\[\begin{align} \left(\frac{a}{2a+b}-\frac{1}{2}\right)+\left(\frac{b}{2b+c}-\frac{1}{2}\right)+\left(\frac{c}{2c+a}-\frac{1}{2}\right) &\leq 1-\frac{3}{2} \\ \frac{2a-(2a+b)}{2(2a+b)}+\frac{2b-(2b+c)}{2(2b+c)}+\frac{2c-(2c+a)}{2(2c+a)} &\leq -\frac{1}{2} \\ -\frac{1}{2}\left( \frac{b}{2a+b}+\frac{c}{2b+c}+\frac{a}{2c+a}\right) &\leq -\frac{1}{2} \\ \frac{b}{2a+b}+\frac{c}{2b+c}+\frac{a}{2c+a} &\geq 1. \end{align}\]
Then we have
\[\frac{b^2}{2ab+b^2}+\frac{c^2}{2bc+c^2}+\frac{a^2}{2ca+a^2} \geq \frac{(a+b+c)^2}{a^2+b^2+c^2+2(ab+bc+ca)}=1. \ _\square\]
For the positive real numbers \(x, y,\) and \(z,\) where \(x+y+z=1,\) prove that
\[\displaystyle \sum_{cyc} \frac{x^3}{(x+y)^2} \ge \dfrac14.\]
What we should actually prove is that \(\frac{x^3}{(x+y)^2}+\frac{y^3}{(y+z)^2}+\frac{z^3}{(z+x)^2}\ge \frac14\). Thus by applying the above inequality on the LHS, we get
\[\begin{split}\dfrac{x^3}{(x+y)^2}+\dfrac{y^3}{(y+z)^2}+\dfrac{z^3}{(z+x)^2}&\ge \dfrac{(x+y+z)^3}{(x+y+y+z+z+x)^2} \\&=\dfrac{(x+y+z)^3}{4(x+y+z)^2} \\&=\dfrac{1}{4}.\end{split}\]
Hence proved. \(_\square\)
Try the following examples.
If \(x,y,z\) are positive real numbers such that \( x+y+z=3\), then find the minimum value of
\[E=\frac { 4yz+9xz+16xy }{ xyz }. \]
Let \(x,y\) and \(z\) be positive reals such that \(x+y+z=6\). Find the minimum value of the expression
\[\dfrac{16yz+36xz+64xy}{xyz}.\]
Find the minimum value of
\[ \frac { { x }^{ 2 }+{ y }^{ 2 } }{ z } +\frac { { z }^{ 2 }+{ x }^{ 2 } }{ y } +\frac { { y }^{ 2 }+{ z }^{ 2 } }{ x }, \]
subject to \( x + y + z = 2015 \) and \( x, y ,z > 0 \).
If \(a,b,c,d,e,f\) are positive real numbers which satisfy
\[a+b+c+d+e+f=7, \]
find the minimum value of
\[\frac { 1 }{ a } +\frac { 4 }{ b } +\frac { 9 }{ c } +\frac { 16 }{ d } +\frac { 25 }{ e } +\frac { 36 }{ f } . \]
There exists a smallest possible positive integer \(N\) such that
\[\dfrac{(x_1+2x_2+\cdots +2014x_{2014})^2}{x_1^2+x_2^2+\cdots +x_{2014}^2}\le N\]
for all real sequences \(\large \{x_i\}_{i=1}^{2014}\).
Find the sum of digits of \(N\).
\(a,b\) and \(c\) are three positive real numbers such that \(a+b+c=n\).
What is the smallest possible value of \(n\) such that
\[(ab+bc+ca)\left(\dfrac{a}{2b+3c}+\dfrac{b}{2c+3a}+\dfrac{c}{2a+3b}\right)\ge 20\]
for all possible ordered triplets \((a,b,c)?\)
\(a\) and \(b\) are real numbers such that \(a^{2} + b^{2} = 1,\) and always satisfy
\[ \frac{1}{1 + a^{2}}+ \frac{1}{1 + b^{2}} + \frac{1}{1 + ab} \geq \frac{x}{1 + \frac{(a+b)^{2}}{z}}.\]
Find \(\left\lfloor \frac{x}{z} \right\rfloor\) when the value of the LHS of the above inequality is the least.
\(\)
Details and Assumptions:
- \(x\) and \(z\) are positive integers.
- \(\lfloor \cdot \rfloor\) denotes the greatest integer function.