Advanced Analysis

Basic Inequalities

1. Conjugate Exponents and \(\ell^p\) Norms

Conjugate Exponents If \(p,q\) are real numbers greater than \(1\), we say that they are conjugate exponents when they satisfy
\[ \frac{1}{p} + \frac{1}{q} = 1. \]
We also take the convention that \(1\) and \(\infty\) are considered conjugate exponents.

\(\ell^p\) Norms Defined For \(p \in [1,\infty)\) and \(x := (x_1,\ldots,x_n)\), define
\[ ||x||_p := \left( \sum_{j=1}^n |x_j|^p \right)^{\frac{1}{p}}\]
for any \(x \in {\mathbb C}^n\). Similarly, define
\[ ||x||_\infty := \max_{j=1,\ldots,n} |x_j|. \]

2. Young's Inequality for Products

Video. Young's Inequality
Theorem
If \(a\) and \(b\) are nonnegative real numbers and if \(p,q\) are conjugate exponents in \((1,\infty)\), then
\[ ab \leq \frac{a^p}{p} + \frac{b^q}{q}\]
with equality if and only if \(a^p = b^q\).
Proof
Meta (Main Idea)
This is a consequence of convexity of the exponential function. Write \(a^p = e^{t_1}\) and \(b^q := e^{t_2}\) for some real \(t_1,t_2\) and apply convexity. Since the exponential function is strictly convex,
\[ e^{\frac{1}{p} t_1 + \frac{1}{q} t_2} \leq \frac{1}{p} e^{t_1} + \frac{1}{q} e^{t_2}\]
with equality if and only if \(t_1 = t_2\).

3. Hölder's Inequality

Video. Hölder's Inequality
Theorem
If \(x := (x_1,\ldots,x_n)\) and \(y := (y_1,\ldots,y_n)\) belong to \({\mathbb C}^n\), then
\[ \left| \sum_{j=1}^n x_j y_j \right| \leq ||x||_p ||y||_q \]
for any conjugate exponents \(p,q \in [1,\infty]\).
Proof
Meta (Main Idea)
If either \(x = 0\) or \(y=0\), then the inequality holds trivially because both sides are zero. Otherwise, as both sides are homogeneous in both \(x\) and \(y\), it may be assumed that \(||x||_p = 1 = ||y||_q\). Use the triangle inequality and apply Young's inequality coordinate-wise.

4. Minkowski's Inequality (Triangle Inequality)

Video. Minkowski Inequality
Theorem
If \(x := (x_1,\ldots,x_n)\) and \(y := (y_1,\ldots,y_n)\) belong to \({\mathbb C}^n\), then for any exponent \(p \in [1,\infty]\),
\[ ||x+y||_p \leq ||x||_p + ||y||_p. \]
Proof
Meta (Main Idea)
If \(p > 1\), let \(q := p/(p-1)\) (the conjugate exponent). If \(x+y = 0\), then the inequality is trivial. Otherwise
\[ |x_j + y_j|^p \leq (|x_j| + |y_j|) |x_j + y_j|^{\frac{p}{q}}\]
Now write
\[{}||x_j+y_j||_p^p{}\]
\[{}\leq \sum_{j=1}^n |x_j| |x_j + y_j|^{\frac{p}{q}}{}\]
\[{}+ \sum_{j=1}^n |y_j| |x_j + y_j|^{\frac{p}{q}}{}\]
and apply Hölder's inequality to both sums.

5. Comparability of the \(\ell^p\) Norms

Video. Comparability of Norms
Theorem
If \(1 \leq p \leq q \leq \infty\), then for any \(x \in {\mathbb C}^n\),
\[ ||x||_q \leq ||x||_p \leq n^{\frac{1}{p} - \frac{1}{q}} ||x||_q. \]
Proof
Meta (Main Idea)
To prove that \(||x||_q \leq ||x||_p\),
\[{}\sum_{j=1}^n |x_j|^q{}\]
\[{}= \sum_{j=1}^n |x_j|^{p} |x_j|^{q-p}{}\]
\[{}\leq \sum_{j=1}^n |x_j|^p ||x||_p^{q-p}{}\]
\[{}= ||x||_p^q.{}\]
For the second inequality, note that
\[ \frac{q}{p} \text{ and } \frac{q}{q-p}\]
are dual exponents. Write \(|x_j|^p = |x_j|^p \cdot 1\) and apply Hölder's inequality with one vector having coordinates \(|x_j|^p\) and the other having all \(1\)s.