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.
\[ ||x||_p := \left( \sum_{j=1}^n |x_j|^p \right)^{\frac{1}{p}}\]
for any \(x \in {\mathbb C}^n\). Similarly, define
\[ ||x||_\infty := \max_{j=1,\ldots,n} |x_j|. \]
2. Young's Inequality for Products
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
3. 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
4. Minkowski's Inequality (Triangle 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