Hölder's Inequality for Sums/Formulation 2/Equality
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Theorem
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that:
- $\dfrac 1 p + \dfrac 1 q = 1$
Let $\GF \in \set {\R, \C}$, that is, $\GF$ represents the set of either the real numbers or the complex numbers.
Let $\sequence {x_n}_{n \mathop \in \N}$ and $\sequence {y_n}_{n \mathop \in \N}$ be (possibly infinite) convergent sequences in $\GF$.
Hölder's Inequality for Sums states that:
- $\ds \sum_{k \mathop \in \N} \size {x_k y_k} \le \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$
We have that:
- $\ds \sum_{k \mathop \in \N} \size {x_k y_k} = \paren {\sum_{k \mathop \in \N} \size {x_k}^p}^{1 / p} \paren {\sum_{k \mathop \in \N} \size {y_k}^q}^{1 / q}$
- $\forall k \in \N: \size {y_k} = c \size {x_k}^{p - 1}$
Proof
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Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.2$ Inequalities: Hölder's Inequality for Sums: $3.2.8$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 36$: Inequalities: Hölder's Inequality