Category:Chebyshev's Sum Inequality
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This category contains pages concerning Chebyshev's Sum Inequality:
Discrete Version
Let $a_1, a_2, \ldots, a_n$ be real numbers such that:
- $a_1 \ge a_2 \ge \cdots \ge a_n$
Let $b_1, b_2, \ldots, b_n$ be real numbers such that:
- $b_1 \ge b_2 \ge \cdots \ge b_n$
Then:
- $\ds \dfrac 1 n \sum_{k \mathop = 1}^n a_k b_k \ge \paren {\dfrac 1 n \sum_{k \mathop = 1}^n a_k} \paren {\dfrac 1 n \sum_{k \mathop = 1}^n b_k}$
Continuous Version
Let $u, v: \closedint 0 1 \to \R$ be integrable functions.
This article, or a section of it, needs explaining. In particular: Is it important in what sense integrable: Darboux, Lebesgue, etc.? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Let $u$ and $v$ both be either increasing or decreasing.
Then:
- $\ds \paren {\int_0^1 u \rd x} \cdot \paren {\int_0^1 v \rd x} \le \int_0^1 u v\rd x$
Pages in category "Chebyshev's Sum Inequality"
The following 8 pages are in this category, out of 8 total.
C
- Chebyshev's Sum Inequality
- Chebyshev's Sum Inequality (Discrete)
- Chebyshev's Sum Inequality/Also known as
- Chebyshev's Sum Inequality/Continuous
- Chebyshev's Sum Inequality/Discrete
- Chebyshev's Sum Inequality/Discrete/Also presented as
- Chebyshev's Sum Inequality/Discrete/Equality
- Chebyshev's Sum Inequality/Discrete/Proof