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Chebyshev's sum inequality


     Another article treats Chebyshev's inequality in probability theory.

In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if

     a_1 \geq a_2 \geq \cdots \geq a_n

and

     b_1 \geq b_2 \geq \cdots \geq b_n,

then

     n \sum_{k=1}^n a_kb_k \geq \left(\sum_{k=1}^n a_k\right)\left(\sum_{k=1}^n b_k\right).

Similarly, if

     a_1 \geq a_2 \geq \cdots \geq a_n

and

     b_1 \leq b_2 \leq \cdots \leq b_n,

then

     n \sum_{k=1}^n a_kb_k \leq \left(\sum_{k=1}^n a_k\right)\left(\sum_{k=1}^n b_k\right).

Chebyshev's sum inequality follows from the rearrangement inequality.

There is also a continuous version of Chebyshev's inequality:

If f and g are real-valued, integrable functions over , both increasing or both decreasing, then

      \int fg \geq \int f \int g.\,

This can be generalized to integrals over any space, as well as products of countable integrals.

    
   
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