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By comparison, Chebyshev's inequality states that all but a 1/N fraction of the sample will lie within √ N standard deviations of the mean. Since there are N samples, this means that no samples will lie outside √ N standard deviations of the mean, which is worse than Samuelson's inequality.
Brezis–Gallouet inequality; Carleman's inequality; Chebyshev–Markov–Stieltjes inequalities; Chebyshev's sum inequality; Clarkson's inequalities; Eilenberg's inequality; Fekete–Szegő inequality; Fenchel's inequality; Friedrichs's inequality; Gagliardo–Nirenberg interpolation inequality; Gårding's inequality; Grothendieck inequality ...
The first Chebyshev function ϑ (x) or θ (x) is given by = where denotes the natural logarithm, with the sum extending over all prime numbers p that are less than or equal to x. The second Chebyshev function ψ (x) is defined similarly, with the sum extending over all prime powers not exceeding x
3. The sum function, = (, …,), is a special case of a function of n variables. This function changes in a bounded way: if variable i is changed, the value of f changes by at most <. Hence, McDiarmid's inequality can also be used and it yields a similar bound:
Chebyshev's sum inequality, about sums and products of decreasing sequences Chebyshev's equioscillation theorem , on the approximation of continuous functions with polynomials The statement that if the function π ( x ) ln x / x {\textstyle \pi (x)\ln x/x} has a limit at infinity, then the limit is 1 (where π is the prime-counting function).
Let P 0,P 1, ...,P m be the first m + 1 orthogonal polynomials [clarification needed] with respect to μ ∈ C, and let ξ 1,...ξ m be the zeros of P m. It is not hard to see that the polynomials P 0 , P 1 , ..., P m -1 and the numbers ξ 1 ,... ξ m are the same for every μ ∈ C , and therefore are determined uniquely by m 0 ,..., m 2 m -1 .
In probability theory, the multidimensional Chebyshev's inequality [1] is a generalization of Chebyshev's inequality, which puts a bound on the probability of the event that a random variable differs from its expected value by more than a specified amount.
There is also a continuous version of Chebyshev's sum inequality: If f and g are real-valued, integrable functions over [a, b], ...