monotonically nonincreasing

smsn for all m>n

Similarly, a real function f(x) is monotonically nonincreasing if

f(x)f(y) for all x>y

Compare this to monotonically decreasing.

Conflict note. In some contexts, such as [1], this is called monotonically decreasing (in turn, our “monotonically decreasing” is called “strictly decreasing”). This is unfortunately counter-intuitive, since a sequence or functionMathworldPlanetmath that is “flat” (such as f(x)=1) is somehow “decreasing.” Beware!

1 Examples

  • (sn)=1,0,-1,-2, is monotonically nonincreasing. It is also monotonically decreasing.

  • (sn)=1,1,1,1, is nonincreasing but not monotonically decreasing.

  • (sn)=(1n+1) is nonincreasing (note that n is nonnegative).

  • (sn)=1,1,2,1,1, is not nonincreasing. It also happens to fail to be monotonically nondecreasing.

  • (sn)=1,2,3,4,5, is not nonincreasing, rather it is nondecreasing (and monotonically increasing).


  • 1 decreasing,” from the NIST Dictionary of Algorithms and Data Structures, Paul E. Black, ed.
Title monotonically nonincreasing
Canonical name MonotonicallyNonincreasing
Date of creation 2013-03-22 12:22:32
Last modified on 2013-03-22 12:22:32
Owner akrowne (2)
Last modified by akrowne (2)
Numerical id 9
Author akrowne (2)
Entry type Definition
Classification msc 40-00
Synonym monotone nonincreasing
Synonym nonincreasing
Related topic MonotonicallyNondecreasing