View source for Majorant and minorant

Two functions, the values of the first being not less and the values of the second being not greater than the corresponding values of a given function (for all admissible values of the independent variable).

For functions represented by a power series, a majorant is e.g. the sum of a power series with positive coefficients which are not less than the absolute values of the corresponding coefficients of the given series.

A majorant (minorant) of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062150/m0621501.png" /> of an ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062150/m0621502.png" /> is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062150/m0621503.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062150/m0621504.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062150/m0621505.png" />) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062150/m0621506.png" />.

In the theory of integral and differential equations, a majorant (minorant) or majorant function (minorant function) for some function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062150/m0621507.png" /> is a continuous function whose [[Dini derivative|Dini derivative]] at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062150/m0621508.png" /> is not less (not greater) than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062150/m0621509.png" /> and is different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062150/m06215010.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062150/m06215011.png" />. The difference between any majorant and any minorant is a non-decreasing function. Any summable function on an interval has absolutely-continuous majorants and minorants which are arbitrarily close to its indefinite [[Lebesgue integral|Lebesgue integral]]. The notion of majorant and minorant can be generalized to the case of additive set functions and also to the case when the derivatives are taken in some generalized sense.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Theory of sets" , Addison-Wesley  (1968)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1952)  (Translated from French)</TD></TR></table>



====Comments====
The terms  "majorant"  and  "minorant"  are rarely used in the sense 3) in English; instead the terms  "upper boundupper bound"  and  "lower boundlower bound"  are in common use.

See also [[Lower bound|Lower bound]]; [[Upper and lower bounds|Upper and lower bounds]].