In mathematics of
special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by ...
, the Neuman–Sándor mean ''M'', of two positive and unequal numbers ''a'' and ''b'', is defined as:
:
This mean interpolates the inequality of the unweighted arithmetic mean ''A'' = (''a'' + ''b'')/2) and of the second Seiffert mean ''T'' defined as:
:
so that ''A'' < ''M'' < ''T''.
The ''M''(''a'',''b'') mean, introduced by
Edward Neuman and József Sándor, has recently been the subject of intensive research and many remarkable inequalities for this mean can be found in the literature. Several authors obtained sharp and optimal bounds for the Neuman–Sándor mean. Neuman and others utilized this mean to study other bivariate means and inequalities.
See also
*
Mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
*
Arithmetic mean
In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the ''mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The colle ...
*
Geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
*
Stolarsky mean In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975.
Definition
For two positive real numbers ''x'', ''y'' the Stolarsky Mean is defined as:
:
\begin
S_p(x,y)
& = ...
*
Identric mean
The identric mean of two positive real numbers ''x'', ''y'' is defined as:
:
\begin
I(x,y)
&=
\frac\cdot
\lim_
\sqrt xi-\eta\\ pt&=
\lim_
\exp\left(\frac-1\right)
\\ pt&=
\begin
x & \textx=y \\ pt\frac \sqrt -y& \text
\end
\end
It can be de ...
* Means in
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
[Gheorghe Toader and Iulia Costin. 2017. Means in Mathematical Analysis: Bivariate Means. 1st Edition. Academic Press. eBook , Paperback . https://www.elsevier.com/books/means-in-mathematical-analysis/toader/978-0-12-811080-5]
References
{{DEFAULTSORT:Neuman-Sándor mean
Means
Special functions