Mean of a function

In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain. In one variable, the mean of a function f(x) over the interval (a,b) is defined by

${\bar {f}}={\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx.$

Recall that a defining property of the average value ${\bar {y}}$ of finitely many numbers $y_{1},y_{2},\dots ,y_{n}$ is that $n{\bar {y}}=y_{1}+y_{2}+\cdots +y_{n}$. In other words, ${\bar {y}}$ is the constant value which when added to itself $n$ times equals the result of adding the $n$ terms of $y_{i}$. By analogy, a defining property of the average value ${\bar {f}}$ of a function over the interval $[a,b]$ is that

$\int _{a}^{b}{\bar {f}}\,dx=\int _{a}^{b}f(x)\,dx$

In other words, ${\bar {f}}$ is the constant value which when integrated over $[a,b]$ equals the result of integrating $f(x)$ over $[a,b]$. But by the second fundamental theorem of calculus, the integral of a constant ${\bar {f}}$ is just

$\int _{a}^{b}{\bar {f}}\,dx={\bar {f}}x{\bigr |}_{a}^{b}={\bar {f}}b-{\bar {f}}a=(b-a){\bar {f}}$

See also the first mean value theorem for integration, which guarantees that if $f$ is continuous then there exists a point $c\in (a,b)$ such that

$\int _{a}^{b}f(x)\,dx=f(c)(b-a)$

The point $f(c)$ is called the mean value of $f(x)$ on $[a,b]$. So we write ${\bar {f}}=f(c)$ and rearrange the preceding equation to get the above definition.

In several variables, the mean over a relatively compact domain U in a Euclidean space is defined by

${\bar {f}}={\frac {1}{{\hbox{Vol}}(U)}}\int _{U}f.$

This generalizes the arithmetic mean. On the other hand, it is also possible to generalize the geometric mean to functions by defining the geometric mean of f to be

$\exp \left({\frac {1}{{\hbox{Vol}}(U)}}\int _{U}\log f\right).$

More generally, in measure theory and probability theory, either sort of mean plays an important role. In this context, Jensen's inequality places sharp estimates on the relationship between these two different notions of the mean of a function.

There is also a harmonic average of functions and a quadratic average (or root mean square) of functions.

A major upgraded theory on mean of constants and mean of function(in China )

From 2006 through 2007, a new theory called "Extensive mean value & Dual-variable-isomorphic convex function" by Tim Liu, appeared in Chinese maths journal and online research repository, implies major theoritical advances in both mean value of constants and mean values(averages) of function.

In these articles, firstly two simple but novel concepts are introduced:

Isomorphic number

Let $X,\,U\subseteq \mathbb {R}$, and there is a strictly monotonic bijection $g\colon X\to U$. For any $x\in X$, $u=g(x)\in U$ is called the isomorphic number(or isomorphic variable) of $x$ generated by mapping $g$.

Dual-variable isomorphic function

Let $X,\,U,\,Y,\,V\subseteq \mathbb {R} ,~D\subseteq X,\,M\subseteq Y$. There are strictly monotonic bijections $g\colon X\to U,~h\colon Y\to V$, and function $f\colon D\to M$(M is the range). Let $E=g(D),~N=h(M)$.

• Function $\varphi \colon =(h\circ f\circ g^{-1})\colon E\to N$ is defined to be the dual-variable-isomorphic function of $f$ generated by mapping $g$, $h$.
• $g$ and $h$ are called the independent variable's generator mapping(function) of $\varphi$, and dependent variable's generator mapping(function) of $\varphi$ respectively.
• $E\subseteq U$ is called the isomorphic domain of $f$ generated by mapping $g$; $N\subseteq V$ is called the isomorphic range of $f$ generated by mapping $h$.

The item "isomorphic" is a borrowed concept from Algebra since in the former definition domain of $u$ is of same structure as that of $x$, and in the latter $\varphi$ is of same structure as of $f$. Simply comparable, The latter has 2 variables to apply bijections and the former only 1.

Isomorphic mean value of constants

After these 2 extensions, the Quasi-arithmetic mean of constants n-tuple of $x$ is redefined as the Isomorphic mean value of the constants:

$M_{f}(x_{1},\dots ,x_{n})=f^{-1}\left({\frac {f(x_{1})+\cdots +f(x_{n})}{n}}\right).$

because the Quasi-arithmetic mean or Generalised f-mean is the image (by $f^{-1}$ on $X$) of the mean of corresponding isomorphic numbers of n-tuple of $x$. This is a conceptual extension of arithmetic mean by a clearer idea than "quasi-" or "generalised", and the Isomorphic mean value is also referred to as extensive mean values of constants

Dual-variable-isomorphic mean value of function

Next, similarly the image (by $h^{-1}$ on $M$) of mean value(average) of dual-variable-isomorphic function $\varphi$ of $f$ generated by $g,h$ on $N$, is defined to be the (dual-variable-)isomorphic mean value of function $f$ on $D$ generated by mapping $g,h$. It can be denoted by $M_{f}{\bigl |}_{g,h}$,

$M_{f}{\bigl |}_{g,h}=h^{-1}\left({\frac {1}{{\hbox{Vol}}(E)}}\int _{E}h\circ f\circ g^{-1}\right)$.

Taking $[a,b]$ as the form of $D$, then:

$M_{f}{\bigl |}_{g,h}=h^{-1}\left({\frac {1}{g(b)-g(a)}}\int _{g(a)}^{g(b)}h\left(f(g^{-1}(u))\right)\mathrm {d} u\right)$.

Since the $h$ is a monotonic bijection, such image of mean value of $\varphi$ by $h^{-1}$ always exists in $M$, and it is not out of $[min{M},max{M}]$ if such boundaries exists. That's why it's a kind of mean value.

Since $g,h$ each can be identity mapping $(y=x)$ independently, or at the same time both be, or it could happen when $g=h$, or $g$ and $h$ could be inverse to each other, the above general form could have up to (but not limited to) 6 forms of variations, thus there are 6 sub-classes of (dual-variable-)isomorphic mean value of function(some of these also could be referred to as "Extensive mean value of function" ):

• Class I isomorphic mean value of function (Extensive mean value of function type I, Dependent-variable-isomorphic mean value of function)

$M_{f}{\bigl |}_{h}=h^{-1}\left({\frac {1}{b-a}}\int _{a}^{b}h{\big (}f(x){\big )}\mathrm {d} x\right)$.

• Class II isomorphic mean value of function (Extensive mean value of function type II, Independent-variable-isomorphic mean value of function)

$M_{f}{\bigl |}_{g}^{II}={\frac {1}{g(b)-g(a)}}\int _{g(a)}^{g(b)}f{\big (}g^{-1}(u){\big )}\mathrm {d} u$.

• Class III isomorphic mean value of function (Same-mapping dual-variable-isomorphic mean value of function)

$M_{f}{\bigl |}_{g,g}=g^{-1}\left({\frac {1}{g(b)-g(a)}}\int _{g(a)}^{g(b)}g\left(f(g^{-1}(u))\right)\mathrm {d} u\right)$.

• Class IV isomorphic mean value of function (General dual-variable-isomorphic mean value of function) where $g$≠$h$

$M_{f}{\bigl |}_{g,h}=h^{-1}\left({\frac {1}{g(b)-g(a)}}\int _{g(a)}^{g(b)}h\left(f(g^{-1}(u))\right)\mathrm {d} u\right)$.

• Arithmetic mean value of function

$M_{f}={\frac {1}{b-a}}\int _{a}^{b}f(x)\mathrm {d} x$.

• $f$’s dual-variable-isomorphic mean value generated by $f,f^{-1}$

$M_{f}{\bigl |}_{f,f^{-1}}=f\left({\frac {1}{f(b)-f(a)}}\int _{f(a)}^{f(b)}f^{-1}(u)\mathrm {d} u\right)$.

Class I isomorphic mean value of function

It's the integral-wise evolved form of Isomorphic mean value of the constants, above mentioned Geometric mean of functions，harmonic average of functions and quadratic average (or root mean square) of functions all belong to this class. It also simplifies to Arithmetic mean value of function when $h$ is identity mapping.

The values of 2 Class I isomorphic mean values of a function $f$ generated by different mapping $h_{1}$, $h_{2}$, can be compared on the aid of the integral corollary form of Jensen's Inequality.

It's pointed out by Tim in his paper, that:

• The geometric mean of $sinx$ on $(0,\pi )$ (2 endpoints are excluded) is 1/2.
• The geometric mean of all parallel chords(as a function) in a circle diametered $D$, is $2D/e$.
• The geometric mean of $y=x$ on $(0,1)$ is $1/e$.
• ...

They are all smaller than the corresponding arithmetic means of the same function.

Class II isomorphic mean value of function

It's the sibling concept of class I isomorphic mean value of function, and it is not easy to perceive if without the introduction of "dual-variable-isomorphic function" while evolution from arithmetic mean value of constants to isomorphic mean value of constants and even up to Class I isomorphic mean value of function is easy.

In Second Mean Value Theorem for Integrals， there's a mean value of function: $f(\xi )$, that is just an instance of Class II isomorphic mean value of function.

It also simplifies to Arithmetic mean value of function when $g$ is identity mapping.

It was discovered that if $f$ is also monotonic and has its inverse function, then $M_{f}{\bigl |}_{g}^{II}$ is centrosymmetric to $M_{g}{\bigl |}_{f}^{II}$ with regards to their mean value's location in their each ranges.

Class III isomorphic mean value of function and Class IV isomorphic mean value of function

These are more complicated than class I and class II isomorphic mean value of function.

Arithmetic mean value of function

It's a indepentent class of dual-variable-isomorphic mean value of function, when $g,h$ are all identity mapping. While it's the special case of Class I through Class IV respectively.

The "dual-variable-isomorphic function" also leads to a sibling theory to "isomorphic mean value of funtion", that is "Dual-variable-isomorphic convex function". One of its usage is to facilitate the comparison of 2 "Isomorphic mean value of constants"(generalised-f mean).

See also

• Mean