Average

with weight $q=(q_1,\dots,q_n)$, $q_i>0$, $\sum q_i=1$, of a set of real numbers $a=(a_1,\dots,a_n)$

A variable

$$\mathfrak M_\phi(a,q)=\phi^{-1}\left(\sum_iq_i\phi(a_i)\right),$$

where $\phi(x)$ is a continuous strictly-monotone function on $\mathbf R$. When $\phi(x)=x^r$, one obtains

$$\mathfrak M_r(a,q)=\left(\sum_iq_ia_i^r\right)^{1/r}$$

and, in particular, when $r=1$, $q_i=1/n$, $i=1,\dots,n,$ $\mathfrak M_r(a,1/n)=\mathfrak A(a)$ will be the arithmetic average of the numbers $a_1,\dots,a_n$, while when $r=-1$, it will be the harmonic average. The concepts of the geometric average $\mathfrak G(a)=(\prod_ia_i)^{1/n}$ and the weighted geometric average

$$\mathfrak G(a,p)=\left(\prod_ia_i^{p_i}\right)^{1/\sum_ip_i}$$

are introduced separately.

One of the basic results of the theory of averages is the inequality $\mathfrak G(a)<\mathfrak A(a)$, except when all $a_i$ are equal to each other. Other results are:

1) $\mathfrak M_\phi(ka,p)=k\mathfrak M_\phi(a,p),k>0$;

2) $\mathfrak M_\psi(a,p)=\mathfrak M_\phi(a,p)$ if and only if $\psi=\alpha\phi+\beta$, $\beta\in\mathbf R$, $\alpha\neq0$;

3) $\mathfrak M_\psi(a,p)\leq\mathfrak M_\phi(a,p)$ if and only if $\phi\circ\psi^{-1}$ is a convex function; in particular $\mathfrak M_r(a,p)\leq\mathfrak M_s(a,p)$ if $r<s$.

The concept of an average can be extended to infinite sequences under the assumption that the corresponding series and products converge, and to other functions. The following is such an example:

$$\mathfrak M_\phi(f,p)=\frac{\phi^{-1}\left(\int\limits_a^bp(x)\phi(f(x))dx\right)}{\int\limits_a^bp(x)dx},$$

given the condition that $f(x)\geq0$ almost everywhere on the corresponding interval and that $p(x)>0$. Thus,

$$\int\limits_a^bf(x)p(x)dx\leq\mathfrak M_\phi(f,p)\int\limits_a^bp(x)dx.$$

References

 [1] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)