1. 1.

   Consider two real functions$f(x),g(x)$, defined and R-integrable on the finite and closed interval$[a,b](a<b)$. If we designate by$\bar{f}$an average of the values ​​of$f(\bar{x})$, so a number such that

   Report issue for preceding element

we have the first formula of the average

which is valid if$g(x)$does not change sign (is constantly$\geq 0$or constantly$\leqq 0$on$[a,b]$) And$f(x)$is arbitrary.

It can be shown that, under the hypothesis of its continuity, the invariance of the sign of$g(x)$is also necessary for the validity of formula (1), for$f(x)$any.

Indeed, let us suppose that$g(x)$is continuous and that it changes sign on$[a,b]$. Without restricting the generality, we can assume that

(because otherwise it is enough to take$-g(x)$instead of$g(x)$) and then there is a point$x_{0}\in(a,b)$such as$g\left(x_{0}\right)<0$. As a result of the continuity
of$g(x)$, if$\varepsilon>0$is quite small we have$a<x_{0}-\varepsilon<x_{0}<x_{0}+\varepsilon<b$And$g(x)$is negative on the interval ($x_{0}-\varepsilon,x_{0}+\varepsilon$).

Now we will build a function$f^{*}(x)$which is:
$1^{\circ}$. Continuous, positive and at most equal to$1\operatorname{on}\left(x_{0}-\varepsilon,x_{0}+\varepsilon\right)$.
$2^{\circ}$. None on$\left[a,x_{0}-\varepsilon\right]$and on$\left[x_{0}+\varepsilon,b\right]$.
The function$f^{*}(x)$East$R$-integrable on$[a,b]$and we have$0\leqq\bar{f}^{*}\leqq 1$.
We can, for example, take
(3)

If we seek to apply formula (1), taking for$f(x)$the function$f^{*}(x)$, we have, taking into account (2),

which shows us that equality (1) is impossible.
We can therefore state the following property:
I. For the first formula of the mean (1) to be true for any continuous function$g(x)$and for any function$R$-integrable$f(x)$, it is necessary and sufficient that$g(x)$does not change sign on$[a,b]$.
2. The integral formula (1) corresponds to the formula for the average "in finite terms"

where the suites$\left(a_{i}\right),\left(b_{i}\right)$are real and

We designate by$\left(c_{i}\right)$the sequel to$n$terms$c_{1},c_{2},\ldots,c_{n},(n>1)$.
The formula for the average (4) is true for any sequence$\left(a_{i}\right)$if the terms of the sequence$\left(b_{i}\right)$are of the same sign (all$\geqq 0$or all of them$\leq 0$).

It is still easy to see that the invariance of the sign of the$\overline{b_{i}}$is necessary for the mean formula (4) to be true for any sequence$\left(a_{i}\right)$. Indeed, let us suppose that the$bi}$are not all of the same sign. We can assume, without restricting the generality, that$\sum_{i=1}^{n}b_{i}\geq 0$(because otherwise we reason in the same way on the sequence ($-bi}$)). Then there is a clue$k$such as$b_{k}<0$. If we then take$a_{k}=1,a_{i}=0$, For$i\neq k$, We have$0\leqq\bar{a}\leqq 1$And$\sum_{i=1}^{n}a_{i}b_{i}=b_{k}<0,\bar{a}\sum_{l=1}^{n}b_{i}\geq 0$and equality (4) is impossible.

We can therefore state the following property:
I'. For the mean formula (4) to be true for any sequence$\left(a_{i}\right)$, it is necessary and sufficient that the terms of the sequence ($bi}$) are of the same sign.

Property I can also be deduced from property I' by a passage to the limit, taking into account the definition of the integral R. Without insisting in more detail, let us say that properties II, III, IV which will follow result in the same way respectively from the properties$\mathrm{II}^{\prime},\mathrm{III}^{\prime},\mathrm{IV}^{\prime}$.
3. C. BONFERRONI demonstrated [1] that the first formula of the mean (1) is true for any monotonic function$f(x)$if$g(x)$is R-integrable and if its integral$G(x)=\int^{x}g(t)dt$remains between$\boldsymbol{0}$And$G(b)$For$x\in[a,b]$. The last property means that we have$0\leqq G(x)\leqq G(b)$For$x\in[a,b]$resp.$G(b)\leqq G(x)\leqq 0$For$x\in[a,b]$following that$0\leqq G(b)$resp.$G(b)\leqq 0$.

It can still be demonstrated that the condition imposed on$g(x)$is necessary. For this, let's take the function

Or$a\leqq\lambda\leqq b$(We have$f(x)=1$on$[a,b]$when$\lambda=b$). This function is monotone (therefore a fortiori R-integrable) and we have$0\leq\bar{f}\leq 1$. Formula (1) then gives us$G(\lambda)=\bar{f}G(b)$For$\lambda\in[a,b]$, which demonstrates the property.

We can therefore state the following property:
II. For the first formula of the mean (1) to be true for any monotonic function$f(x)$, it is necessary and sufficient that the integral$G(x)=\int^{x}g(t)dt$, of the function$R$-integrable$g(x)$, remains between 0 and$G(b)$For$x\in[a,b]$
4. C. Bonferroni obtains the sufficiency of the condition of property II by a passage to the limit of the corresponding property relative to formula (4).

If we consider the partial sums$s_{i}=b_{1}+b_{2}+\ldots+b_{i}$,$i=1,2,\ldots,n$of the sequel$\left(b_{i}\right)$, the mean formula (4) is verified for any monotonic sequence$\left(a_{i}\right)$if the terms of the sequence$\left(s_{i}\right)$remain included (in the broad sense) between 0 and$s_{n}$.

C. Bonferroni's demonstration is as follows. Note that if the$s_{i},i=1,2,\ldots,n$are between 0 and$s_{n},s_{n}-s_{i},i=1,2,\ldots,n$are also between 0 and$s_{n}$. I have the monotony of the sequence ($a_{i}$) shows us,
on the one hand, that$\bar{a}$is between$a_{1}$And$year}$and, on the other hand, that using the Abel transformation formula, we have

The formula for the mean (4) follows immediately.
The necessity of the condition results from taking the monotonic sequence$\left(a_{i}\right)$, Or$a_{1}=a_{2}=\ldots=a_{i}=1,a_{i+1}=a_{i+2}=\ldots=a_{n}=0$.

We can state the following property
II'. For the mean formula (4) to be true for any monotonic sequence$\left(a_{i}\right)$it is necessary and sufficient that the terms of the sequence$\left(s_{i}\right)$partial sequels of the sequence$\left(b_{i}\right)$remain between 0 and$s_{n}$.
5. Let us now consider the second formula for the mean
where t

This formula is valid for any function$f(x)$R-integrable if the function$g(x)$is monotonous on$[a,b]$.

Let's suppose that$g(x)$has a continuous derivative$g^{\prime}(x)$on$[a,b]$. We can then demonstrate that the monotonicity of$g(x)$is necessary for formula (5) to hold for$f(x)$any. Indeed, if$g^{\prime}(x)$is a continuous function and if we set$F(x)=\int_{a}^{x}f(t)dt$, We have

and formula (5) becomes

so comes back to the first formula of the average$(F(\xi)=\bar{F})$.
In our case the monotony of$g(x)$on$[a,b]$is equivalent to the fact that the derivative$g^{\prime}(x)$does not change sign on$[a,b]$. The necessity of the condition we have in mind results as in nr. 1. We must only take the function$g^{\prime}(x)$instead of$g(x)$and for$F(x)$. a function$f^{*}(x)$
suitable. Since by construction$F(x)$is an integral, to satisfy the conditions$1^{\circ},2^{\circ}$from No. 1, it is sufficient, for example, to take for$F(x)$1st function

This amounts, moreover, to taking for$f(x)$the function

We can state the following property:
III. For the mean formula (5) to be true for any function$g(x)$having a continuous derivative on$[a,b]$and for any function$f(x)R$-integrable on$[a,b]$, it is necessary and sufficient that$g(x)$be monotonous on$[a,b]$.
6. Formula (5) also corresponds to a formula for the average "in finite terms"
where$\vec{r}$is an average value of the$n-1$first terms of the sequence ($r_{i}$) partial sequences$r_{i}=a_{1}+a_{2}+\ldots+a_{i},\quad i=1,2,\ldots,n$of the following ($a_{i}$), SO

The mean formula (6) is true for any sequence$\left(a_{i}\right)$if the sequel$\left(b_{i}\right)$is monotonic. The demonstration is simple, since if we notice that

we return to the first formula of the average.
The necessity of the monotony of the sequence ($b_{i}$), so that (6) remains true for any sequence ($a_{i}$), immediately results in taking$a_{i}=1,a_{i+1}=-1$And$a_{k}=0$, For$k\neq i,i+1$, successively for$i=1,2,\ldots,n-1$.

We can therefore state the following property:
III'. For the mean formula (6) to be true for any sequence:$\left(a_{i}\right)$, it is necessary and sufficient that the following$\left(b_{i}\right)$be monotonous.
7. C. Bonferroni in his cited work [1] also demonstrated that the formula for the mean (5) is valid for any function$f(x)$whose integral$F(x)=\int f(t)dt$is monotonous if$g(x)$remains understood (in the broad sense): between$g(a)$And$g(b)$For$x\in[a,b]$.

Assuming$g(x)$continues, the stated condition is also necessary. Indeed, suppose that$g(x)$let's continue and take for$f(x)$function (3), where$x_{0}$is a point of$(a,b)$And$\varepsilon$a sufficiently small positive number. Assuming that the mean formula (5) holds, we have one of the equalities
$\frac{1}{2\varepsilon}\int_{x_{\sigma}\varepsilon}^{x_{0}+\varepsilon}g(x)dx=\left\{\begin{array}[]{l}g(b),\text{ pour }\xi\leqq x_{0}-\varepsilon,\\ \frac{\left(\xi-x_{0}+\varepsilon\right)g(a)+\left(x_{0}+\varepsilon-\xi\right)g(b)}{2\varepsilon},\text{ pour }\xi\in\left(x_{0}-\varepsilon,x_{0}+\varepsilon\right),\\ g(a),\text{ pour }\xi\geq x_{0}+\xi,\end{array}\right.$
We therefore see that for$\varepsilon$positive and quite small,
(7)