Presented on April 14, 1969

1. 1.

   Let us consider a real linear (therefore additive and homogeneous) functional$R[f]$, defined on a vector space S formed by real and continuous functions$f(x)$, defined on a given interval I (of non-zero length) of the real axis. We always assume that S contains all the polynomials. The space$S$can coincide with the set of all continuous functions$f:\mathrm{I}\rightarrow\mathbf{R}$but can also be more restricted. In the following sections, the set will be specified when necessary.$S$and the nature of its elements.

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The degree of accuracy of$R[f]$is the whole$m\geqq-1$who enjoys the following property:

The degree of accuracy may not exist, but if it does, it is well-defined. The existence of a degree of accuracy equal to$m$is equivalent to the fact that the linear functional$R[f]$vanishes on any polynomial of degree$m$but is different from zero on at least one polynomial of degree$m+1$.

When necessary, the nature of the linear functional will be further specified.$R[f]$.

Let us recall the following definition of the simplicity of the linear functional$R[f]$ :

The linear functional$R[f]$is said to be of simple form if there exists an integer$m\geqq-1$independent of the function$f(x)$, such as for all$f(x)\in\mathrm{S}$we have

Or$K$is a non-zero constant independent of the function$f(x)$And$\xi_{\alpha},\alpha=1,2,\ldots,m+2$are$m+2$distinct points of the interval I, generally depending on the function$f(x)$.

The number$m$is determined completely and is precisely the degree of accuracy of$R[f]$Moreover, we have$K=R[x^{m+1}]$.

In formula (1) we denote by$\left[y_{1},y_{2},\ldots,y_{r};f\right]$the difference divided, of order$r-1$of the function$f(x)$on the points or nodes (distinct or not)$y_{1},y_{2},\ldots,y_{r}$.

The theory of higher-order convex functions allows us to find various criteria for the simplicity of the linear functional$R[f]$For example, such a criterion can be stated as follows:

Theorem 1. A necessary and sufficient condition for the linear functional$R[f]$degree of accuracy$m$, or of the simple form is that one has$R[f]\neq 0$for any function$f(x)\in\mathrm{S}$convex of order$m$.

A function$f(x)$is said to be convex of order$m$on I if all its differences divided$\left[x_{1},x_{2},\ldots,x_{m_{+2}};f\right]$, of order$m+1$, on nodes$x_{1},x_{2},\ldots,x_{m_{+2}}\in\mathrm{I}$distinct, are positive. If these divided differences are all non-negative, the function is said to be non-concave of order$m$(on I). Finally, if the differences divided by order$m+1$of the function$f(x)$are all negative respectively all non-positive, this function is said to be concave respectively non-convex of order$m$By passing the function$f(x)$to the function -$f(x)$, the properties concerning concave and non-convex functions of order respectively$m$are generally deduced from the corresponding properties of convex and non-concave functions of order$m$.

When$R[f]$is the degree of accuracy$m$and is of the simple form we have

for any function$f(x)\in\mathrm{S}$convex of order$m$Indeed, first$x^{m+1}$is indeed convex of order$m$Then, if$R\left[x^{m+1}\right]R[f]\leqq 0$for a function$f(x)$convex of order$m$, for the function$f_{1}(x)=\left\{R\left[x^{m+1}\right]\right\}^{2}f(x)$-$-R\left[x^{m+1}\right]R[f]x^{m+1}$, which is still convex of order$m$we would have$R\left[f_{1}\right]=0$which, according to theorem 1, is impossible.

Under the same conditions if$f(x)$is a non-concave function of order$m$we have

Indeed, for all$\varepsilon>0$, the function$f(x) + ε x^{m+1}$is convex of order$m$and so we have$R\left[x^{m+1}\right]\left\{R[f]+\varepsilon R\left[x^{m+1}\right]\right\}=R\left[x^{m+1}\right]R[f]++\varepsilon\left\{R\left[x^{m+1}\right]\right\}^{2}>0$, hence, by making it tend$ε$towards 0, we deduce inequality (3).

For the properties of higher-order convex functions, for the notion of simplicity of a linear functional, and for various other properties used in this work, see my previous work. For example, my work in "Mathematica" [4].

If$m\geqq 0$, one can even assert that the points$\xi_{\alpha},\alpha=1,2,\ldots$,$m+2$in (1) are inside the interval I.

If$m\geqq 0$, if$R[f]$is the degree of accuracy$m$, is of the simple form and if$f(x)$to a derivative$f^{(m+1)}(x)$order$m+1$inside I, we have

Or$\xi$is a point inside the interval I.
Formulas (1), (4) allow, in the case of simplicity, the delimiting of the functional$R[f]$when we know the boundaries of the divided difference of order$m+1$of the function$f(x)$, or the boundaries of the$(m+1)th$derivative$f^{(m+1)}(x)$, assumed to exist, of this function.
2. Suppose that the linear functional$R[f]$either defined on the set S of continuous functions on I and having a derivative$f^{(m+1)}(x)$order$m+1$on the interior of I. In addition, we assume$m\geqq 0$, that$R[f]$either degree of accuracy$m$and of the simple form. So if$\xi$is a given point inside I, the functional

is linear and vanishes on any polynomial of degree$m+1$By taking$f(x)=x^{m+2}$and taking into account the formula for the means (4), we see that there is a well-defined value$c$(from within interval I) of$\xi$for which the functional (5) also vanishes on any polynomial of degree$m+2$The number$c$is given by the equation

We have
Lemma 1. Under the previous assumptions, the linear functional

is defined on S and has a degree of accuracy$m+2$
It suffices to demonstrate that $R[x^{m+3}]$is not zero.
Taking into account (6), we deduce

If we ask

Or$z$is an independent parameter of$x$, We have

So we have$P^{(m+1)}(x)>0$For$x\neq-z$It follows that the polynomial (9) is convex of order$m$(everywhere). We have, according to (2),

regardless of$z$It follows that the discriminant of this quadratic trinomial is negative, therefore

and equation (8) shows us that

Lemma 1 follows.
We will see later that the linear functional (7) is of the simple form.
3. We will now assume that the interval$\mathbf{I}$reduces to the bounded and closed interval$[a,b](a<b)$and that the elements$f(x)$of S have a derivative$(m+1)$nth continues on$[a,b]$.

We always assume$m\geqq 0$
So then $R[f]$a linear functional defined on S, of degree of accuracy$m$and of the simple form and consider the linear functional (7), the number$c$being determined by equation (6). We then have$a<c<<b$.

We have
Lemma 2. Under the previous assumptions, if there exists an integer$k$,$0\leqq k\leqq m+1$such as the linear functional$R[f]$either limited in relation to the standard

We have

for any function$f(x)\in\mathrm{S}$non-concave of order$m+2$.

Let us consider the functions

Or$\lambda$is an independent parameter of$x$and included between$a$And$b$
The function $\varphi_{m_{+3},\lambda}(x)$belongs to S and is non-concave of order$m+2$for everything$\lambda$. We have

We will demonstrate that inequality (12) holds for this function, so if we set$f(x)=\varphi_{m_{+3},\lambda}(x)$. Indeed,

and if we take (6) into account, we have

But the functions

are non-concave in order$m$since their derivatives of order$m+1$are
$(m+2)!\left(\frac{x-\lambda+|x-\lambda|}{2}\right)\geqq 0,\quad(m+2)!\left(\frac{|x-\lambda|-x+\lambda}{2}\right)\geqq 0$
respectively.
So we have

and, taking into account (10),

for everything$\lambda$between$a$And$b$From a theorem of our work cited [4] (theorem 15) it follows
that the linear functional$R_{1}[f]$is of the simple form, therefore inequality (12) is true for all function$f(x)\in\mathrm{S}$non-concave of order$m+2$(and even without any possibility of equality if$f(x)$is convex of order$m+2$).

Lemma 2 is therefore proven.
Note: We wrote the norm that appears in Lemma 2 in the form (11). It could easily be replaced by another norm
that linearly contains only max$|f(x)|$and max$\left|f\left({}^{m+1}\right)(x)\right|$, by virtue of certain delimitations given by J. HADAMARD for intermediate-order derivatives.
4. We can now demonstrate the

Theorem 2. If the following hypotheses are verified:

1. 1.

   $m$is a non-negative integer.

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2. 2.

   S is the function space$f(x)$having a continuous derivative of order$m+1$on the bounded and closed interval$[a,b](a,<b)$(which leads to the continuity of$f(x)$and all its derivatives of orders$1,2,\ldots,m+1$on$[a,b]$).

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3. 3.

   $R[f]$is a linear functional defined on S, with degree of accuracy$m$, of the simple form and bounded with respect to the norm (11) for some integer$k,0\leqq k\leqq m+1$.

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4. 4.

   $c$is the point determined by equation (6) (We then have$a<c<b$).

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5. 5.

   The function$f(x)$also checks one of the following 4 properties:
   A. is non-concave of order$m+1$and non-concave of order$m+2$,
   B. is non-convex of order$m+1$and non-concave of order$m+2$,
   C. is non-concave of order$m+1$and non-convex of order$m+2$,
   D. is non-convex of order$m+1$and non-convex of order$m+2$,
   then the average formula (4) is verified in the cases$A$And$D$by at least one point$\xi$of the interval$[c,b]$and in the cases$B,C$for at least one point$\xi$of the interval$[a,c]$.

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It suffices to demonstrate this here in case A. In this case, the function

is non-increasing on$[a,b]$and is surely zero at at least one point inside the interval$[a,b]$So we have$g(a)\geqq 0,g(b)\leqq 0$and from lemma 2 it follows that$g(c)\geqq 0$The desired property follows from this. Note also that the points$\xi$which satisfy (4) form an interval and the property obtained means that this interval has at least one point in common with$[c,b]$In particular, when the function$f(x)$is convex of order$m+1$, the point$\xi$of (4) is unique and belongs to the interval$[c,b]$.