1. 1.

   Before we deal with the inequalities that we want to establish in this work, it is necessary to summarize some properties of the orthogonal polynomials that we will use later.

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A sequence of polynomials in$x$

Or

is an orthogonal sequence if:
$1^{\circ}$. We can find a linear operation$U(f)$, defined in the field of polynomials, such that
$2^{\circ}$. We have
a)
$U\left(P_{i}P_{j}\right)=0,i\neq j$
b)$U\left(P_{i}^{2}\right)>0$,
whatever$i,j=0,1,2,\ldots$
It follows that the orthogonal sequence is completely characterized by the moments

and then we have

With the help of$2^{\circ}$we easily show that the polynomials (1) are completely determined and we can calculate them explicitly. We know, moreover, that the operation$U(f)$is necessarily of the form ² )

Or$\alpha(x)$is a non-decreasing function in ($-\infty$,$+\infty$).
It can be assumed that only the$2r$moments$c_{0},c_{1},\ldots c_{2r-1}$are given ($\alpha(x)$then only takes a finite number of distinct values). In this case we take in$\left.2^{\circ}a\right)i,j=0,1\ldots,r$and in$\left.2^{\circ}b\right)i=0,1\ldots$,$r-1$. We then have a finite orthogonal sequence$P_{0},P_{1},\ldots,P_{r}$. But, as we can easily see, such a sequence is always the section of an infinite sequence (1).
2. Let (1) be an orthogonal sequence. We know that the zeros of the polynomial$P_{i}(i>0)$are all real and distinct. Two consecutive polynomials$P_{i-1},P_{i}$never have common zeros and the zeros of$P_{i}$separate those of$P_{i-1}$.

Let$x_{1}<x_{2}<\ldots<x_{m}$the zeros of the polynomial$P_{m}$. Consider the system of$2m$equations
(2)$\quad c_{i}=\lambda_{1}x_{1}^{i}+\lambda_{2}x_{2}^{i}+\ldots+\lambda_{m}x_{m}^{i},\quad i=0.1\ldots,2m-1$,
in the unknowns$\lambda_{1},\lambda_{2},\ldots,\lambda_{m},x_{1},x_{2},\ldots,x_{m}$. We immediately find that the$x_{i}$are precisely the zeros of the polynomial$P_{m}$and the$\lambda_{i}$, which are then determined completely by the$m$first equations, are positive numbers. We will say that these numbers$\lambda_{i}$are the weights of the polynomial$P_{m}$. These weights also enjoy the property that the operation

is an operation$U(f)$corresponding to the finite orthogonal sequence$P_{0}$,$P_{1},\ldots,P_{m}$. These results are easily obtained by noting that the orthogonality condition$2^{\circ}$a) gives us$U\left(x^{i}P_{m}\right)=0,i=0,1,\ldots$,$m-1$, which precisely express that the$x_{i}$are the zeros of$P_{m}$and that system (2) is then compatible in$\lambda_{1},\lambda_{2},\ldots,\lambda_{m}$. The positivity of the weights easily results from the fact that these numbers are inversely proportional to the numbers$P_{m-1}\left(x_{i}\right)P_{m}^{\prime}\left(x_{i}\right)$. We can also notice that
done

because the operation$U(f)$is positive.

Let us also note that in an orthogonal sequence (1) we can arbitrarily take the polynomials$P_{m-1},\mathrm{P}_{m}$with the only conditions of reality and separation of their zeros. We can also arbitrarily take the polynomial$P_{m}$with real and distinct zeros and the positive weights of this polynomial. By these data the polynomials$P_{0},P_{1},\ldots,P_{m}$of the sequence (1) are completely determined.

Now consider the polynomial$P_{m}+\varrho P_{m-1}$Or$\varrho$is a constant$\not\subset 0$. We can easily see that the zeros$y_{1}<y_{2}<\ldots<y_{n}$of this polynomial are all real and distinct. We have the separation properties

The number$\varrho$is completely determined by the fact that$y_{1}$has any given value$<x_{1}$Or$y_{m}$any given value$>x_{m}$. We can also notice that the zeros of$P_{m-1}$always separate those of$P_{m}+\varrho P_{m-1}$.

The system
(4)$\quad c_{i}=v_{1}y_{1}^{i}+v_{2}y_{2}^{i}+\ldots+v_{m}y_{m}^{i}\quad,\quad i=0,1,\ldots,2m-2$
is still compatible in$\nu_{i}$We still have dinner that numbers$\nu_{1},\nu_{2},\ldots,\nu_{m}$. determined by this system, are the weights of the polynomial$P_{m}+\varrho P_{m-1}$, These weights are positive regardless of o. Indeed, formula (3) is still applicable. We have

done

where we posed$Q(x)=P_{m}(x)\not-\varrho P_{m-1}(x)$.
It can also be easily shown that we thus obtain all the solutions of the system (4).
3. Let us now return to the convex functions. We will first give an interpretation of the divided difference$\left[x_{1},x_{2},\ldots,x_{n+2};f\right]$of order$n+1$. We can write

where the$p_{i}$do not depend on the function$f(x)$. If we suppose$x_{1}<x_{2}<\ldots<x_{n+2}$, the well-known expression, by the quotient of two
determinants, of the divided difference, shows us that the coefficients$p_{1},p_{2},\ldots,p_{n+2}$are alternately positive and negative, the latter always being positive$\mathrm{f},p_{n+2}>0$. On the other hand, these coefficients are, up to a constant factor, determined by the relations

We can therefore state the following properties:
If$n=2m-3$is odd and$x_{1}<x_{2}<\ldots<x_{n+2}$, we can write

and then$x_{2},x_{4},\ldots,x_{2m-2}$are the zeros and$\mu_{1},\mu_{2},\ldots,\mu_{m-1}$the weights of the polynomial$P_{m-1}$of the orthogonal sequence$P_{0},P_{1},\ldots,P_{m}$, Or$x_{1},x_{3},\ldots,x_{2m-1}$are the zeros and$\lambda_{1},\lambda_{2},\ldots,\lambda_{m}$the weights of the polynomial$P_{m}$.

If$n=2m-2$esi pair and$x_{1}<x_{2}<\ldots<x_{2m}$, we can write

and then$x_{1},x_{3},\ldots,x_{2m-1}$are the zeros and$\nu_{1},\nu_{2},\ldots,\nu_{m}$the weights of a polynomial$P_{m}+\varrho P_{m-1}$, Or$\varrho>0$And$P_{m-1},P_{m}$belong to the orthogonal sequence$P_{0},P_{1},\ldots,P_{m-1},P_{m},\ldots$, determined by the zeros$x_{2},x_{4},\ldots$,$x_{2m}$and the weights$\lambda_{1},\lambda_{2},\ldots,\lambda_{m}du$polynomial$P_{m}$.
4. Suppose$n=2m-1$odd and write the inequality

Or$r>m,\lambda_{1},\lambda_{2},\ldots,\lambda_{r}$are positive and$x_{1}<x_{2}<\ldots<x_{r}$,$y_{1}<y_{2}<\ldots<y_{m}$.

Let us look for the necessary and sufficient conditions for inequality (5) to be verified for any function$f(x)$, non-concave of order$n$, defined on the points$x_{i},y_{j},i=1,2,\ldots,r,j=1,2,\ldots,m$.

Referring to the previous Note ³ ), we see that conditions
(6)$\quad\sum_{i=1}^{r}\lambda_{i}x_{i}^{s}=\sum_{j=1}^{m}\mu_{j}y_{i}^{s}\quad,s=0,1,\ldots,n(n=2m-1)$
are necessary. It is therefore necessary that the$y_{i}$let the zeros and the$\mu_{j}$the weights of the polynomial$P_{m}$of the orthogonal sequence$P_{0},P_{1},\ldots,P_{m},\ldots,P_{r}$, determined by the zeros$x_{i}$and the weights$\lambda_{i}$of the polynomial$P_{r}$.

We propose to demonstrate that equalities (6) are also sufficient. We will do this demonstration by induction. The property is true for$r=m+1$, because then inequality (5) is an inequality of definition of non-concavity of order$n$. Indeed, the difference between the first and second members is, up to a constant positive factor, a divided difference of order$n+1$. Let us now show that assuming the property if in the first member of (5) is true there is$r-1$terms ($r>m+1$), it will result true for$r$terms. By hypothesis, if we determine the$y_{j}^{\prime}$and the$\mu_{j}^{\prime}>0$by equalities

We have

SO

But, we know that the$y_{j}^{\prime}$are distinct and all included between$x_{1}$And$x_{r-1}$. We can therefore determine the$y_{j}^{\prime\prime}$and the$\mu_{j}^{\prime\prime}>0$by equalities

We then have

But, (7) and (9) compared with (6) show us that$\mu_{i}^{\prime\prime}=\mu_{i}$,$y_{j}^{\prime\prime}=y_{j}$and then, from (8) and (10), inequality (5) results.

We can therefore state the following property:
If$y_{1},y_{2},\ldots,y_{m}$are the zeros and$\mu_{1},\mu_{2},\ldots,\mu_{m}$the weights of the polynomial$P_{m}$of the orthogonal sequence$P_{0},P_{1},\ldots,P_{m},\ldots,P_{r}$determined by zeros$x_{1},x_{2},\ldots,x_{r}$and the weights$\lambda_{1},\lambda_{2},\ldots,\lambda_{r}(r>m)$of the polynomial$P_{r}$, inequality (5) is verified for any non-concape function of odd order$n=2m-1$, defined on the points$x_{i},y_{j}$.

If, moreover, the function is convex of order$n=2m-1$on the points$x_{i},y_{j}$we have in (5) the sign$>$.

The last part of the statement is immediately justified.
5. If$m=1,n=1$ ; inequality (5) becomes

which is the classical Jensen inequality for ordinary non-concave functions (of order 1) ⁴ ).

Consider a finite and closed interval$(a,b)$and either$f(x)$a continuous non-concave function of order$1,3,5,\ldots$, therefore of any odd order.