given and the coefficientsp_(i)p_{i}being independent of the functionff.
The necessary and sufficient conditions that the coefficients must satisfyp_(i)p_{i}, for this to be so, are easily obtained, for example, using the formula that we can call the fundamental formula for transforming divided differences. This formula is written^(2){ }^{2})
f={[0","quad" pour "x=x_(1)","x_(2)","dots","x_(j+n)],[(x-x_(j+1))(x-x_(j+2))dots(x-x_(j+n))","" pour "x=x_(j+n+1)","x_(j+n+2)","dots","x_(m)],[quad j=1;2","dots","m-n-I]:}f=\left\{\begin{array}{l}
0, \quad \text { pour } x=x_{1}, x_{2}, \ldots, x_{j+n} \\
\left(x-x_{j+1}\right)\left(x-x_{j+2}\right) \ldots\left(x-x_{j+n}\right), \text { pour } x=x_{j+n+1}, x_{j+n+2}, \ldots, x_{m} \\
\quad j=1 ; 2, \ldots, m-n-\mathbf{I}
\end{array}\right.
we get
When the functionffis defined in an interval containing the points (2) the conditions (3) are no longer necessary forn > 1n>1. For inequality (I) to hold, whatever the functionffdefined in an interval containing within it the points (2), it is necessary and sufficient that this inequality is true for any polynomial of degreennand for any function of the form(|x-lambda|+x-lambda)^(n),lambda(|x-\lambda|+x-\lambda)^{n}, \lambdabeing a constant. Indeed, any non-concave function of ordernnis the limit of a sequence of functions which are, at an additive polynomial of degreennnear, sums of such functions(|x-lambda|+x-lambda)^(n)(|x-\lambda|+x-\lambda)^{n}with positive coefficients^(3){ }^{3}).
We thus find the necessary and sufficient equalities.
which are none other thanA_(1)=A_(2)=dots=A_(n+1)=0A_{1}=A_{2}=\ldots=A_{n+1}=0, expressing that the first member of (I) is identically zero for any polynomial of degreeeta\eta.
We then find the necessary and sufficient inequalities
^(3){ }^{3}) Tiberiu Popoviciu, On the extension of convex functions of higher order, Bull. Math. Soc. Roumaine des Sci., 36, 75-108 (1934).
2. We can generalize the inequality ( x ) by also introducing the values ​​of the derivatives offfto the pointsx_(i)x_{i}. For simplicity, let us assumeffdefined in an interval containing the points (2). Ifn > 1n>1the functionff, non-concave of ordernn, has continuous derivativest^('),t^(''),dots,t^((n-1))t^{\prime}, t^{\prime \prime}, \ldots, t^{(n-1)}and derivatives to the left of the d and to the right of ordern,f_(g)^((n)),f_(d)^((n))*[f_(g)^((n))=(f^((n-1)))_(g)^('),f_(d)^((n))=(f^((n-1)))_(d)^(')]n, f_{g}^{(n)}, f_{d}^{(n)} \cdot\left[f_{g}^{(n)}=\left(f^{(n-1)}\right)_{g}^{\prime}, f_{d}^{(n)}=\left(f^{(n-1)}\right)_{d}^{\prime}\right]in every interior point.
We can then look for the necessary and sufficient conditions so that we have
Or0 <= k_(i) <= n0 \leqq k_{i} \leqq nAndf^((n))f^{(n)}denotes one of the derivativesf_(g)^((n)),f_(d)^((n))f_{g}^{(n)}, f_{d}^{(n)}(not necessarily the same for allx_(i)x_{i}), whatever the functionff, non-concave of ordernnin an interval containing within it the pointsbar(x)_(i)\bar{x}_{i}. We can easily see that the necessary and sufficient conditions are still that the inequality holds for any polynomial of degreennand for any function of the form(|x-lambda|+x-lambda)^(n)(|x-\lambda|+x-\lambda)^{n}. These conditions can therefore be written
The first member is the limit of a divided difference of ordern+In+I, whenk_(i)+Ik_{i}+Iof hisn+2n+2points tend towardsx_(i),i=I,2,dots,mx_{i}, i=I, 2, \ldots, m. It is easy to obtain the explicit form of this generalized divided difference.
3. We can also establish, sometimes, the exactness of inequality (4) by expressing the first member in the form of a sum of
divided differences of ordern+1n+1(of the generalized form (5)) chosen appropriately. We will establish in this way an interesting particular inequality. Letx_(1) < x_(2) < dots < x_(n)nx_{1}<x_{2}<\ldots<x_{n} ndistinct points of the real axis and let
Letxi_(1) < xi_(2) < dots < xi_(n-I)\xi_{1}<\xi_{2}<\ldots<\xi_{n-\mathrm{I}}the zeros, all real, of the derivativevarphi^(')(x)\varphi^{\prime}(x)of the polynomialvarphi(x)\varphi(x). We have
We have the following formulas [x_(1),x_(2),dots,x_(n);f]=sum_(i=1)^(n)(f(x_(i)))/(varphi^(')(x_(i))),[xi_(1),xi_(2),dots,xi_(n-1);'^(')]=nsum_(i=1)^(n-1)(f^(')(xi_(i)))/(varphi^('')(xi_(i)))\left[x_{1}, x_{2}, \ldots, x_{n} ; f\right]=\sum_{i=1}^{n} \frac{f\left(x_{i}\right)}{\varphi^{\prime}\left(x_{i}\right)},\left[\xi_{1}, \xi_{2}, \ldots, \xi_{n-1} ; \prime^{\prime}\right]=n \sum_{i=1}^{n-1} \frac{f^{\prime}\left(\xi_{i}\right)}{\varphi^{\prime \prime}\left(\xi_{i}\right)},[x_(1),x_(2),dots,x_(n),xi_(j),xi_(j);f]=sum_(i=1)^(n)(f(x_(i)))/((x_(i)-xi_(j))^(2)varphi^(')(x_(i)))+(f^(')(xi_(j)))/(varphi(xi_(j)))\left[x_{1}, x_{2}, \ldots, x_{n}, \xi_{j}, \xi_{j} ; f\right]=\sum_{i=1}^{n} \frac{f\left(x_{i}\right)}{\left(x_{i}-\xi_{j}\right)^{2} \varphi^{\prime}\left(x_{i}\right)}+\frac{f^{\prime}\left(\xi_{j}\right)}{\varphi\left(\xi_{j}\right)},
j=1,2,dots,n-1j=1,2, \ldots, n-1
We deduce from this
(6)-sum_(j=1)^(n)(varphi(xi_(j)))/(varphi^('')(xi_(j)))[x_(1),x_(2),dots,x_(n),xi_(j),xi_(j);f]=-\sum_{j=1}^{n} \frac{\varphi\left(\xi_{j}\right)}{\varphi^{\prime \prime}\left(\xi_{j}\right)}\left[x_{1}, x_{2}, \ldots, x_{n}, \xi_{j}, \xi_{j} ; f\right]=
Formula (6) can therefore be written (n-I)/(n)[x_(1),x_(2),dots,x_(n);f]-(I)/(n)[xi_(1),xi_(2),dots,xi_(n-1);f^(')]=\frac{n-\mathrm{I}}{n}\left[x_{1}, x_{2}, \ldots, x_{n} ; f\right]-\frac{\mathrm{I}}{n}\left[\xi_{1}, \xi_{2}, \ldots, \xi_{n-1} ; f^{\prime}\right]=
and we can state the following theorem
Ifx_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n}arennreal axis points andxi_(1),xi_(2),dots,xi_(n-1)\xi_{1}, \xi_{2}, \ldots, \xi_{n-1}are the zeros of the derivative of the polynomialvarphi(x)=(x-x_(1))(x-x_(2))dots(x-x_(n))\varphi(x)=\left(x-x_{1}\right)\left(x-x_{2}\right) \ldots\left(x-x_{n}\right), any functionff, non-concave of ordernnin an interval containing within it the pointsxx, verifies inequality
(7)quad(n-1)[x_(1),x_(2),dots,x_(n);f] >= [xi_(1),xi_(2),dots,xi_(n-1);t^(')]\quad(n-1)\left[x_{1}, x_{2}, \ldots, x_{n} ; f\right] \geq\left[\xi_{1}, \xi_{2}, \ldots, \xi_{n-1} ; t^{\prime}\right].
We have demonstrated the property in the case where the pointsx_(i)x_{i}are distinct. It also remains true, as a result of continuity, when these points are not distinct [the divided differences then being of the generalized form (5)].
Remarks I. We can easily see that if in addition the functionffis convex of ordernn, the equality in (7) can only hold ifx_(1)=x_(2)=dots=x_(n)x_{1}=x_{2}=\ldots=x_{n}.
II. The restriction that thex_(i)x_{i}are within the definition interval offfis not essential. The result remains the same if one or both ends of this interval coincide with a simple zero ofvarphi(x)\varphi(x). The result is true even without restriction if the function is differentiable a sufficient number of times at the ends of the interval.
4. From the previous theorem we can draw some simple conclusions. Let us denote byxi_(I)^((i)),xi_(2)^((j)),dots,xi_(n-i)^((i))\xi_{\mathrm{I}}^{(i)}, \xi_{2}^{(j)}, \ldots, \xi_{n-i}^{(i)}the zeros of the order derivativeiiofvarphi(x)\varphi(x). We then have, under the same conditions, (n-1)![x_(1),x_(2),dots,x_(n);f] >= (n-2)![xi_(1),xi_(2),dots,xi_(n-1);f^(')] >= dots(n-1)!\left[x_{1}, x_{2}, \ldots, x_{n} ; f\right] \geqq(n-2)!\left[\xi_{1}, \xi_{2}, \ldots, \xi_{n-1} ; f^{\prime}\right] \geqq \ldots dots >= (n-i-I)![xi_(I)^((i)),xi_(2)^((i)),dots,xi_(n-i)^((i));f^((i))] >= dots >= [xi_(I)^((n-I));f^((n-I))]\ldots \geq(n-i-\mathrm{I})!\left[\xi_{\mathrm{I}}^{(i)}, \xi_{2}^{(i)}, \ldots, \xi_{n-i}^{(i)} ; f^{(i)}\right] \geq \ldots \geq\left[\xi_{\mathrm{I}}^{(n-\mathrm{I})} ; f^{(n-\mathrm{I})}\right].
But,
so:
If the functionffis non-concave of ordernnin an interval containing the pointsx_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n}, we have the inequality
If f is convex of ordernn, equality is only possible forx_(1)=x_(2)=dots=x_(n)x_{1}=x_{2}=\ldots=x_{n}. Let us take, in particular, the functionf=x^(n+r-1)f=x^{n+r-1}. So ifrris a natural number
, the difference divided[x_(1),x_(2),dots,x_(n);//]\left[x_{1}, x_{2}, \ldots, x_{n} ; /\right]is equal to the well-known symmetric function
Let us denote by Wr' the same symmetric functions ofxi_(1),xi_(2),dots,xi_(n-1)\xi_{1}, \xi_{2}, \ldots, \xi_{n-1}and, in general, byW_(r)^((i))W_{r}^{(i)} les fonctions symétriques correspondantes de xi_(1)^((i))\xi_{1}^{(i)}, xi_(2)^((i)),dots,xi_(n-i)^((i))\xi_{2}^{(i)}, \ldots, \xi_{n-i}^{(i)}. Remarquons que la fonction x^(n+r-1)x^{n+r-1} est convexe d'ordre nn dans l'intervalle (-oo,+oo)(-\infty,+\infty) si rr est tun nombre naturel pair et est convexe d'ordre nn dans ( 0,+oo0,+\infty ) si rr est un nombre naturel impair. Nous pouvons donc énoncer la propriété suivante:
Si x_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n} sont les zéros, tous réels, d'un polynome de degré nn et xi_(r)^((i)),xi_(2)^((i)),dots,xi_(n-i)^((i))\xi_{\mathrm{r}}^{(i)}, \xi_{2}^{(i)}, \ldots, \xi_{n-i}^{(i)} sont les zéros de la iema dérivée de ce polynome (xi_(r)^((n-1))=:}{:(x_(1)+x_(2)+dots+x_(n))/(n))\left(\xi_{\mathrm{r}}^{(n-1)}=\right. \left.\frac{x_{1}+x_{2}+\ldots+x_{n}}{n}\right), nous avons les inégalités
pour tout nombre naturel pair r >= 2r \geq 2.
Si, de plus, les zéros x_(1),x_(2),dots,x_(n)x_{1}, x_{2}, \ldots, x_{n} sont non-négatifs, ces inégalités sont vraies aussi pour r >= 3r \geq 3 impair.
Le signe >=\geq ne devient == dans ces inégalités que si x_(1)=x_(2)=dots=x_(n)x_{1}=x_{2}=\ldots=x_{n}.
L'inégalité (7) est à rapprocher de l'inégalité de M. K. T o d à 4)
valable pour toute fonction non-concave d'ordre I^(5)I^{5} ). Pour f=x^(p)f=x^{p} où, p >=p \geq I ou p < 0p<0, cette inégalité revient à celle de MM. H. E. B i a y ^(6){ }^{6} ) ẹt S. Kake y a ^(7){ }^{7} )
qui est à rapprocher de (8).
Cernăufi, le 27 février 1940.
^(1)){ }^{1)} La note III est sous presse dans Mathematica, 16, 74-86. La note IV doit paraître dans cette même revue. ^(2){ }^{2} ) Pour les notations voir mes travaux antérieurs.
^(4){ }^{4} ) K. Toda, On certain functional inequalities, Journal of the Hiroshima. Univ., A,, 427-40 (1934). ^(5){ }^{5} ) Voir aussi la note III, loc. cil. (1). ^(6){ }^{6} ) H. E. B r a y, On the zeros of a polynomial and of its devivatives, Aner. Journal of Math., 53, 864-872 (1931). ^(7){ }^{7} ) S.K a ke y a, On an inequality between the roots of an equation and its derivative, Proceedings Phys.-Math. Soc. Japan (3), 15, 149-154 (1933).
