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T. Popoviciu, Sur un problème de maximum de Stieltjes, (French) C. R. Acad. Sci., Paris 202, 1645-1647 (1936).
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Theory of functions. - On a problem of Sticlijes maximum. Note by Mr. Tibère Popoviciu.
- Assumptions made about the functionf(x).-f(x)f(x) .-f(x)is a real, defined and uniform function on the set E formed bymmfinite and closed intervals (a_(i),b_(i)a_{i}, b_{i}) ,a_(i) < b_(i)(i=1,dots,m);b_(i) <= a_(i+1)(i=1,dots,m-1)a_{i}<b_{i}(i=1, \ldots, m); b_{i} \leqq a_{i+1}(i=1, \ldots, m-1)of the real axis. We will say thatf(x)f(x)is a function (C) if, in each interval(a_(i),b_(i)):I^(@)\left(a_{i}, b_{i}\right): \mathrm{I}^{\circ}it is non-negative in the open interval, but there are points as close as one wants toa_(i)a_{i}and ofbi)bi}where the function is positive;2^(@)2^{\circ}it is continuous ina_(i)a_{i}and inb_(i);3^(@)f(a_(i))=f(b_(i))=0;4^(@)b_{i}; 3^{\circ} f\left(a_{i}\right)=f\left(b_{i}\right)=0 ; 4^{\circ}it is exponentially concave in the whole interval. This last property means
that
f(x_(2)) > f(x_(1))^((x_(3)-x_(2))/(x_(3)-x_(1)))f(x_(3))^((x_(2)-x_(1))/(x_(3)-x_(1)))quad(a_(i) <= x_(1) < x_(2) < x_(3) <= b_(i))f\left(x_{2}\right)>f\left(x_{1}\right)^{\frac{x_{3}-x_{2}}{x_{3}-x_{1}}} f\left(x_{3}\right)^{\frac{x_{2}-x_{1}}{x_{3}-x_{1}}} \quad\left(a_{i} \leqq x_{1}<x_{2}<x_{3} \leqq b_{i}\right)
so thatlog f(x)\log f(x)is concave.
2. Assumptions made on the pointsx_(i)x_{i}. - We will say that thennpointsx_(1)x_{1},x_(2),dots,x_(n)x_{2}, \ldots, x_{n}of the real axis have a given distribution[k_(1),k_(2),dots,k_(m)]if\left[k_{1}, k_{2}, \ldots, k_{m}\right] \mathrm{si}, among these points, there arek_(i)k_{i}in the meantime(a_(i),b_(i))(i=1,dots,m)\left(a_{i}, b_{i}\right)(i=1, \ldots, m). We havek_(i) >= 0,k_(1)+k_(2)+dots+k_(m)=nk_{i} \geqq 0, k_{1}+k_{2}+\ldots+k_{m}=n. We will say that two different distributions[k_(1),k_(2),dots,k_(m)]\left[k_{1}, k_{2}, \ldots, k_{m}\right]And[k_(1)^('),k_(2)^('),dots,k_(m)^(')]\left[k_{1}^{\prime}, k_{2}^{\prime}, \ldots, k_{m}^{\prime}\right]ofnnpointsx_(1) < x_(2) < dots < x_(n)x_{1}<x_{2}<\ldots<x_{n}(none of which coincide with aa_(i)a_{i}or abi)bi}) Andx_(1)^('),x_(2)^('),dots,x_(n)^(')x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n}^{\prime}respectively, are consecutive if we can choose the pointsx_(i)^(')x_{i}^{\prime}so that we have
x_(1) < x_(1)^(') < x_(2) < x_(2)^(') < dots < x_(n) < x_(n)^(')quadx_{1}<x_{1}^{\prime}<x_{2}<x_{2}^{\prime}<\ldots<x_{n}<x_{n}^{\prime} \quadOrquadx_(1)^(') < x_(1) < x_(2)^(') < x_(2) < dots < x_(n)^(') < x_(n)\quad x_{1}^{\prime}<x_{1}<x_{2}^{\prime}<x_{2}<\ldots<x_{n}^{\prime}<x_{n}.
We will also say that two distributions[l_(1),l_(2),dots,l_(m)]\left[l_{1}, l_{2}, \ldots, l_{m}\right]And[h_(1),h_(2),dots,h_(m)]\left[h_{1}, h_{2}, \ldots, h_{m}\right]ofnnpointsx_(1) < x_(2) < dots < n_(n)x_{1}<x_{2}<\ldots<n_{n}and ofn-1n-1pointsy_(1),y_(2),dots,y_(n-1)y_{1}, y_{2}, \ldots, y_{n-1}respectively, are consecutive if we can choose the pointsy_(i)y_{i}so that we havex_(1) < y_(1) < x_(2) < dots < y_(n-1) < x_(n)x_{1}<y_{1}<x_{2}<\ldots<y_{n-1}<x_{n}.
3. Position of the problem. - The functionf(x)f(x)being a function ( C ) and the pointsx_(i)x_{i}having a given distribution, we studied the maximum of the expression
2. Assumptions made on the pointsx_(i)x_{i}. - We will say that thennpointsx_(1)x_{1},x_(2),dots,x_(n)x_{2}, \ldots, x_{n}of the real axis have a given distribution[k_(1),k_(2),dots,k_(m)]if\left[k_{1}, k_{2}, \ldots, k_{m}\right] \mathrm{si}, among these points, there arek_(i)k_{i}in the meantime(a_(i),b_(i))(i=1,dots,m)\left(a_{i}, b_{i}\right)(i=1, \ldots, m). We havek_(i) >= 0,k_(1)+k_(2)+dots+k_(m)=nk_{i} \geqq 0, k_{1}+k_{2}+\ldots+k_{m}=n. We will say that two different distributions[k_(1),k_(2),dots,k_(m)]\left[k_{1}, k_{2}, \ldots, k_{m}\right]And[k_(1)^('),k_(2)^('),dots,k_(m)^(')]\left[k_{1}^{\prime}, k_{2}^{\prime}, \ldots, k_{m}^{\prime}\right]ofnnpointsx_(1) < x_(2) < dots < x_(n)x_{1}<x_{2}<\ldots<x_{n}(none of which coincide with aa_(i)a_{i}or abi)bi}) Andx_(1)^('),x_(2)^('),dots,x_(n)^(')x_{1}^{\prime}, x_{2}^{\prime}, \ldots, x_{n}^{\prime}respectively, are consecutive if we can choose the pointsx_(i)^(')x_{i}^{\prime}so that we have
x_(1) < x_(1)^(') < x_(2) < x_(2)^(') < dots < x_(n) < x_(n)^(')quadx_{1}<x_{1}^{\prime}<x_{2}<x_{2}^{\prime}<\ldots<x_{n}<x_{n}^{\prime} \quadOrquadx_(1)^(') < x_(1) < x_(2)^(') < x_(2) < dots < x_(n)^(') < x_(n)\quad x_{1}^{\prime}<x_{1}<x_{2}^{\prime}<x_{2}<\ldots<x_{n}^{\prime}<x_{n}.
We will also say that two distributions[l_(1),l_(2),dots,l_(m)]\left[l_{1}, l_{2}, \ldots, l_{m}\right]And[h_(1),h_(2),dots,h_(m)]\left[h_{1}, h_{2}, \ldots, h_{m}\right]ofnnpointsx_(1) < x_(2) < dots < n_(n)x_{1}<x_{2}<\ldots<n_{n}and ofn-1n-1pointsy_(1),y_(2),dots,y_(n-1)y_{1}, y_{2}, \ldots, y_{n-1}respectively, are consecutive if we can choose the pointsy_(i)y_{i}so that we havex_(1) < y_(1) < x_(2) < dots < y_(n-1) < x_(n)x_{1}<y_{1}<x_{2}<\ldots<y_{n-1}<x_{n}.
3. Position of the problem. - The functionf(x)f(x)being a function ( C ) and the pointsx_(i)x_{i}having a given distribution, we studied the maximum of the expression
D(x_(1),x_(2),dots,x_(n);f)=f(x_(1)),f(x_(2)),dots,f(x_(n))prod_(i > j)^(1,2,dots,n)(x_(i)-x_(j))^(2)\mathrm{D}\left(x_{1}, x_{2}, \ldots, x_{n} ; f\right)=f\left(x_{1}\right), f\left(x_{2}\right), \ldots, f\left(x_{n}\right) \prod_{i>j}^{1,2, \ldots, n}\left(x_{i}-x_{j}\right)^{2}
- Results. - Under these conditions, we found that:
I.D(x_(1),x_(2),dots,x_(n);f)\mathrm{D}\left(x_{1}, x_{2}, \ldots, x_{n} ; f\right)has a maximum reached for at least one point systemx_(i)x_{i}.
II. This maximum is reached for a single system of n points, distinct and different from the pointsa_(i)a_{i}Andb_(i)b_{i}. We designate byxi_(n,1) < xi_(n,2) < dots < xi_(n,n)\xi_{n, 1}<\xi_{n, 2}<\ldots<\xi_{n, n}the points for which the maximum is reached. We will say that this is the maximizing system of our problem.
III. The correspondence between a function ( C ) and its maximizing system is continuous. Therefore, at anyepsi > 0\varepsilon>0, corresponds to aeta > 0\eta>0such that ifg(x)g(x)is another (C) function verifying|f-g| < eta|f-g|<\etaon E, we have
|xi_(n,i)-xi_(n,i)^(')| < epsiquad(i=1,dots,n)\left|\xi_{n, i}-\xi_{n, i}^{\prime}\right|<\varepsilon \quad(i=1, \ldots, n)
xi_(n,i)^(')\xi_{n, i}^{\prime}being the maximizing system ofg(x)g(x)corresponding to the same problem.
IV. Maximizing systemsxi_(n,i),xi_(n,i)^(')\xi_{n, i}, \xi_{n, i}^{\prime}corresponding, for the same functionf(x)f(x), to two consecutive distributions ofnnpoints, separate. We therefore have
xi_(n,1) < xi_(n,1)^(') < xi_(n,2) < dots < xi_(n,n) < xi_(n,n)^(')quad\xi_{n, 1}<\xi_{n, 1}^{\prime}<\xi_{n, 2}<\ldots<\xi_{n, n}<\xi_{n, n}^{\prime} \quadOrquadxi_(n,1)^(') < xi_(n,1) < xi_(n,2)^(') < dots < xi_(n,n)^(') < xi_(n,n)\quad \xi_{n, 1}^{\prime}<\xi_{n, 1}<\xi_{n, 2}^{\prime}<\ldots<\xi_{n, n}^{\prime}<\xi_{n, n}.
V. Maximizing systemsxi_(n,i),xi_(n-1,i)\xi_{n, i}, \xi_{n-1, i}corresponding, for the same functionf(x)f(x), at two consecutive distributions of net of n-1 points, separate. We therefore have
IV. Maximizing systemsxi_(n,i),xi_(n,i)^(')\xi_{n, i}, \xi_{n, i}^{\prime}corresponding, for the same functionf(x)f(x), to two consecutive distributions ofnnpoints, separate. We therefore have
xi_(n,1) < xi_(n,1)^(') < xi_(n,2) < dots < xi_(n,n) < xi_(n,n)^(')quad\xi_{n, 1}<\xi_{n, 1}^{\prime}<\xi_{n, 2}<\ldots<\xi_{n, n}<\xi_{n, n}^{\prime} \quadOrquadxi_(n,1)^(') < xi_(n,1) < xi_(n,2)^(') < dots < xi_(n,n)^(') < xi_(n,n)\quad \xi_{n, 1}^{\prime}<\xi_{n, 1}<\xi_{n, 2}^{\prime}<\ldots<\xi_{n, n}^{\prime}<\xi_{n, n}.
V. Maximizing systemsxi_(n,i),xi_(n-1,i)\xi_{n, i}, \xi_{n-1, i}corresponding, for the same functionf(x)f(x), at two consecutive distributions of net of n-1 points, separate. We therefore have
xi_(n,1) < xi_(n-1,1) < xi_(n,2) < dots < xi_(n-1,n-1) < xi_(n,n).\xi_{n, 1}<\xi_{n-1,1}<\xi_{n, 2}<\ldots<\xi_{n-1, n-1}<\xi_{n, n} .
We can say that two maximizing systems ofnnpoints or elsennAndn-1n-1points always separate, unless this separation is visibly impossible.
5. The Stieltjes problem. - The special case
5. The Stieltjes problem. - The special case
f(x)=prod_(i=1)^(m)|x-a_(i)|^(sigma_(i))|x-b_(i)|^(rho_(i))quad(sigma_(i),rho_(i)" positifs ")f(x)=\prod_{i=1}^{m}\left|x-a_{i}\right|^{\sigma_{i}}\left|x-b_{i}\right|^{\rho_{i}} \quad\left(\sigma_{i}, \rho_{i} \text { positifs }\right)
was examined by Th. Stieltjes, who obtained properties I and II in another way^((1)){ }^{(1)}In this case, the polynomialP(x)=prod_(i=1)^(n)(x-xi_(n,i))\mathrm{P}(x)=\prod_{i=1}^{n}\left(x-\xi_{n, i}\right)verifies a linear differential equation of the form
y^('')+(sum_(i=1)^(m)(sigma_(i))/(x-a_(i))+(rho_(i))/(x-b_(i)))y^(')+(psi(x))/((x-a_(1))(x-b_(1))dots(x-a_(m))(x-b_(m)))y=0y^{\prime \prime}+\left(\sum_{i=1}^{m} \frac{\sigma_{i}}{x-a_{i}}+\frac{\rho_{i}}{x-b_{i}}\right) y^{\prime}+\frac{\psi(x)}{\left(x-a_{1}\right)\left(x-b_{1}\right) \ldots\left(x-a_{m}\right)\left(x-b_{m}\right)} y=0
psi(x)\psi(x)being a certain polynomial of degree2m-22 m-2.
We obtained properties IV and V by showing that it is sufficient to demonstrate this for this particular case and even only for certain particular values ​​of the constantssigma_(i)\sigma_{i}Andrho_(i)\rho_{i}.
We obtained properties IV and V by showing that it is sufficient to demonstrate this for this particular case and even only for certain particular values ​​of the constantssigma_(i)\sigma_{i}Andrho_(i)\rho_{i}.
Property IV is a very broad generalization of some results of F. Klein (^(2){ }^{2}) and EB van Vleck (^(3){ }^{3}). Property V is related, form=Im=\mathrm{I}, to the theory of Jacobi polynomials (^(4){ }^{4}).
- (^(1){ }^{1}) Acta Mathematica, 6, 1885, p. 321.
(^(2){ }^{2}) Mathematische Annalen, 18, 1881, p. 237.
(^(3){ }^{3}) Bulletin of the Amer. Math. Soc.,2^(e)2^{\mathrm{e}}series, 4, 1898, p. 426.
^((1)){ }^{(1)}Th. J. Stieltjes, Reports, 100, 1885, p. 620.