Abstract
Authors
Keywords
?
Paper coordinates
T. Popoviciu, Sur une inegalité, Mathematica, 23 (1947-1948), pp. 127-128 (in French).
About this paper
Journal
Mathematica
Publisher Name
DOI
Print ISSN
Online ISSN
[MR0027024]
google scholar link
??
Paper (preprint) in HTML form
Original text
Rate this translation
Your feedback will be used to help improve Google Translate
ON AN INEQUALITY
BY
TIBERIU POPOVICIU
Requested on April 27, 1948
In the "New Annals of Mathematics"^((1)){ }^{(1)}LA Le Cointe demonstrates that ifa_(1)^('),a_(2)^('),dots,a_(2m)^(')a_{1}^{\prime}, a_{2}^{\prime}, \ldots, a_{2 m}^{\prime}are the2m2 mpositive numbersa_(1),a_(2),dots,a_(2m)a_{1}, a_{2}, \ldots, a_{2 m}arranged in non-descending (or non-ascending) order, we have
a_(1)^(')a_(2)^(')dotsa_(m)^(')+a_(m+1)^(')a_(m+2)^(')dotsa_(2m)^(') >= a_(1)a_(2)dotsa_(m)+a_(m+1)a_(m+2)dotsa_(2m)a_{1}^{\prime} a_{2}^{\prime} \ldots a_{m}^{\prime}+a_{m+1}^{\prime} a_{m+2}^{\prime} \ldots a_{2 m}^{\prime} \geqq a_{1} a_{2} \ldots a_{m}+a_{m+1} a_{m+2} \ldots a_{2m}
We will give a generalization of this property by demonstrating the
Theorem. - Letvarphi(x),psi(x)\varphi(x), \psi(x)two continuous and increasing functions in the interval.(a,b)(a, b)Andvarphi^(-1),psi^(-1)\varphi^{-1}, \psi^{-1}their inverse functions. Let us set
{:[M(x_(1),x_(2),dots,x_(n))=varphi^(-1)((sum_(i=1)^(n)varphi(x_(i)))/(n))],[N(x_(1),x_(2),dots,x_(m))=psi^(-1)((sum_(i=1)^(m)psi(x_(i)))/(m))]:}\begin{aligned} & \mathrm{M}\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\varphi^{-1}\left(\frac{\sum_{i=1}^{n} \varphi\left(x_{i}\right)}{n}\right) \\ & \mathrm{N}\left(x_{1}, x_{2}, \ldots, x_{m}\right)=\psi^{-1}\left(\frac{\sum_{i=1}^{m} \psi\left(x_{i}\right)}{m}\right) \end{aligned}
Let's still bea_(1),a_(2),dots,a_(nm)nma_{1}, a_{2}, \ldots, a_{nm} nmnumbers belonging to the interval(a,b)(a, b)Anda_(1)^('),a_(2)^('),dots,a_(nm)^(')a_{1}^{\prime}, a_{2}^{\prime}, \ldots, a_{nm}^{\prime}these same numbers arranged in non-increasing (or non-descending) order.
Then we have inequality
(1)
(1)
{:[M((N)(a_(1)^('),a_(2)^('),dots,a_(m)^(')),N(a_(m+1)^('),a_(m+2)^('),dots,a_(2m)^(')),dots:}],[{:N(a_((n-1)m+1)^('),a_((n-1)m+2)^('),dots,a_(nm)^('))) >= resp. <= ],[M((N)(a_(1),a_(2),dots,a_(m)),N(a_(m+1),a_(m+2),dots,a_(2m)),dots,:}],[{:N(a_((n-1)m+1),a_((n-1)m+2),dots,a_(nm)))]:}\begin{gathered}
\mathrm{M}\left(\mathrm{~N}\left(a_{1}^{\prime}, a_{2}^{\prime}, \ldots, a_{m}^{\prime}\right), \mathrm{N}\left(a_{m+1}^{\prime}, a_{m+2}^{\prime}, \ldots, a_{2 m}^{\prime}\right), \ldots\right. \\
\left.\mathrm{N}\left(a_{(n-1) m+1}^{\prime}, a_{(n-1) m+2}^{\prime}, \ldots, a_{n m}^{\prime}\right)\right) \geqq \mathrm{resp} . \leqq \\
\mathrm{M}\left(\mathrm{~N}\left(a_{1}, a_{2}, \ldots, a_{m}\right), \mathrm{N}\left(a_{m+1}, a_{m+2}, \ldots, a_{2 m}\right), \ldots,\right. \\
\left.\mathrm{N}\left(a_{(n-1) m+1}, a_{(n-1) m+2}, \ldots, a_{n m}\right)\right)
\end{gathered}
depending on the functionvarphi(psi^(-1))\varphi\left(\psi^{-1}\right)is non-concave resp. non-convex.
(^(1){ }^{1}) T. 2. 372-374 (1843).
(^(1){ }^{1}) T. 2. 372-374 (1843).
Let us designate byffthe functionvarphi(psi^(-1))\varphi\left(\psi^{-1}\right)and let us suppose, to fix the ideas, that this function is non-concave. The functionsvarphi^(-1),psi^(-1)\varphi^{-1}, \psi^{-1}being also increasing, the inequality to be demonstrated amounts to
sum_(i=1)^(n)f((sum_(j=1)^(m)(psi(a_((i-1)m+j)^(')))/(m))/(m)) >= sum_(i=1)^(n)f((sum_(j=1)^(m)(psi(a_((i-1)m+j)))/(m))/(m))\sum_{i=1}^{n} f\left(\frac{\sum_{j=1}^{m} \frac{\psi\left(a_{(i-1) m+j}^{\prime}\right)}{m}}{m}\right) \geqq \sum_{i=1}^{n} f\left(\frac{\sum_{j=1}^{m} \frac{\psi\left(a_{(i-1) m+j}\right)}{m}}{m}\right)
which is a consequence of the following property of Messrs. GH Hardy, JE Littlewood and G. Polya(^(2))\left({ }^{2}\right).
Iff(x)f(x)is a non-concave function and if(x_(i) >= x_(i+1),y_(i) >= y_(i+1),i=:}1,2,dots,n-1)\left(x_{i} \geq x_{i+1}, y_{i} \geqq y_{i+1}, i=\right. 1,2, \ldots, n-1)
{:[x_(1)+x_(2)+cdots+x_(i) >= y_(1)+y_(2)+cdots+y_(i)","quad i=1","2","dots","n-1","],[x_(1)+x_(2)+cdots+x_(n)=y_(1)+y_(2)+cdots+y_(n)]:}\begin{aligned}
& x_{1}+x_{2}+\cdots+x_{i} \geqq y_{1}+y_{2}+\cdots+y_{i}, \quad i=1,2, \ldots, n-1, \\
& x_{1}+x_{2}+\cdots+x_{n}=y_{1}+y_{2}+\cdots+y_{n}
\end{aligned}
we have inequality
sum_(i=1)^(n)f(x_(i)) >= sum_(i=1)^(n)f(y_(i))\sum_{i=1}^{n} f\left(x_{i}\right) \geqq \sum_{i=1}^{n} f\left(y_{i}\right)
In the case where the functionvarphi(psi^(-1))\varphi\left(\psi^{-1}\right)is convex (resp. concave) it is easy to find the cases where the equality is valid in (1).
The property of Le Cointe corresponds to the case wheren=2,varphi=x,psi=log xn=2, \varphi=x, \psi=\log x.
By particularizing the functionsvarphi,psi\varphi, \psiwe find various specific statements.
By particularizing the functionsvarphi,psi\varphi, \psiwe find various specific statements.
- ^((2)){ }^{(2)}„Some simple inequalities satisfied by convex function", Messenger of Math., 58, 145-152 (1929). See also "Inequalities", p. 89.