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T. Popoviciu, Sur quelques inegalités entre les fonction convexes (I), Comptes Rendus des séances de l’Académie des Sciences de Roumanie, 2 (1938), pp. 449-454 (in French).
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112. ON SOME INEQUALITIES BETWEEN CONVEX FUNCTIONS
(FIRST NOTE)
By tiberiu popoviciu, Mc. AS r.
(Meeting of March 4, 1938).
By tiberiu popoviciu, Mc. AS r.
(Meeting of March 4, 1938).
- Eithervarphi=varphi(x)\varphi=\varphi(x)a defined, uniform, continuous and convex function (of order I) in the closed interval ( 0,1 ). We assume thatvarphi(0)=0,varphi(1)=1quad0 <= varphi(x) <= 1\varphi(0)=0, \varphi(1)=1 \quad 0 \leqq \varphi(x) \leqq 1. It follows thatvarphi(x)\varphi(x)is a positive and increasing function for0 < x <= I0<x \leqq \mathrm{I}. The right derivativevarphi^(')(x)\varphi^{\prime}(x)exists, is increasing for0 <= x < I0 \leqq x<\mathrm{I}, but may not be bounded in (0,I0, \mathrm{I}).
Let us also consider a functionf=f(x)f=f(x), definite, uniform, continuous and non-concave of order0,1,dots,n0,1, \ldots, nin the meantime(0,1)^(1)(0,1)^{1}). We assume thatf(0)=a >= 0,f(I)=b <= I,a < bf(0)=a \geqq 0, f(\mathrm{I})=b \leqq \mathrm{I}, a<band we will designate by(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}the set of functions verifying all these properties. We see that(E_(a)^(b))_(n)sub(E_(a)^(b))_(n-1)\left(\mathrm{E}_{a}^{b}\right)_{n} \subset\left(\mathrm{E}_{a}^{b}\right)_{n-1}, SO(E_(a)^(b))_(n)sub(E_(a)^(b))_(o),n > 0\left(\mathrm{E}_{a}^{b}\right)_{n} \subset\left(\mathrm{E}_{a}^{b}\right)_{o}, n>0. In this work we
we will supposen > 0n>0. We will also indicate the results forn=0n=0, which are almost all known.
The problem we will examine in this work is the following: Given the functionvarphi\varphi, determine the maximum ofA_(varphi)(f)\mathrm{A}_{\varphi}(f)whenA(f)\mathrm{A}(f)is given andffruns through the whole(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}.
We asked
A=A(f)=int_(0)^(1)fdx,quadA_(varphi)=A_(varphi)(f)=int_(0)^(1)varphi(f)dx\mathrm{A}=\mathrm{A}(f)=\int_{0}^{1} f d x, \quad \mathrm{~A}_{\varphi}=\mathrm{A}_{\varphi}(f)=\int_{0}^{1} \varphi(f) d x
We will then make some applications, generalizing certain known results.
2. Let us call an elementary function of degreennany function whose(n-1)^("ème ")(n-1)^{\text {ème }}èderivative is a polygonal line. Such a function is therefore of the form
2. Let us call an elementary function of degreennany function whose(n-1)^("ème ")(n-1)^{\text {ème }}èderivative is a polygonal line. Such a function is therefore of the form
{:(I)g(x)=P(x)+sum_(i=1)^(m)c_(i)[(x-x_(i)+|x-x_(i)|)/(2(I-x_(i)))]^(n):}\begin{equation*}
g(x)=\mathrm{P}(x)+\sum_{i=1}^{m} c_{i}\left[\frac{x-x_{i}+\left|x-x_{i}\right|}{2\left(\mathrm{I}-x_{i}\right)}\right]^{n} \tag{I}
\end{equation*}
P(x)=a_(o)+a_(1)x+a_(2)x^(2)+dots+a_(n)quadx^(n-1),0 <= x_(1) < x_(2) < dots < x_(m) < I\mathrm{P}(x)=a_{o}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} \quad x^{n-1}, 0 \leqslant x_{1}<x_{2}<\ldots<x_{m}<\mathrm{I}.
We assume thatc_(i)!=0,i=1,2,dots,mc_{i} \neq 0, i=1,2, \ldots, mand we will then say that the elementary functiong(x)g(x)is atmmvertices. We will say that the vertices are at the pointsx_(1),x_(2),dots,x_(m)x_{1}, x_{2}, \ldots, x_{m}. Ifx_(1)=0x_{1}=0there is a summit at the pointo^(2)o^{2}). For the elementary functiong(x)g(x)hasmmsummits belong to(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}it is necessary and sufficient that one has
a_(o)=a,a_(r) >= 0,r=1,2,dots,n-1,c_(i) > 0,i=1,2,dots,m,P(1)+sum_(i=1)^(m)c_(i)=ba_{o}=a, a_{r} \geqq 0, r=1,2, \ldots, n-1, c_{i}>0, i=1,2, \ldots, m, \mathrm{P}(1)+\sum_{i=1}^{m} c_{i}=b.
In the following we only consider elementary functions belonging to(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}.
We assume thatc_(i)!=0,i=1,2,dots,mc_{i} \neq 0, i=1,2, \ldots, mand we will then say that the elementary functiong(x)g(x)is atmmvertices. We will say that the vertices are at the pointsx_(1),x_(2),dots,x_(m)x_{1}, x_{2}, \ldots, x_{m}. Ifx_(1)=0x_{1}=0there is a summit at the pointo^(2)o^{2}). For the elementary functiong(x)g(x)hasmmsummits belong to(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}it is necessary and sufficient that one has
a_(o)=a,a_(r) >= 0,r=1,2,dots,n-1,c_(i) > 0,i=1,2,dots,m,P(1)+sum_(i=1)^(m)c_(i)=ba_{o}=a, a_{r} \geqq 0, r=1,2, \ldots, n-1, c_{i}>0, i=1,2, \ldots, m, \mathrm{P}(1)+\sum_{i=1}^{m} c_{i}=b.
In the following we only consider elementary functions belonging to(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}.
According to the previous definition, any polynomial of degreen-In-Iis an elementary function of degreennwith o vertices. A polynomial of effective degreennis an elementary function of degreennat . I vertex, the vertex being at point 0 .
Among the I-vertex functions we find the following
{:(2)h_(lambda)(x)=a+(b-a)[(x-lambda+|x-lambda|)/(2(I-lambda))]^(n)","quad(0 <= lambda < I).:}\begin{equation*}
h_{\lambda}(x)=a+(b-a)\left[\frac{x-\lambda+|x-\lambda|}{2(I-\lambda)}\right]^{n}, \quad(0 \leqq \lambda<I) . \tag{2}
\end{equation*}
In the casen=1n=1all one-vertex functions are of this form.
Function (1) can be written
g(x)=((b-a)P(x)+a[P(I)-b])/(b-a)+(I)/(b-a)sum_(i=1)^(m)c_(i)h_(x_(i))(x),g(x)=\frac{(b-a) \mathrm{P}(x)+a[\mathrm{P}(\mathrm{I})-b]}{b-a}+\frac{\mathrm{I}}{b-a} \sum_{i=1}^{m} c_{i} h_{x_{i}}(x),
which shows us that it is the (generalized) arithmetic mean of a polynomial of degreenn-I and ofmmfunctions of the form (2).
Any function of(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}is the limit of a uniformly convergent sequence of elementary functions of(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}. This property can also be specified in the following way. Ifffbelongs to(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}, the derivativesf^('),f^(''),dots,f^((n-1))f^{\prime}, f^{\prime \prime}, \ldots, f^{(n-1)}exist foro <= x < I\mathrm{o} \leqq x<\mathrm{I}. The functionffis then the limit of a uniformly convergent sequence of elementary functions of(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}, these functions all having the same first termP(x)=a+xf^(')(0)+(x^(2))/(2!)f^('')(0)+dots+(x^(n-1))/((n-1)!)f^((n-1))(0)P(x)=a+x f^{\prime}(0)+\frac{x^{2}}{2!} f^{\prime \prime}(0)+\ldots+\frac{x^{n-1}}{(n-1)!} f^{(n-1)}(0). Moreover, we can assume that the value of the integralAAis the same for all these functions.
These considerations apply, in particular, to the functionvarphi\varphi, which belongs to(E_(0)^(1))_(1)\left(\mathrm{E}_{0}^{1}\right)_{1}, and serve to establish the following assertions with complete rigor.
3. Letvarphi\varphigiven andg(x)g(x)an elementary function of(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}havingm > 1m>1vertices and given by formula (I). Let us construct the function.
3. Letvarphi\varphigiven andg(x)g(x)an elementary function of(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}havingm > 1m>1vertices and given by formula (I). Let us construct the function.
g^(**)(x)=P(x)+sum_(i=1)^(m-2)c_(i)[(x-x_(i)+|x-x_(i)|)/(2(I-x_(i)))]^(n)+c[(x-mu+|x-mu|)/(2(I-mu))]^(n)g^{*}(x)=\mathrm{P}(x)+\sum_{i=1}^{m-2} c_{i}\left[\frac{x-x_{i}+\left|x-x_{i}\right|}{2\left(\mathrm{I}-x_{i}\right)}\right]^{n}+c\left[\frac{x-\mu+|x-\mu|}{2(\mathrm{I}-\mu)}\right]^{n}
and determine the constantsc,muc, \muso that we haveg^(**)(I)=bg^{*}(\mathrm{I})=b,A(g^(**))=A(g)\mathrm{A}\left(g^{*}\right)=\mathrm{A}(g). We find thatc=c_(m-1)+c_(m)c=c_{m-1}+c_{m}Andmu=(c_(m-1)x_(m-1)+c_(m)x_(m))/(c_(m-1)+c_(m))\mu=\frac{c_{m-1} x_{m-1}+c_{m} x_{m}}{c_{m-1}+c_{m}}, SOx_(m-1) < mu < x_(m),c > 0x_{m-1}<\mu<x_{m}, c>0Andg^(**)g^{*}is indeed an elementary function of degreennhasm-1m-1summits belonging to(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}.
There is a numbermu^('),mu < mu^(') < I\mu^{\prime}, \mu<\mu^{\prime}<Isuch that we have
(3)quadg^(**)(x) <= g(x)\quad g^{*}(x) \leqq g(x)following thatx <= mu^(')x \leqq \mu^{\prime}, Forx_(m-1) < x < I^(4)x_{m-1}<x<I^{4})
(3)quadg^(**)(x) <= g(x)\quad g^{*}(x) \leqq g(x)following thatx <= mu^(')x \leqq \mu^{\prime}, Forx_(m-1) < x < I^(4)x_{m-1}<x<I^{4})
Consider the function
varphi_(lambda)(x)=(x-lambda+|x-lambda|)/(2),quad0 <= lambda < 1\varphi_{\lambda}(x)=\frac{x-\lambda+|x-\lambda|}{2}, \quad 0 \leqq \lambda<1
Ifg(x_(m-1)) < lambda < bg\left(x_{m-1}\right)<\lambda<b, we have
Avarphi_(lambda)(g^(**))-Avarphi_(lambda)(g)=int_(i^(**))^(1)g^(**)dx-int_(t)^(1)gdx+lambda(t^(**)-t)\mathrm{A} \varphi_{\lambda}\left(g^{*}\right)-\mathrm{A} \varphi_{\lambda}(g)=\int_{i^{*}}^{1} g^{*} d x-\int_{t}^{1} g d x+\lambda\left(t^{*}-t\right)
Orlambda=g(t)=g^(**)(t^(**))\lambda=g(t)=g^{*}\left(t^{*}\right)and property (3) shows us that this function oflambda\lambdacancels out forlambda=g(x_(m-1))\lambda=g\left(x_{m-1}\right)Andlambda=b\lambda=b, is increasing forg(x_(m-1)) < lambda < g(mu^('))g\left(x_{m-1}\right)<\lambda<g\left(\mu^{\prime}\right)and decreasing forg(mu^(')) < lambda < bg\left(\mu^{\prime}\right)<\lambda<b. It therefore follows that
{:(4)Avarphi_(lambda)(g^(**)) > Avarphi_(lambda)(g)","quad g(x_(m-1)) < lambda < b:}\begin{equation*}
\mathrm{A} \varphi_{\lambda}\left(g^{*}\right)>\mathrm{A} \varphi_{\lambda}(g), \quad g\left(x_{m-1}\right)<\lambda<b \tag{4}
\end{equation*}
- We now propose to demonstrate thatA_(varphi)(g^(**))>>A_(varphi)(g)\mathrm{A}_{\varphi}\left(g^{*}\right)> >\mathrm{A}_{\varphi}(g). The inequality with the sign>=\geqqresults from (4), but it is a matter of proving that equality is not possible.
Consider the non-decreasing and non-concave function,
Phi_(epsi)(x)={[varphi(x)","," dans "(0","I-epsi)],[varphi(I-epsi)+(x-I+epsi)varphi(I-epsi)","," dans "(I-epsi","I)","]:}\Phi_{\varepsilon}(x)=\left\{\begin{array}{lr}
\varphi(x), & \text { dans }(0, \mathrm{I}-\varepsilon) \\
\varphi(\mathrm{I}-\varepsilon)+(x-\mathrm{I}+\varepsilon) \varphi(\mathrm{I}-\varepsilon), & \text { dans }(\mathrm{I}-\varepsilon, \mathrm{I}),
\end{array}\right.
Orepsi\varepsilonis a fairly small positive number andvarphi^(')\varphi^{\prime}is the right derivative ofvarphi\varphi. We have
{:(5)Phi_(z)(x)=int_(0)^(1)varphi_(t)(x)dPhi_(epsi)^(')(t)+alpha x:}\begin{equation*}
\Phi_{z}(x)=\int_{0}^{1} \varphi_{t}(x) d \Phi_{\varepsilon}^{\prime}(t)+\alpha x \tag{5}
\end{equation*}
the integral being Stieltjes andalpha\alphaa constant of no importance to us^(5){ }^{5}).
It can be easily demonstrated that
A_(Phi Sigma)(g^(**))-A_(Phi epsi)(g)=int_(0)^(1){(A)varphi_(l)(g^(**))-Apsi_(t)(g)}dPhi_(epsi)^(')(t)\mathrm{A}_{\Phi \Sigma}\left(g^{*}\right)-\mathrm{A}_{\Phi \varepsilon}(g)=\int_{0}^{1}\left\{\mathrm{~A} \varphi_{l}\left(g^{*}\right)-\mathrm{A} \psi_{t}(g)\right\} d \Phi_{\varepsilon}^{\prime}(t)
- This formula is analogous to that given for non-convex functions by Messrs. W. Blascke and G. Pick, see: „Distanzschätzungen im Funktionenraum II". Math. Ann. Bd. 77 (1916) p. 277-300. The introduction of the functionPhi_(epsi)\boldsymbol{\Phi}_{\varepsilon}is only necessary ifb=1b=1. It may happen in fact that the right derivative ofvarphi(x)\varphi(x)is not of bounded variation.
which results from the fact that the integral ofStie1tjesS t i e 1 t j e sof the second member exists.
From this formula and the definition ofPhi_(epsi)\Phi_{\varepsilon}it follows thatA_(Phi)(g^(**))\mathrm{A}_{\Phi}\left(g^{*}\right)--APhi_(epsi)(g)\mathrm{A} \Phi_{\varepsilon}(g)is positive and does not decrease whenepsi\varepsilondecreases. We have therefore
lim_(3rarr0)[A_(Phi epsi)(g^(**))-A_(Phi epsi)(g)]=A_(varphi)(g^(**))-A_(varphi)(g) > 0\lim _{3 \rightarrow 0}\left[\mathrm{~A}_{\Phi \varepsilon}\left(g^{*}\right)-\mathrm{A}_{\Phi \varepsilon}(g)\right]=\mathrm{A}_{\varphi}\left(g^{*}\right)-\mathrm{A}_{\varphi}(g)>0
- Now consider a functionffof(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}. Let the elementary function of degreennat the summit
h(x)=a+xf^(')(o)+(x^(2))/(2!)f^('')(o)+dots+(x^(n-1))/((n-I)!)f^((n-1))(o)+d[(x-rho+|x-rho|)/(2(I-rho))]^(n)h(x)=a+x f^{\prime}(\mathrm{o})+\frac{x^{2}}{2!} f^{\prime \prime}(\mathrm{o})+\ldots+\frac{x^{n-1}}{(n-\mathrm{I})!} f^{(n-1)}(\mathrm{o})+d\left[\frac{x-\rho+|x-\rho|}{2(\mathrm{I}-\rho)}\right]^{n}
Ord,rhod, \rhoare completely determined by the conditionsh(I)=bh(I)=b,A(h)=A(f).h(x)\mathrm{A}(h)=\mathrm{A}(f) . h(x)belongs to(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}and we haveA_(varphi)(h) >= A_(varphi)(f)\mathrm{A}_{\varphi}(h) \geqq \mathrm{A}_{\varphi}(f)We propose to demonstrate that equality is only possible ifh=fh=f.
Let us take formula (5); we have
A_(Phi s)(h)-A_(Phi s)(f)=int_(0)^(1){A_(varphi t)(h)-A_(varphi t)(f)}dPhi_(epsi)^(')(t)\mathrm{A}_{\Phi s}(h)-\mathrm{A}_{\Phi s}(f)=\int_{0}^{1}\left\{\mathrm{~A}_{\varphi t}(h)-\mathrm{A}_{\varphi t}(f)\right\} d \Phi_{\varepsilon}^{\prime}(t)
The functionAvarphi_(i)(h)-Avarphi_(t)(f)\mathrm{A} \varphi_{i}(h)-\mathrm{A} \varphi_{t}(f)is continuous inttand is>= 0\geqq 0.
It is easily demonstrated that ifffdoes not reduce to an elementary function of degreennat o or I vertex, this function is not identically zero. We therefore haveA_(Phi widehat(epsi))(h)-A_(Phi widehat(epsi))(f) > 0\mathrm{A}_{\Phi \widehat{\varepsilon}}(h)-\mathrm{A}_{\Phi \widehat{\varepsilon}}(f)>0, at least forepsi\varepsilonquite small. The requested property results as above.
It is easily demonstrated that ifffdoes not reduce to an elementary function of degreennat o or I vertex, this function is not identically zero. We therefore haveA_(Phi widehat(epsi))(h)-A_(Phi widehat(epsi))(f) > 0\mathrm{A}_{\Phi \widehat{\varepsilon}}(h)-\mathrm{A}_{\Phi \widehat{\varepsilon}}(f)>0, at least forepsi\varepsilonquite small. The requested property results as above.
So finally:
If the functionvarphi\varphiand the numberA(f)\mathrm{A}(f)are given, the maximum ofA_(varphi)(f)\mathrm{A}_{\varphi}(f)In(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}can only be achieved for an elementary function of degreennat most I summit.
If the functionvarphi\varphiand the numberA(f)\mathrm{A}(f)are given, the maximum ofA_(varphi)(f)\mathrm{A}_{\varphi}(f)In(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}can only be achieved for an elementary function of degreennat most I summit.
In the casen=1n=1the functionh(x)h(x)is completely determined by the value of the integral A . We therefore have the following property:
Ifvarphi\varphiis given and f is a continuous, non-decreasing, non-concave function in(O,I)(\mathrm{O}, \mathrm{I}), we have
{:(6)int_(0)^(1)varphi(f)dx <= (b+a-2(A))/(b-a)varphi(a)+(2((A)-a))/((b-a)^(2))int_(a)^(b)varphi(x)dx","quadA=int_(0)^(1)fdx:}\begin{equation*}
\int_{0}^{1} \varphi(f) d x \leqq \frac{b+a-2 \mathrm{~A}}{b-a} \varphi(a)+\frac{2(\mathrm{~A}-a)}{(b-a)^{2}} \int_{a}^{b} \varphi(x) d x, \quad \mathrm{~A}=\int_{0}^{1} f d x \tag{6}
\end{equation*}
equality being possible only for the function
t=a+(b-a)(x-lambda+|x-lambda|)/(2(I-lambda)),lambda=(b+a-2(A))/(b-a)t=a+(b-a) \frac{x-\lambda+|x-\lambda|}{2(\mathrm{I}-\lambda)}, \lambda=\frac{b+a-2 \mathrm{~A}}{b-a}
- We also say that such a function is (n+1n+1)-times monotonous.
- We have already considered such functions in a previous work, see: Tiberiu Popoviciu, „On the extension of convex functions of higher order". Bull. Math. Soc. Roum. Sc., t. 36 (1934), pp. 75-108.
- In the casem=2m=2, we obviously have
- The position of the pointmu^(')\mu^{\prime}in the meantime (mu,r\mu, \mathrm{r}) depends on the values ​​ofc_(m-1),c_(m)c_{m-1}, c_{m}. In particular, ifn=1n=1we always havemu < mu^(') < x_(m)\mu<\mu^{\prime}<x_{m}.