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T. Popoviciu, Sur quelques inegalités entre les fonctions convexes (III), Comptes Rendus de l’Instit. des Sci. de Roumanie, 3 (1939) no. 4, pp. 396-402 (in French).
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Comptes Rendus de l’Instit. des Sci. de Roumanie
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[Zbl 0021.30305, JFM 65.0215.01]
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REPORTS
SESSIONS
OF
THE ROMANIAN INSTITUTE OF SCIENCES
FORMER
ACADEMY OF SCIENCES OF ROMANIA
PUBLISHED BY THE EDITORIAL BOARD OF THE INSTITUTE
ACADEMY OF SCIENCES OF ROMANIA
PUBLISHED BY THE EDITORIAL BOARD OF THE INSTITUTE
Published in Bucharest on July 1, 1939
MINUTES OF THE MEETINGS
OF
THE ROMANIAN INSTITUTE OF SCIENCES
FORMER ACADEMY OF SCIENCES OF ROMANIA
SUMMARY
371. ON SOME INEQUALITIES BETWEEN CONVEX FUNCTIONS
(THIRD NOTE)
By TIBERIU POPOVICIU, Mc. ISR
(Session of May 6, 1939)
I. Let us pose, retaining the notations of the two previous notes^(1){ }^{1}),
(Session of May 6, 1939)
I. Let us pose, retaining the notations of the two previous notes^(1){ }^{1}),
B(varphi)=A_(varphi)-varphi(A).\mathrm{B}(\varphi)=\mathrm{A}_{\varphi}-\varphi(\mathrm{A}) .
We have the classical inequalityB(varphi) >= 0\mathrm{B}(\varphi) \geqq 0.
We will look for a contrary inequality, so the maximum ofB(varphi)\mathrm{B}(\varphi), whenvarphi\varphiis given andffare functions of(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}for onenngiven.
We will look for a contrary inequality, so the maximum ofB(varphi)\mathrm{B}(\varphi), whenvarphi\varphiis given andffare functions of(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}for onenngiven.
In this Note we will examine the casen=1n=1and we will also say a few words about the casen=0n=0. The casen > In>Iwill be studied in the fifth Note.
Let us remember thatvarphi\varphiis continuous and convex in the closed interval(0.1)(0.1). We can assume thatvarphi(0)=0,varphi(1)=I,0 <= varphi <= I\varphi(0)=0, \varphi(1)=I, 0 \leqq \varphi \leqq I, without restricting the generality, sinceB(varphi)B(\varphi)does not vary when adding tovarphi\varphia linear function. The functionvarphi\varphiis then positive and grow-
health for0 < x <= I0<x \leqq \mathrm{I}. The whole(E_(a)^(b))_(n)\left(\mathrm{E}_{a}^{b}\right)_{n}is formed by the functionsff, continuous and non-concave of order0,1,dots,n0.1, \ldots, nin the interval ( 0,1 ) such thatf(O)=a >= oquad f(I)=b <= If(\mathrm{O})=a \geqq \mathrm{o} \quad f(\mathrm{I})=b \leqq \mathrm{I}. The functionvarphi\varphiis a convex function of(E_(0)^(1))_(1)\left(\mathrm{E}_{0}^{1}\right)_{1}.
Note that in all our formulas only the values ​​ofvarphi\varphiIn(a,b)(a, b). It is therefore sufficient to assume thatvarphi\varphiis a function that is continuous and convex in the closed interval (a,ba, b). If the right derivativevarphi_(d)^(')(a)\varphi_{d}^{\prime}(a)inhashasand the left derivativevarphi_(g)^(')(b)\varphi_{g}^{\prime}(b)are finished, we can modify the functionvarphi\varphiin the intervals(o,a)(\mathrm{o}, a), (b,Ib, \mathrm{I}) so that it becomes, up to a linear additive function, a convex function of(E_(0)^(1))_(1)\left(\mathrm{E}_{0}^{1}\right)_{1}.
Let us also note that we have analogous inequalities, but in opposite directions, ifvarphi\varphiinstead of being convex is concave. The inequalities are true, of course, also whenvarphi\varphiis only non-concave (or non-convex). Only the uniqueness of the maxima (or minima) can possibly be faulty.
2. The maximum problem posed above has been solved forn=on=oby MK K no pp^(1){ }^{1}). In this case we have
2. The maximum problem posed above has been solved forn=on=oby MK K no pp^(1){ }^{1}). In this case we have
{:(I)B(varphi) <= max_((a,b))[((b-x)varphi(a)+(x-a)varphi(b))/(b-a)-varphi(x)]:}\begin{equation*}
\mathrm{B}(\varphi) \leqq \max _{(a, b)}\left[\frac{(b-x) \varphi(a)+(x-a) \varphi(b)}{b-a}-\varphi(x)\right] \tag{I}
\end{equation*}
which immediately results from the inequality
{:[((b-A)varphi(a)+(A-a)varphi(b))/(b-a)-A_(varphi)=(I)/(b-a)int_(o)^(1)[(f-a)varphi(a)-(b-a)varphi(f)+],[+(b-f)varphi(b)]dx >= 0]:}\begin{gathered}
\frac{(b-\mathrm{A}) \varphi(a)+(\mathrm{A}-a) \varphi(b)}{b-a}-\mathrm{A}_{\varphi}=\frac{\mathrm{I}}{b-a} \int_{o}^{1}[(f-a) \varphi(a)-(b-a) \varphi(f)+ \\
+(b-f) \varphi(b)] d x \geqq 0
\end{gathered}
The maximum is reached for a single valuex_(1)x_{1}ofxx, since the function of the second member of (I) is concave. The maximum of B (%) cannot be reached by a function of(E_(a)^(b))_(o)\left(\mathrm{E}_{a}^{b}\right)_{o}, but only by certain limit functions of this family. This is the function
f={[a",",0 <= x <= x(b-x_(1))/(b-a_(1))],[b",",(b-x_(1))/(b-a) < x <= 1]:}f= \begin{cases}a, & 0 \leqq x \leqq x \frac{b-x_{1}}{b-a_{1}} \\ b, & \frac{b-x_{1}}{b-a}<x \leqq 1\end{cases}
It can be noted thatx_(1)x_{1}, never coincides withaaOrbb. This number can be(:ou:)(a+b)/(2)\langle o u\rangle \frac{a+b}{2}. So that we havex_(1) >= (a+b)/(2)x_{1} \geqq \frac{a+b}{2}it is necessary and sufficient that
(varphi(b)-varphi(a))/(b-a) >= varphi_(g)^(')((a+b)/(2))\frac{\varphi(b)-\varphi(a)}{b-a} \geqq \varphi_{g}^{\prime}\left(\frac{a+b}{2}\right)
or
(2)quad(1)/(b-a)int_(a)^(1)varphi_(g)^(')(x)dx >= varphi_(g)^(')((a+b)/(2))\quad \frac{1}{b-a} \int_{a}^{1} \varphi_{g}^{\prime}(x) d x \geqq \varphi_{g}^{\prime}\left(\frac{a+b}{2}\right),
by designating byvarphi_(g)^(')\varphi_{g}^{\prime}the left derivative ofvarphi\varphi. So that it is so for all the values ​​ofa,ba, bit is necessary and sufficient thatvarphi\varphibe a function of(E_(o)^(1))_(2)\left(\mathrm{E}_{o}^{1}\right)_{2}, therefore non-concave of order 2.
(2)quad(1)/(b-a)int_(a)^(1)varphi_(g)^(')(x)dx >= varphi_(g)^(')((a+b)/(2))\quad \frac{1}{b-a} \int_{a}^{1} \varphi_{g}^{\prime}(x) d x \geqq \varphi_{g}^{\prime}\left(\frac{a+b}{2}\right),
by designating byvarphi_(g)^(')\varphi_{g}^{\prime}the left derivative ofvarphi\varphi. So that it is so for all the values ​​ofa,ba, bit is necessary and sufficient thatvarphi\varphibe a function of(E_(o)^(1))_(2)\left(\mathrm{E}_{o}^{1}\right)_{2}, therefore non-concave of order 2.
The demonstration of this fact is not difficult. The functionvarphi_(g)^(')\varphi_{g}^{\prime}is increasing and positive. I say that it must be continuous in the open interval (O,I\mathrm{O}, \mathrm{I}). So, in fact,x_(0),O < x_(0) < Ix_{0}, \mathrm{O}<x_{0}<\mathrm{I}, a point of possible discontinuity. Or also
0 < epsi < (varphi_(g)^(')(x_(0)+0)-varphi_(g)^(')(x_(0)-0))/(2)0<\varepsilon<\frac{\varphi_{g}^{\prime}\left(x_{0}+0\right)-\varphi_{g}^{\prime}\left(x_{0}-0\right)}{2}
We can then find aeta > 0\eta>0, such that we have0 < x_(0)-eta < x_(0)++2eta < 10<x_{0}-\eta<x_{0}+ +2 \eta<1Andvarphi_(g)^(')(x_(0)+2eta) < epsi+varphi_(g)^(')(x_(0)+0)\varphi_{g}^{\prime}\left(x_{0}+2 \eta\right)<\varepsilon+\varphi_{g}^{\prime}\left(x_{0}+0\right). We have, by takinga=x_(0)-eta,b=x_(0)+2etaa=x_{0}-\eta, b=x_{0}+2 \eta,
{:[(1)/(3eta)int_(x_(0)-eta)^(x_(0)+2*eta)varphi_(g)^(')(x)dx-varphi_(g)^(')(x_(0)+(eta)/(2)) < (varphi_(g)(x_(0)-0+2[epsi+varphi_(g)^(')(x_(0)+0)])/(3)-],[quad-varphi_(g)^(')(x_(0)+0)=-(varphi_(g)^(')(x_(0)+0)-varphi_(g)^(')(x_(0)-0))/(3)+(2epsi)/(3) < 0]:}\begin{aligned}
& \frac{1}{3 \eta} \int_{x_{0}-\eta}^{x_{0}+2 \cdot \eta} \varphi_{g}^{\prime}(x) d x-\varphi_{g}^{\prime}\left(x_{0}+\frac{\eta}{2}\right)<\frac{\varphi_{g}\left(x_{0}-0+2\left[\varepsilon+\varphi_{g}^{\prime}\left(x_{0}+0\right)\right]\right.}{3}- \\
& \quad-\varphi_{g}^{\prime}\left(x_{0}+0\right)=-\frac{\varphi_{g}^{\prime}\left(x_{0}+0\right)-\varphi_{g}^{\prime}\left(x_{0}-0\right)}{3}+\frac{2 \varepsilon}{3}<0
\end{aligned}
which is in contradiction with (2). The functionvarphi_(g)^(')\varphi_{g}^{\prime}must therefore be continuous^(1){ }^{1}). We also see that the functionvarphi_(g)^(')+alpha x\varphi_{g}^{\prime}+\alpha xstill verifies property (2), regardlessalpha\alpha. It is easily demonstrated on (2) thatvarphi_(g)^(')+alpha x\varphi_{g}^{\prime}+\alpha xcannot reach its maximum in an interval(a,b)(a, b)that at the endsaaOrbb, unless it is constant in (a,ba, b). According to a remark by MS S aks^(2){ }^{2}), the functionvarphi_(g)^(')\varphi_{g}^{\prime}is non-concave of orderII, SOvarphi\varphiis non-concave of order 2.
- It follows thatvarphi\varphihas a continuous derivative.
- S. Saks „O funckjach wypuklych i podharmoniczhych” Mathesis Polska, 6, 43-64 (1931).
We demonstrate in the same way that to havex_(1) <= (a+b)/(2)x_{1} \leqq \frac{a+b}{2}, for all values ​​ofaaAndbb, it is necessary and sufficient thatvarphi\varphibe non-convex of order 2.
Forvarphi=log x,a > 0\varphi=\log x, a>0, inequality (1) gives us
(A)/(G) <= (c-I)/(c^((c-1-log c)/((c-1)log c))*log c)=(I-J)/(e*log((I)/(c)))((I)/(c))^((1)/(1-c)),quad c=(a)/(b)\frac{\mathrm{A}}{\mathrm{G}} \leqq \frac{c-\mathrm{I}}{c^{\frac{c-1-\log c}{(c-1) \log c}} \cdot \log c}=\frac{\mathrm{I}-\mathrm{J}}{e \cdot \log \frac{\mathrm{I}}{c}}\left(\frac{\mathrm{I}}{c}\right)^{\frac{1}{1-c}}, \quad c=\frac{a}{b}
OrG=exp.int_(0)^(1)log fdx\mathrm{G}=\exp . \int_{0}^{1} \log f d xis the geometric mean offf. We have demonstrated this inequality in a previous work^(1){ }^{1}).
3. Let us now move on to the study of the casen=1n=1. Formula (6) and the results of the first note allow us to state the following property
3. Let us now move on to the study of the casen=1n=1. Formula (6) and the results of the first note allow us to state the following property
Ifvarphi\varphiis a function of the form indicated andffa function of(E_(a)^(b))_(1)\left(\mathrm{E}_{a}^{b}\right)_{1}, we have inequality
(3)A_(psi)-varphi(A)幺_((a,(a+b)/(2)))max_(a)[varphi(a)+(2(x-a))/((b-a)^(2))int_(a)^(b)[varphi(t)-varphi(a)]dt-varphi(x)]\mathrm{A}_{\psi}-\varphi(\mathrm{A}) \underset{\left(a, \frac{a+b}{2}\right)}{幺} \max _{a}\left[\varphi(a)+\frac{2(x-a)}{(b-a)^{2}} \int_{a}^{b}[\varphi(t)-\varphi(a)] d t-\varphi(x)\right]幺
equality being possible only for a single valuex_(2)x_{2}ofxxand for function
(4)f=a+(b-a)(x-lambda+|x-lambda|)/(2(1-lambda)),lambda=(b+a-2x_(2))/(b-a)f=a+(b-a) \frac{x-\lambda+|x-\lambda|}{2(1-\lambda)}, \lambda=\frac{b+a-2 x_{2}}{b-a}only.
(3)A_(psi)-varphi(A)幺_((a,(a+b)/(2)))max_(a)[varphi(a)+(2(x-a))/((b-a)^(2))int_(a)^(b)[varphi(t)-varphi(a)]dt-varphi(x)]\mathrm{A}_{\psi}-\varphi(\mathrm{A}) \underset{\left(a, \frac{a+b}{2}\right)}{幺} \max _{a}\left[\varphi(a)+\frac{2(x-a)}{(b-a)^{2}} \int_{a}^{b}[\varphi(t)-\varphi(a)] d t-\varphi(x)\right]幺
equality being possible only for a single valuex_(2)x_{2}ofxxand for function
(4)f=a+(b-a)(x-lambda+|x-lambda|)/(2(1-lambda)),lambda=(b+a-2x_(2))/(b-a)f=a+(b-a) \frac{x-\lambda+|x-\lambda|}{2(1-\lambda)}, \lambda=\frac{b+a-2 x_{2}}{b-a}only.
The numberx_(2)x_{2}never coincides withaaand maybe> ou <>o u<that(2a+b)/(3)\frac{2 a+b}{3}. To havex_(2) >= (2a+b)/(3)x_{2} \geqq \frac{2 a+b}{3}it is necessary and sufficient that one has(2)/((b-a)^(2))int_(a)^(b)[varphi(t)-varphi(a)]dt-varphi_(g)^(')((2a+b)/(3))=(2)/((b-a)^(2))int_(a)^(b)(b-t)varphi_(g)^(')(t)dt-\frac{2}{(b-a)^{2}} \int_{a}^{b}[\varphi(t)-\varphi(a)] d t-\varphi_{g}^{\prime}\left(\frac{2 a+b}{3}\right)=\frac{2}{(b-a)^{2}} \int_{a}^{b}(b-t) \varphi_{g}^{\prime}(t) d t-
-varphi_(g)^(')((2a+b)/(3)) >= 0.-\varphi_{g}^{\prime}\left(\frac{2 a+b}{3}\right) \geqq 0 .
- Tiberiu Popovici „Asupra mediilor aritmetice şi geometrice" Gazeta Matematică, 40, 155-160 (1934). M. Knop also reported this inequality, as an application, with a slight error in the second member.
We demonstrate exactly as above for inequality (2) that:
So thatx_(2)x_{2}either>= (2a+b)/(3)\geqq \frac{2 a+b}{3}, whatever a andbb, it is necessary and sufficient that the functionvarphi\varphiis non-concave of order 2, therefore a function of(E_(0)^(1))_(2)\left(\mathrm{E}_{0}^{1}\right)_{2}.
So thatx_(2)x_{2}either>= (2a+b)/(3)\geqq \frac{2 a+b}{3}, whatever a andbb, it is necessary and sufficient that the functionvarphi\varphiis non-concave of order 2, therefore a function of(E_(0)^(1))_(2)\left(\mathrm{E}_{0}^{1}\right)_{2}.
So thatx_(2)x_{2}either<= (2a+b)/(3)\leqq \frac{2 a+b}{3}, whateveraaAndbb, it is necessary and sufficient that the functionvarphi\varphibe non-convex of order 2.
The numberx_(2)x_{2}may also coincide with(a+b)/(2)\frac{a+b}{2}. We then have the following statement.
Ifvarphi\varphiis a function of the form indicated,ffa function of(E_(a)^(b))_(1)\left(\mathrm{E}_{a}^{b}\right)_{1}and if
{:(5)(2)/((b-a)^(2))int_(a)^(b)[varphi(t)-varphi(a)]dt-varphi_(g)^(')((a+b)/(2)) >= 0:}\begin{equation*}
\frac{2}{(b-a)^{2}} \int_{a}^{b}[\varphi(t)-\varphi(a)] d t-\varphi_{g}^{\prime}\left(\frac{a+b}{2}\right) \geqq 0 \tag{5}
\end{equation*}
we have inequality
A_(varphi)-varphi(A) <= (I)/(b-a)int_(a)^(b)varphi(t)dt-varphi((a+b)/(2))\mathrm{A}_{\varphi}-\varphi(\mathrm{A}) \leqq \frac{\mathrm{I}}{b-a} \int_{a}^{b} \varphi(t) d t-\varphi\left(\frac{a+b}{2}\right)
equality being possible only for the functionf=a+(b-a)xf=a+(b-a) x.
But, it should be noted that this cannot happen for all values ​​of a and b. The property results from the above ifvarphi\varphiis not non-concave of order 2. Ifvarphi\varphiis non-concave of order 2 the derivativevarphi^(')\varphi^{\prime}exists, is continuous, increasing and non-concave of order I. Letxi,0 < xi < 1\xi, 0<\xi<1a point where the second derivativevarphi^('')(xi)\varphi^{\prime \prime}(\xi)exists and is> 0>0. There is obviously such a point.^(1){ }^{1}). So thenepsi\varepsilona positive number and< (2varphi^('')(xi))/(3)<\frac{2 \varphi^{\prime \prime}(\xi)}{3}Andeta > 0\eta>0such as0 < xi-eta < xi+eta < 10<\xi-\eta<\xi+\eta<1and that
But, it should be noted that this cannot happen for all values ​​of a and b. The property results from the above ifvarphi\varphiis not non-concave of order 2. Ifvarphi\varphiis non-concave of order 2 the derivativevarphi^(')\varphi^{\prime}exists, is continuous, increasing and non-concave of order I. Letxi,0 < xi < 1\xi, 0<\xi<1a point where the second derivativevarphi^('')(xi)\varphi^{\prime \prime}(\xi)exists and is> 0>0. There is obviously such a point.^(1){ }^{1}). So thenepsi\varepsilona positive number and< (2varphi^('')(xi))/(3)<\frac{2 \varphi^{\prime \prime}(\xi)}{3}Andeta > 0\eta>0such as0 < xi-eta < xi+eta < 10<\xi-\eta<\xi+\eta<1and that
(varphi^(')(xi)-varphi^(')(xi-eta))/(eta) > varphi^('')(xi)-epsiquad,quad(varphi^(')(xi+eta)-varphi^(')(xi))/(eta) < varphi^('')(xi)+epsi.\frac{\varphi^{\prime}(\xi)-\varphi^{\prime}(\xi-\eta)}{\eta}>\varphi^{\prime \prime}(\xi)-\varepsilon \quad, \quad \frac{\varphi^{\prime}(\xi+\eta)-\varphi^{\prime}(\xi)}{\eta}<\varphi^{\prime \prime}(\xi)+\varepsilon .
If we takea=xi-eta,b=xi+etaa=\xi-\eta, b=\xi+\etaand if we take into account the fact thatvarphi^(')\varphi^{\prime}remains, in the interval (xi-eta,xi+eta\xi-\eta, \xi+\eta), not above the seg-
straight lines joining the points of the curvey=varphi^(')(x)y=\varphi^{\prime}(x)Forx=xi-etax=\xi-\eta,xi,xi+eta\xi, \xi+\eta, we find that
{:[(I)/(2eta^(2))int_(xi-eta)^(xi+eta)(xi+eta-t)varphi^(')(t)dt-varphi^(')(xi) <= (eta)/(I2)[(varphi^(')(xi+eta)-varphi^(')(xi))/(eta)-:}],[{:-(5varphi^(')(xi)-varphi^(')(xi-eta))/(eta)] < (eta)/(I2)[-4varphi^('')(xi)+6epsi] < 0]:}\begin{aligned} \frac{\mathrm{I}}{2 \eta^{2}} \int_{\xi-\eta}^{\xi+\eta}(\xi+\eta-t) \varphi^{\prime}(t) d t-\varphi^{\prime}(\xi) \leqq \frac{\eta}{\mathrm{I} 2}\left[\frac{\varphi^{\prime}(\xi+\eta)-\varphi^{\prime}(\xi)}{\eta}-\right. \\ \left.-\frac{5 \varphi^{\prime}(\xi)-\varphi^{\prime}(\xi-\eta)}{\eta}\right]<\frac{\eta}{\mathrm{I} 2}\left[-4 \varphi^{\prime \prime}(\xi)+6 \varepsilon\right]<0 \end{aligned}
which demonstrates the property.
We do not insist on this question. In general, it can be demonstrated that if for aaagiven the numberbbis close enough toaawe necessarily havex_(2) < (a+b)/(2)x_{2}<\frac{a+b}{2}.
4. Let us make some applications of the previous formulas.
I^(0).varphi=x^(p),p > I,a=0,b=I\mathrm{I}^{0} . \varphi=x^{p}, p>\mathrm{I}, a=0, b=\mathrm{I}. Inequality (5) is written2^(p) >= p(p+I)2^{p} \geqq p(p+\mathrm{I})and we can state the following property.
{:[(I)/(2eta^(2))int_(xi-eta)^(xi+eta)(xi+eta-t)varphi^(')(t)dt-varphi^(')(xi) <= (eta)/(I2)[(varphi^(')(xi+eta)-varphi^(')(xi))/(eta)-:}],[{:-(5varphi^(')(xi)-varphi^(')(xi-eta))/(eta)] < (eta)/(I2)[-4varphi^('')(xi)+6epsi] < 0]:}\begin{aligned} \frac{\mathrm{I}}{2 \eta^{2}} \int_{\xi-\eta}^{\xi+\eta}(\xi+\eta-t) \varphi^{\prime}(t) d t-\varphi^{\prime}(\xi) \leqq \frac{\eta}{\mathrm{I} 2}\left[\frac{\varphi^{\prime}(\xi+\eta)-\varphi^{\prime}(\xi)}{\eta}-\right. \\ \left.-\frac{5 \varphi^{\prime}(\xi)-\varphi^{\prime}(\xi-\eta)}{\eta}\right]<\frac{\eta}{\mathrm{I} 2}\left[-4 \varphi^{\prime \prime}(\xi)+6 \varepsilon\right]<0 \end{aligned}
which demonstrates the property.
We do not insist on this question. In general, it can be demonstrated that if for aaagiven the numberbbis close enough toaawe necessarily havex_(2) < (a+b)/(2)x_{2}<\frac{a+b}{2}.
4. Let us make some applications of the previous formulas.
I^(0).varphi=x^(p),p > I,a=0,b=I\mathrm{I}^{0} . \varphi=x^{p}, p>\mathrm{I}, a=0, b=\mathrm{I}. Inequality (5) is written2^(p) >= p(p+I)2^{p} \geqq p(p+\mathrm{I})and we can state the following property.
Ifffis a function of(F_(0)^(1))_(1)\left(\mathrm{F}_{0}^{1}\right)_{1}we have inequality
int_(0)^(1)f^(p)dx-(int_(0)^(1)fdx)^(p) <= {[(p-I)[(2)/(p(p+I))]^((p)/(p-1))," si ",I < p <= p^(')],[(I)/(p+I)-(I)/(2^(p))," si ",p >= p_(1)]:}\int_{0}^{1} f^{p} d x-\left(\int_{0}^{1} f d x\right)^{p} \leqq\left\{\begin{array}{lll}(p-\mathrm{I})\left[\frac{2}{p(p+\mathrm{I})}\right]^{\frac{p}{p-1}} & \text { si } & \mathrm{I}<p \leqq p^{\prime} \\ \frac{\mathrm{I}}{p+\mathrm{I}}-\frac{\mathrm{I}}{2^{p}} & \text { si } & p \geqq p_{1}\end{array}\right.
Orp_(1)p_{1}is the root, between 4.79 and 4.8, of the equation2^(p)=p(p+1)2^{p}=p(p+1).
In the first case equality is only possible if
int_(0)^(1)f^(p)dx-(int_(0)^(1)fdx)^(p) <= {[(p-I)[(2)/(p(p+I))]^((p)/(p-1))," si ",I < p <= p^(')],[(I)/(p+I)-(I)/(2^(p))," si ",p >= p_(1)]:}\int_{0}^{1} f^{p} d x-\left(\int_{0}^{1} f d x\right)^{p} \leqq\left\{\begin{array}{lll}(p-\mathrm{I})\left[\frac{2}{p(p+\mathrm{I})}\right]^{\frac{p}{p-1}} & \text { si } & \mathrm{I}<p \leqq p^{\prime} \\ \frac{\mathrm{I}}{p+\mathrm{I}}-\frac{\mathrm{I}}{2^{p}} & \text { si } & p \geqq p_{1}\end{array}\right.
Orp_(1)p_{1}is the root, between 4.79 and 4.8, of the equation2^(p)=p(p+1)2^{p}=p(p+1).
In the first case equality is only possible if
f=(x-lambda+|x-lambda|)/(2(I-lambda))quad,quad lambda=I-2[(2)/(p(p+I))]^((1)/(p-1))f=\frac{x-\lambda+|x-\lambda|}{2(\mathrm{I}-\lambda)} \quad, \quad \lambda=\mathrm{I}-2\left[\frac{2}{p(p+\mathrm{I})}\right]^{\frac{1}{p-1}}
in the second case only iff=xf=x.
2^(0).varphi=x^(p),0 < p < I,a=0,b=I2^{0} . \varphi=x^{p}, 0<p<\mathrm{I}, a=0, b=\mathrm{I}. The functionvarphi\varphiis concave of order 1 and convex of order 2. We deduce that
2^(0).varphi=x^(p),0 < p < I,a=0,b=I2^{0} . \varphi=x^{p}, 0<p<\mathrm{I}, a=0, b=\mathrm{I}. The functionvarphi\varphiis concave of order 1 and convex of order 2. We deduce that
Ifffis a function of(E_(0)^(1))_(1)\left(\mathrm{E}_{0}^{1}\right)_{1}, we have the inequality
(int_(0)^(1)fdx)^(p)-int_(0)^(1)f^(p)dx <= (I-p)[(p(p+I))/(2)]^((p)/(1-p))\left(\int_{0}^{1} f d x\right)^{p}-\int_{0}^{1} f^{p} d x \leqq(\mathrm{I}-p)\left[\frac{p(p+\mathrm{I})}{2}\right]^{\frac{p}{1-p}}ifquad0 < p < I\quad 0<p<\mathrm{I}
equality being possible only for the function
(int_(0)^(1)fdx)^(p)-int_(0)^(1)f^(p)dx <= (I-p)[(p(p+I))/(2)]^((p)/(1-p))\left(\int_{0}^{1} f d x\right)^{p}-\int_{0}^{1} f^{p} d x \leqq(\mathrm{I}-p)\left[\frac{p(p+\mathrm{I})}{2}\right]^{\frac{p}{1-p}}ifquad0 < p < I\quad 0<p<\mathrm{I}
equality being possible only for the function
f=(x-lambda+|x-lambda|)/(2(1-lambda)),quad lambda=1-2[(p(p+1))/(2)]^((1)/(1-p))f=\frac{x-\lambda+|x-\lambda|}{2(1-\lambda)}, \quad \lambda=1-2\left[\frac{p(p+1)}{2}\right]^{\frac{1}{1-p}}
We have, in particular,
int_(0)^(1)f^(2)dx-(int_(0)^(1)fdx)^(2) <= (1)/(9),quadsqrt(int_(0)^(1)fdx)-int_(0)^(1)sqrtfdx <= (3)/(16)\int_{0}^{1} f^{2} d x-\left(\int_{0}^{1} f d x\right)^{2} \leqq \frac{1}{9}, \quad \sqrt{\int_{0}^{1} f d x}-\int_{0}^{1} \sqrt{f} d x \leqq \frac{3}{16}
equality being possible only for functions
f=(3x-1+|3x-1|)/(4)quad,quad f=(32 x-23+|32 x-23|)/(18)f=\frac{3 x-1+|3 x-1|}{4} \quad, \quad f=\frac{32 x-23+|32 x-23|}{18}
respectively．The first inequality is known and we will take it up again－ in the fifth note．
3^(0)3^{0}．We can also considervarphi=x^(p),p < 0,a > 0\varphi=x^{p}, p<0, a>0.In this casevarphi\varphiis convex of orderIIand concave of order 2. Let, in particular, be the casep=-1p=-1．Let us designate byH=1//int_(0)^(1)(dx)/(t)H=1 / \int_{0}^{1} \frac{d x}{t}the harmonic mean offf．We deduce the following property:
3^(0)3^{0}．We can also considervarphi=x^(p),p < 0,a > 0\varphi=x^{p}, p<0, a>0.In this casevarphi\varphiis convex of orderIIand concave of order 2. Let, in particular, be the casep=-1p=-1．Let us designate byH=1//int_(0)^(1)(dx)/(t)H=1 / \int_{0}^{1} \frac{d x}{t}the harmonic mean offf．We deduce the following property:
Ifffis a function of(E_(a)^(b))_(1),a > 0,A\left(\mathrm{E}_{a}^{b}\right)_{1}, a>0, \mathrm{~A}and H the arithmetic mean and the harmonic mean offf,we have inequality
(I)/(H)-(I)/((A)) <= (I)/(b)|(I)/(sqrtc)-(sqrt2)/(I-c)sqrt(I-c+c log c)|^(2)quad,quad c=(a)/(b),\frac{\mathrm{I}}{\mathrm{H}}-\frac{\mathrm{I}}{\mathrm{~A}} \leqq \frac{\mathrm{I}}{b}\left|\frac{\mathrm{I}}{\sqrt{c}}-\frac{\sqrt{2}}{\mathrm{I}-c} \sqrt{\mathrm{I}-c+c \log c}\right|^{2} \quad, \quad c=\frac{a}{b},
equality being possible only for a single function of the form (4) where
lambda=(I+c)/(I-c)-sqrt((2c)/(I-c+c log c)),quad c=(a)/(b)\lambda=\frac{\mathrm{I}+c}{\mathrm{I}-c}-\sqrt{\frac{2 c}{\mathrm{I}-c+c \log c}}, \quad c=\frac{a}{b}
4^(0)4^{0}．Let us consider again the casevarphi=log x,a > 0\varphi=\log x, a>0．This function is concave of order 1 and convex of order 2 ．We deduce the following property．
Ifffis a function of(E_(a)^(b))_(1),a > 0,A\left(\mathrm{E}_{a}^{b}\right)_{1}, a>0, \mathrm{~A}and G the arithmetic mean and the geometric mean of the functionff,we have inequality
(A)/(G) <= (2c(c-I-log c))/((c-I)^(2)*e^((1+c)/(1-c)))((I)/(c))^((2c)/((c-1)^(2)))quad,quad c=(a)/(b)\frac{\mathrm{A}}{\mathrm{G}} \leqq \frac{2 c(c-\mathrm{I}-\log c)}{(c-\mathrm{I})^{2} \cdot e^{\frac{1+c}{1-c}}}\left(\frac{\mathrm{I}}{c}\right)^{\frac{2 c}{(c-1)^{2}}} \quad, \quad c=\frac{a}{b}
equality being possible only for a single function of the form (4) where
lambda=(I+c)/(I-c)-(I-c)/(c-I-log c)quad,quad c=(a)/(b)\lambda=\frac{\mathrm{I}+c}{\mathrm{I}-c}-\frac{\mathrm{I}-c}{c-\mathrm{I}-\log c} \quad, \quad c=\frac{a}{b}
For Firm which
dtuan if is dollow：4046
- Memorial of Mathematical Sciences. Fasc. XLIV, p. 12. Sequences of functions in general. Real domain, by ML Léau,
I. I ask the reader to refer to the two preceding notes for the hypotheses and notations. See this CR, 2, 449-454, 454-458 (1938). - JK Knopp,,Über die maximalen Abstände und Verhältnisse verschiedener Mittelwerte'' Math. Zeitschrift, 39, 768-776 (1935). The assumptions made aboutvarphi\varphiby MK Knopp are a little more restrictive (the existence of the second derivativevarphi^('')\varphi^{\prime \prime}).
- varphi^('')\varphi^{\prime \prime}exists, except perhaps on a set that is at most countable.