On some inequalities between convex functions (I)

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Tiberiu Popoviciu

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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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1938 e -Popoviciu- Comptes Rendus des séances de l’Acad. des Sci. de Roumanie – Sur quelques inegalités entre les fonction convexes (I)

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Comptes Rendus des séances de l’Académie des Sciences de Roumanie

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1938 e -Popoviciu- Comptes Rendus des séances de l_Acad. des Sci. de Roumanie - Sur quelques inegali

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112. ON SOME INEQUALITIES BETWEEN CONVEX FUNCTIONS

(FIRST NOTE)
By tiberiu popoviciu, Mc. AS r.
(Meeting of March 4, 1938).

Either $φ = φ (x)$ a defined, uniform, continuous and convex function (of order I) in the closed interval ( 0,1 ). We assume that $φ (0) = 0, φ (1) = 1 0 ≦ φ (x) ≦ 1$ . It follows that $φ (x)$ is a positive and increasing function for $0 < x ≦ I$ . The right derivative $φ^{'} (x)$ exists, is increasing for $0 ≦ x < I$ , but may not be bounded in ( $0, I$ ).

Let us also consider a function

f = f (x)

, definite, uniform, continuous and non-concave of order

0, 1, \dots, n

in the meantime

(0, 1)^{1}

). We assume that

f (0) = a ≧ 0, f (I) = b ≦ I, a < b

and we will designate by

{(E_{a}^{b})}_{n}

the set of functions verifying all these properties. We see that

{(E_{a}^{b})}_{n} \subset {(E_{a}^{b})}_{n - 1}

, SO

{(E_{a}^{b})}_{n} \subset {(E_{a}^{b})}_{o}, n > 0

. In this work we

we will suppose

n > 0

. We will also indicate the results for

n = 0

, which are almost all known.

The problem we will examine in this work is the following: Given the function

φ

, determine the maximum of

A_{φ} (f)

when

A (f)

is given and

f

runs through the whole

{(E_{a}^{b})}_{n}

.

We asked

A = A (f) = \int_{0}^{1} f d x, A_{φ} = A_{φ} (f) = \int_{0}^{1} φ (f) d x

We will then make some applications, generalizing certain known results.
2. Let us call an elementary function of degree

n

any function whose

(n - 1)^{ème}

derivative is a polygonal line. Such a function is therefore of the form

\begin{matrix} (I) & g (x) = P (x) + \sum_{i = 1}^{m} c_{i} {[\frac{x - x_{i} + | x - x_{i} |}{2 (I - x_{i})}]}^{n} \end{matrix}

P (x) = a_{o} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n - 1}, 0 ⩽ x_{1} < x_{2} < \dots < x_{m} < I

.
We assume that

c_{i} \neq 0, i = 1, 2, \dots, m

and we will then say that the elementary function

g (x)

is at

m

vertices. We will say that the vertices are at the points

x_{1}, x_{2}, \dots, x_{m}

. If

x_{1} = 0

there is a summit at the point

o^{2}

). For the elementary function

g (x)

has

m

summits belong to

{(E_{a}^{b})}_{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} = b

.
In the following we only consider elementary functions belonging to

{(E_{a}^{b})}_{n}

.

According to the previous definition, any polynomial of degree

n - I

is an elementary function of degree

n

with o vertices. A polynomial of effective degree

n

is an elementary function of degree

n

at . I vertex, the vertex being at point 0 .

Among the I-vertex functions we find the following

\begin{matrix} (2) & h_{λ} (x) = a + (b - a) {[\frac{x - λ + | x - λ |}{2 (I - λ)}]}^{n}, (0 ≦ λ < I) . \end{matrix}

In the case

n = 1

all one-vertex functions are of this form.

Function (1) can be written

g (x) = \frac{(b - a) P (x) + a [P (I) - b]}{b - a} + \frac{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 degree

n

-I and of

m

functions of the form (2).

Any function of

{(E_{a}^{b})}_{n}

is the limit of a uniformly convergent sequence of elementary functions of

{(E_{a}^{b})}_{n}

. This property can also be specified in the following way. If

f

belongs to

{(E_{a}^{b})}_{n}

, the derivatives

f^{'}, f^{''}, \dots, f^{(n - 1)}

exist for

o ≦ x < I

. The function

f

is then the limit of a uniformly convergent sequence of elementary functions of

{(E_{a}^{b})}_{n}

, these functions all having the same first term

P (x) = a + x f^{'} (0) + \frac{x^{2}}{2!} f^{''} (0) + \dots + \frac{x^{n - 1}}{(n - 1)!} f^{(n - 1)} (0)

. Moreover, we can assume that the value of the integral

A

is the same for all these functions.

These considerations apply, in particular, to the function

φ

, which belongs to

{(E_{0}^{1})}_{1}

, and serve to establish the following assertions with complete rigor.
3. Let

φ

given and

g (x)

an elementary function of

{(E_{a}^{b})}_{n}

having

m > 1

vertices and given by formula (I). Let us construct the function.

g^{*} (x) = P (x) + \sum_{i = 1}^{m - 2} c_{i} {[\frac{x - x_{i} + | x - x_{i} |}{2 (I - x_{i})}]}^{n} + c {[\frac{x - μ + | x - μ |}{2 (I - μ)}]}^{n}

and determine the constants

c, μ

so that we have

g^{*} (I) = b

,

A (g^{*}) = A (g)

. We find that

c = c_{m - 1} + c_{m}

And

μ = \frac{c_{m - 1} x_{m - 1} + c_{m} x_{m}}{c_{m - 1} + c_{m}}

, SO

x_{m - 1} < μ < x_{m}, c > 0

And

g^{*}

is indeed an elementary function of degree

n

has

m - 1

summits belonging to

{(E_{a}^{b})}_{n}

.

There is a number

μ^{'}, μ < μ^{'} < I

such that we have
(3)

g^{*} (x) ≦ g (x)

following that

x ≦ μ^{'}

, For

x_{m - 1} < x < I^{4}

)

Consider the function

φ_{λ} (x) = \frac{x - λ + | x - λ |}{2}, 0 ≦ λ < 1

If

g (x_{m - 1}) < λ < b

, we have

A φ_{λ} (g^{*}) - A φ_{λ} (g) = \int_{i^{*}}^{1} g^{*} d x - \int_{t}^{1} g d x + λ (t^{*} - t)

Or

λ = g (t) = g^{*} (t^{*})

and property (3) shows us that this function of

λ

cancels out for

λ = g (x_{m - 1})

And

λ = b

, is increasing for

g (x_{m - 1}) < λ < g (μ^{'})

and decreasing for

g (μ^{'}) < λ < b

. It therefore follows that

\begin{matrix} (4) & A φ_{λ} (g^{*}) > A φ_{λ} (g), g (x_{m - 1}) < λ < b \end{matrix}

We now propose to demonstrate that $A_{φ} (g^{*}) >> A_{φ} (g)$ . The inequality with the sign $≧$ results from (4), but it is a matter of proving that equality is not possible.

Consider the non-decreasing and non-concave function,

Φ_{ε} (x) = {\begin{array}{lr} φ (x), & dans (0, I - ε) \\ φ (I - ε) + (x - I + ε) φ (I - ε), & dans (I - ε, I), \end{array}

Or

ε

is a fairly small positive number and

φ^{'}

is the right derivative of

φ

. We have

\begin{matrix} (5) & Φ_{z} (x) = \int_{0}^{1} φ_{t} (x) d Φ_{ε}^{'} (t) + α x \end{matrix}

the integral being Stieltjes and

α

a constant of no importance to us

^{5}

).

It can be easily demonstrated that

A_{Φ Σ} (g^{*}) - A_{Φ ε} (g) = \int_{0}^{1} {A φ_{l} (g^{*}) - A ψ_{t} (g)} d Φ_{ε}^{'} (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 function $Φ_{ε}$ is only necessary if $b = 1$ . It may happen in fact that the right derivative of $φ (x)$ is not of bounded variation.
which results from the fact that the integral of $S t i e 1 t j e s$ of the second member exists.

From this formula and the definition of

Φ_{ε}

it follows that

A_{Φ} (g^{*})

--

A Φ_{ε} (g)

is positive and does not decrease when

ε

decreases. We have therefore

lim_{3 \to 0} [A_{Φ ε} (g^{*}) - A_{Φ ε} (g)] = A_{φ} (g^{*}) - A_{φ} (g) > 0

Now consider a function $f$ of ${(E_{a}^{b})}_{n}$ . Let the elementary function of degree $n$ at the summit
$h (x) = a + x f^{'} (o) + \frac{x^{2}}{2!} f^{''} (o) + \dots + \frac{x^{n - 1}}{(n - I)!} f^{(n - 1)} (o) + d {[\frac{x - ρ + | x - ρ |}{2 (I - ρ)}]}^{n}$
Or $d, ρ$ are completely determined by the conditions $h (I) = b$ , $A (h) = A (f) . h (x)$ belongs to ${(E_{a}^{b})}_{n}$ and we have $A_{φ} (h) ≧ A_{φ} (f)$ We propose to demonstrate that equality is only possible if $h = f$ .

Let us take formula (5); we have

A_{Φ s} (h) - A_{Φ s} (f) = \int_{0}^{1} {A_{φ t} (h) - A_{φ t} (f)} d Φ_{ε}^{'} (t)

The function

A φ_{i} (h) - A φ_{t} (f)

is continuous in

t

and is

≧ 0

.
It is easily demonstrated that if

f

does not reduce to an elementary function of degree

n

at o or I vertex, this function is not identically zero. We therefore have

A_{Φ \hat{ε}} (h) - A_{Φ \hat{ε}} (f) > 0

, at least for

ε

quite small. The requested property results as above.

So finally:
If the function

φ

and the number

A (f)

are given, the maximum of

A_{φ} (f)

In

{(E_{a}^{b})}_{n}

can only be achieved for an elementary function of degree

n

at most I summit.

In the case

n = 1

the function

h (x)

is completely determined by the value of the integral A . We therefore have the following property:

If

φ

is given and f is a continuous, non-decreasing, non-concave function in

(O, I)