Notes on higher order convex functions (V)

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Tiberiu Popoviciu

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T. Popoviciu, Notes sur les fonctions convexes d’ordre supérieur (V), Bull. de la Sect. Sci. de l’Acad. Roum., 22 (1940), pp. 351-356 (in French).

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1940 f -Popoviciu- Bull. Sect. Sci. Acad. Roum. – Notes sur les fonctions convexes d’ordre superieur (V)

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1940 f -Popoviciu- Bull. Sect. Sci. Acad. Roum. - Notes sur les fonctions convexes d_ordre superieur

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\begin{aligned} ACADEMIE ROUMAINE \\ BULLETIN DE LA SECTION SCIENTIFIQUE \\ TOMEXXIT-ème \end{aligned}

NOTES ON HIGHER ORDER CONVEX FUNCTIONS (V)
BY

TIBERIU POPOVICIU

Note presented by Mr. S. Stoilow, Mc. AR, in the session of February 23, 1940 INEQUALITIES VERIFIED BY AN ORDER FUNCTION

n

E'T BY ITS DERIVATIVES

In the two previous notes (III and IV $^{1}$ ) we studied the inequalities of the form

\begin{matrix} (I) & \sum_{i = 1}^{m} p_{i} f (x_{i}) ≧ 0, \end{matrix}

verified by any non-concave function of order

n

, points
(2)

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

given and the coefficients

p_{i}

being independent of the function

f

.
The necessary and sufficient conditions that the coefficients must satisfy

p_{i}

, for this to be so, are easily obtained, for example, using the formula that we can call the fundamental formula for transforming divided differences. This formula is written

^{2}

)

\sum_{i = 1}^{m} p_{i} f (x_{i}) = \sum_{i = 1}^{n + 1} A_{i} [x_{1}, x_{2}, \dots, x_{i}; f] + \sum_{i = 1}^{n - n - 1} C_{i} [x_{i}, x_{i + 1}, \dots, x_{i + n + 1}; f],

Or

A_{i}, C_{i}

are independent of the function

f

.
We then obtain sufficient conditions for the inequality (r) err writing

\begin{matrix} (3) & A_{1} = A_{2} = \dots = A_{n + I} = 0, C_{i} \geq 0, i = I, 2, \dots, m - n - I . \end{matrix}

These conditions are also necessary if the function is defined only on points (2):

We can, moreover, easily obtain the coefficients

A_{i}, C_{i}

by suitably particularizing the function

f

. By first taking

f = (x - x_{1}) (x - x_{2}) \dots (x - x_{j - 1}), j = 1, 2, \dots, n + 1, (pour j = 1, f = 1)

we get

A_{j} = \sum_{i = 1}^{m} p_{i} (x_{i} - x_{j}) \dots (x_{i} - x_{j - 1}) = \sum_{i = j}^{m} p_{i} (x_{i} - x_{1}) \dots (x_{i} \dots x_{j - 1})

j = I, 2, \dots, n + I .

If we then take

f = {\begin{cases} 0, pour x = x_{1}, x_{2}, \dots, x_{j + n} \\ (x - x_{j + 1}) (x - x_{j + 2}) \dots (x - x_{j + n}), pour x = x_{j + n + 1}, x_{j + n + 2}, \dots, x_{m} \\ j = 1; 2, \dots, m - n - I \end{cases}

we get

When the function

f

is defined in an interval containing the points (2) the conditions (3) are no longer necessary for

n > 1

. For inequality (I) to hold, whatever the function

f

defined in an interval containing within it the points (2), it is necessary and sufficient that this inequality is true for any polynomial of degree

n

and for any function of the form

(| x - λ | + x - λ)^{n}, λ

being a constant. Indeed, any non-concave function of order

n

is the limit of a sequence of functions which are, at an additive polynomial of degree

n

near, sums of such functions

(| x - λ | + x - λ)^{n}

with positive coefficients

^{3}

).

We thus find the necessary and sufficient equalities.

\sum_{i = 1}^{m} p_{i} = \sum_{i = 1}^{m} p_{i} x_{i} = \dots = \sum_{i = 1}^{m} p_{i} x_{i}^{n} = 0,

which are none other than

A_{1} = A_{2} = \dots = A_{n + 1} = 0

, expressing that the first member of (I) is identically zero for any polynomial of degree

η

.

We then find the necessary and sufficient inequalities

\sum_{i = r + 1}^{m} p_{i} (x_{i}

^{3}

) Tiberiu Popoviciu, On the extension of convex functions of higher order, Bull. Math. Soc. Roumaine des Sci., 36, 75-108 (1934).
2. We can generalize the inequality ( x ) by also introducing the values of the derivatives of

f

to the points

x_{i}

. For simplicity, let us assume

f

defined in an interval containing the points (2). If

n > 1

the function

f

, non-concave of order

n

, has continuous derivatives

t^{'}, t^{''}, \dots, t^{(n - 1)}

and derivatives to the left of the d and to the right of order

n, f_{g}^{(n)}, f_{d}^{(n)} \cdot [f_{g}^{(n)} = {(f^{(n - 1)})}_{g}^{'}, f_{d}^{(n)} = {(f^{(n - 1)})}_{d}^{'}]

in every interior point.

We can then look for the necessary and sufficient conditions so that we have

\begin{matrix} (4) & \sum_{i = 1}^{m} \sum_{j = 0}^{k_{i}} p_{i j} f^{(j)} (x_{i}) \geq 0 \end{matrix}

Or

0 ≦ k_{i} ≦ n

And

f^{(n)}

denotes one of the derivatives

f_{g}^{(n)}, f_{d}^{(n)}

(not necessarily the same for all

x_{i}

), whatever the function

f

, non-concave of order

n

in an interval containing within it the points

{\bar{x}}_{i}

. We can easily see that the necessary and sufficient conditions are still that the inequality holds for any polynomial of degree

n

and for any function of the form

(| x - λ | + x - λ)^{n}

. These conditions can therefore be written

\begin{aligned} \sum_{i = 1}^{m} \sum_{j = 0}^{k_{i}} p_{i j} r (r - 1) \dots (r - j + 1) x_{i}^{γ - j} = 0, r = 0, 1, \dots, n \\ \sum_{i = 1}^{m} \sum_{j = 0}^{k_{i}} p_{i j} n (n - 1) \dots (n - j + 1) {(x_{i} - x)}^{n - j} \geq 0, x \in (x_{i}, x_{r + 1}), \end{aligned}

r = I; 2, \dots, m - I .

If, moreover,

\sum_{j = 0}^{k_{i}} | p_{i j} | \neq 0, i = 1, 2, \dots, m,

we can say that for a convex function of order

n

the sign

>

takes place in (4).

In the case

\sum_{i = 1}^{m} (k_{i} + 1) = n + 2

, the inequality can be written

\begin{matrix} (5) & [\underset{k_{1} + I}{\underset{⏟}{x_{1}, x_{1}, \dots, x_{1}}}, \underset{k_{2} + I}{\underset{⏟}{x_{2}, x_{2}, \dots, x_{2}}}, \dots, \underset{k_{m} + I}{\underset{⏟}{x_{m}, x_{m}, \dots, x_{m}}}; f] \geq 0 . \end{matrix}

The first member is the limit of a divided difference of order

n + I

, when

k_{i} + I

of his

n + 2

points tend towards

x_{i}, i = I, 2, \dots, m

. It is easy to obtain the explicit form of this generalized divided difference.
3. We can also establish, sometimes, the exactness of inequality (4) by expressing the first member in the form of a sum of
divided differences of order

n + 1

(of the generalized form (5)) chosen appropriately. We will establish in this way an interesting particular inequality. Let

x_{1} < x_{2} < \dots < x_{n} n

distinct points of the real axis and let

φ (x) = (x - x_{1}) (x - x_{2}) \dots (x - x_{n}) .

Let

ξ_{1} < ξ_{2} < \dots < ξ_{n - I}

the zeros, all real, of the derivative

φ^{'} (x)

of the polynomial

φ (x)

. We have

x_{1} < ξ_{1} < x_{2} < ξ_{2} < \dots < ξ_{n - 1} < x_{n} .

We have the following formulas

[x_{1}, x_{2}, \dots, x_{n}; f] = \sum_{i = 1}^{n} \frac{f (x_{i})}{φ^{'} (x_{i})}, [ξ_{1}, ξ_{2}, \dots, ξ_{n - 1};'^{'}] = n \sum_{i = 1}^{n - 1} \frac{f^{'} (ξ_{i})}{φ^{''} (ξ_{i})}

,

[x_{1}, x_{2}, \dots, x_{n}, ξ_{j}, ξ_{j}; f] = \sum_{i = 1}^{n} \frac{f (x_{i})}{{(x_{i} - ξ_{j})}^{2} φ^{'} (x_{i})} + \frac{f^{'} (ξ_{j})}{φ (ξ_{j})}

,

j = 1, 2, \dots, n - 1

We deduce from this
(6)

- \sum_{j = 1}^{n} \frac{φ (ξ_{j})}{φ^{''} (ξ_{j})} [x_{1}, x_{2}, \dots, x_{n}, ξ_{j}, ξ_{j}; f] =

= \sum_{i = 1}^{n} \frac{f (x_{i})}{φ^{'} (x_{i})} (- \sum_{j = 1}^{n - 1} \frac{φ (ξ_{j})}{{(x_{i} - ξ_{j})}^{2} φ^{''} (ξ_{j})}) - \sum_{j = 1}^{n - 1} \frac{f^{'} (ξ_{j})}{φ^{''} (ξ_{j})} .

The polynomial

ψ (x) = φ (x) - \frac{1}{n} (x - \frac{x_{2} + x_{2} + \dots + x_{n}}{n}) φ^{'} (x)

being of degree

n - 2

We have

\frac{ψ (x)}{φ^{'} (x)} = \sum_{j = 1}^{n - 1} \frac{ψ (ξ_{j})}{(x - ξ_{j}) φ^{''} (ξ_{j})} = \sum_{j = 1}^{n - 1} \frac{φ (ξ_{j})}{(x - ξ_{j}) φ^{''} (ξ_{j})}

And

\begin{aligned} - \sum_{i = 1}^{n - 1} \frac{p (ξ_{i})}{{(x_{i} - ξ_{i})}^{2} φ^{''} (ξ_{i})} & = {[\frac{ψ (x)}{φ^{'} (x)}]}_{x = x_{i}}^{'} = \\ = & {[\frac{φ (x)}{φ^{'} (x)} - \frac{1}{n} (x - \frac{x_{1} + x_{2} + \dots + x_{n}}{n})]}_{x = x_{i}}^{'} = \frac{n - 1}{n} . \end{aligned}

Formula (6) can therefore be written

\frac{n - I}{n} [x_{1}, x_{2}, \dots, x_{n}; f] - \frac{I}{n} [ξ_{1}, ξ_{2}, \dots, ξ_{n - 1}; f^{'}] =

= - \sum_{j = 1}^{n - 1} \frac{φ (ξ_{j})}{φ^{''} (ξ_{j})} [x_{1}, x_{2}, \dots, x_{n}, ξ_{j}, ξ_{j}; f]

We have

- \frac{φ (ξ_{i})}{φ^{''} (ξ_{j})} > 0, j = I, 2, \dots, n - I

and we can state the following theorem
If

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

are

n

real axis points and

ξ_{1}, ξ_{2}, \dots, ξ_{n - 1}

are the zeros of the derivative of the polynomial

φ (x) = (x - x_{1}) (x - x_{2}) \dots (x - x_{n})

, any function

f

, non-concave of order

n

in an interval containing within it the points

x

, verifies inequality
(7)

(n - 1) [x_{1}, x_{2}, \dots, x_{n}; f] \geq [ξ_{1}, ξ_{2}, \dots, ξ_{n - 1}; t^{'}]

.

We have demonstrated the property in the case where the points

x_{i}

are distinct. It also remains true, as a result of continuity, when these points are not distinct [the divided differences then being of the generalized form (5)].

Remarks I. We can easily see that if in addition the function

f

is convex of order

n

, the equality in (7) can only hold if

x_{1} = x_{2} = \dots = x_{n}

.
II. The restriction that the

x_{i}

are within the definition interval of

f

is not essential. The result remains the same if one or both ends of this interval coincide with a simple zero of

φ (x)

. The result is true even without restriction if the function is differentiable a sufficient number of times at the ends of the interval.
4. From the previous theorem we can draw some simple conclusions. Let us denote by

ξ_{I}^{(i)}, ξ_{2}^{(j)}, \dots, ξ_{n - i}^{(i)}

the zeros of the order derivative

i

of

φ (x)

. We then have, under the same conditions,

(n - 1)! [x_{1}, x_{2}, \dots, x_{n}; f] ≧ (n - 2)! [ξ_{1}, ξ_{2}, \dots, ξ_{n - 1}; f^{'}] ≧ \dots

\dots \geq (n - i - I)! [ξ_{I}^{(i)}, ξ_{2}^{(i)}, \dots, ξ_{n - i}^{(i)}; f^{(i)}] \geq \dots \geq [ξ_{I}^{(n - I)}; f^{(n - I)}]

.
But,

[ξ_{I}^{(n - 1)}; f^{(n - 1)}] = f^{(n - 1)} (\frac{x_{1} + x_{2} + \dots + x_{n}}{n})

so:
If the function

f

is non-concave of order

n

in an interval containing the points

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

, we have the inequality

(n - 1)! [x_{1}, x_{2}, \dots, x_{n}; f] \geq f^{(n - 1)} (\frac{x_{1} + x_{2} + \dots + x_{n}}{n})

If f is convex of order

n

, equality is only possible for

x_{1} = x_{2} = \dots = x_{n}

. Let us take, in particular, the function

f = x^{n + r - 1}

. So if

r

is a natural number
, the difference divided

[x_{1}, x_{2}, \dots, x_{n}; /]

is equal to the well-known symmetric function

W_{r} = \sum_{α_{1} + α_{2} + \dots + α_{i l} = r .} x_{1}^{α_{1}} x_{2 i}^{α_{2}} \dots x_{n}^{α_{n}}

Let us denote by Wr' the same symmetric functions of

ξ_{1}, ξ_{2}, \dots, ξ_{n - 1}

and, in general, by

W_{r}^{(i)}

les fonctions symétriques correspondantes de

ξ_{1}^{(i)}

,

ξ_{2}^{(i)}, \dots, ξ_{n - i}^{(i)}

. Remarquons que la fonction

x^{n + r - 1}

est convexe d'ordre

n

dans l'intervalle

(- \infty, + \infty)

si

r

est tun nombre naturel pair et est convexe d'ordre

n

dans (

0, + \infty

) si

r

est un nombre naturel impair. Nous pouvons donc énoncer la propriété suivante:

Si

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

sont les zéros, tous réels, d'un polynome de degré

n

et

ξ_{r}^{(i)}, ξ_{2}^{(i)}, \dots, ξ_{n - i}^{(i)}

sont les zéros de la iema dérivée de ce polynome

(ξ_{r}^{(n - 1)} = \frac{x_{1} + x_{2} + \dots + x_{n}}{n})

, nous avons les inégalités

\begin{matrix} (8) & \frac{W_{r}}{(\binom{n + r - I}{r})} \geq \frac{W_{r}^{'}}{(\binom{n + r - 2}{r})} \geq \dots \geq \frac{W_{r}^{(i)}}{(\binom{n + r - i - I}{r})} \geq \dots \geq {(\frac{x_{1} + x + \dots + x_{n}}{n})}^{'} \end{matrix}

pour tout nombre naturel pair

r \geq 2

.
Si, de plus, les zéros

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

sont non-négatifs, ces inégalités sont vraies aussi pour

r \geq 3

impair.

Le signe

\geq

ne devient

=

dans ces inégalités que si

x_{1} = x_{2} = \dots = x_{n}

.
L'inégalité (7) est à rapprocher de l'inégalité de M. K. T o d à 4)

\frac{f (x_{1}) + f (x_{2}) + \dots + f (x_{n})}{n} \geq \frac{f (ξ_{1}) + f (ξ_{2}) + \dots + f (ξ_{n - 1})}{n - I}

valable pour toute fonction non-concave d'ordre

I^{5}

). Pour

f = x^{p}

où,

p \geq

I ou

p < 0

, cette inégalité revient à celle de MM. H. E. B i a y

^{6}

) ẹt S. Kake y a

^{7}

)

\frac{x_{I}^{p} + x_{2}^{p} + \dots + x_{n}^{p}}{n} \geq \frac{{\dot{ξ}}_{I}^{p} + ξ_{2}^{p} + \dots + ξ_{n - I}^{p}}{n - I}, (x_{i} > 0)

qui est à rapprocher de (8).
Cernăufi, le 27 février 1940.

$^{1)}$ La note III est sous presse dans Mathematica, 16, 74-86. La note IV doit paraître dans cette même revue.
$^{2}$ ) Pour les notations voir mes travaux antérieurs.
$^{4}$ ) K. Toda, On certain functional inequalities, Journal of the Hiroshima. Univ., A,, 427-40 (1934).
$^{5}$ ) Voir aussi la note III, loc. cil. (1).
$^{6}$ ) H. E. B r a y, On the zeros of a polynomial and of its devivatives, Aner. Journal of Math., 53, 864-872 (1931).
$^{7}$ ) S.K a ke y a, On an inequality between the roots of an equation and its derivative, Proceedings Phys.-Math. Soc. Japan (3), 15, 149-154 (1933).

1940

Notes on higher order convex functions (V)

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NOTES ON HIGHER ORDER CONVEX FUNCTIONS (V)
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TIBERIU POPOVICIU

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Notes on higher order convex functions (V)

Abstract

Authors

Keywords

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PDF

About this paper

Journal

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DOI

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NOTES ON HIGHER ORDER CONVEX FUNCTIONS (V) BY

TIBERIU POPOVICIU

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NOTES ON HIGHER ORDER CONVEX FUNCTIONS (V)
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