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175
S HR LES F 0 NC TI0 NS C0 N VEX ES ET LES I N É G A L I TÉS ENTRE R E
LES VALEIJ RSg M0 YEN NES'
R
J. L. W. Y. JENSEN
coPEmR*Oos.
i . Functions as:ces and concares. DeJnition. Examples.
In his famous Ana lyse algé bri qu e (note II, pp. 4ç 2-Çq) Cxc-car
demonstrates that "the geometric mean between several numbers is always less
than their algebraic mean". The method employed by Axncev is extremely
elegant, and is ä pa 8é without change in all treatises on algebraic analysis. It
consists, as is well known, in the fact t h a t , from the inequality
where a and h are rositive numbers, we are led to the analogous inequality for four
numbers, namely
and the following, for 8, i 6, ... 2 numbers, after which this number, by an
artifice, is reduced to an arbitrary lower number, n. This simple method was my
starting point in the following investigations, which lead, by what is in fact a very
simple and elementary route, to general and not unimportant results.
' Lecture delivered at the Danish Mathematical Society on January
Acte mothemntieo. 30. Imprimé le t9 déeezisbre t90b.
9°I
176
J. L. W. ¥ . Jensen.
I'll introduce the definition snivants. When a Î'onction p(z), real, finite and
uniform, of the real variable z, satisfies in a certain inter- valle t h e inequality
we say that p(z) is a connected function in this interval.
If, on the other hand, Cf z) satisfies the inequality
(-)
P(-) + r(z) ñï*r("" !')
we say that p(z) is a concave function.
Furthermore, we assume that these inequalities do not reduce to t h e
g i v e n interval, to equality
(j)
In this case, the function p(z) is said to be -linear- in the given interval. This
expression has been adopted because 1 equality (3) is satisfied by
It follows from these definitions that -1-linear functions form a transition
from the class of convex functions to that of concave functions, and must be
considered as limiting cases common t o both classes.
From the definitions given, it immediately follows that qi:e - p(s) is concave, when p(z) is connected, and vice versa. In what follows, it would be
auÎfisant to consider only convexB functions, since it' s so easy to switch from one
clause t o another. As, however, it may be advantageous to consider both
clauses of functions, con- cave functions will be mentioned from time to
t i m e ; but it should be remembered, even when this is not 8always recalled, that
every pro-position relating to convex functions corresponds to an analogous
proposition for concave functions8.
As t h e main purpose of this research is to present a series of
rlalités d'un caractère général, comprenant comme cas particuliers presque toutes
les inégalités' jusqu'ici connus, nous allons développer d'abord les théorèmes
néeesanires propres a ce but, avant d ' entreprendre l'étude plus approfondie des
fonctions convexes elles-mémes.
On convex functions and inequalities between mean values.
17 7
It will first be necessary to state some propositions that immediately
follow from the definitions, and to give some examples of the classes of
functions defined.
A sum of convex or -linear functions is connected when at least one of the
functions is connected. If p(z) is connected, and c is a positive constant, ep (z) is
connected. These propositions are also true for concave functions. Examples:
I°. z' is a connected function in any interval. So a+ ôz -]- es' is connected
or concave, depending on whether c is positive or negative.
2º. l - l is
linear "
in any interval that does not include the
zero, but connected otherwise.
So
where c
are positive constants, is connected or -linear
in any interval,
connected if the interval includes one or more dea values z, , z"......................, z"
-linear-, if that doesn't happen.
3º. Inequality
which occurs for o N b > o, and for any value of p greater than ä i , is
If z and y are positive, and z > y, we deduce from this inequality
(g)
")' -
So z*, for positive z values and a p value greater than
i , is a related function.
and y to - and
From the last inequality it follows, by changing z
,
2sy
pat a very elementary method.
Acta mclhe'natiaa. 30. Imyritzt é te 20 üé ceæ6ze I 90o.
2ö
178
J. L.
,". À'. Jensen.
So z *, when j' > i , is a related function }'for positive values.
tives de z . If, on the other hand, we put ttans ("} z^ and j/" 'i lii rlace of z and y,
we obtain
z" is therefore a concave function, for positive z, when o < y < i . From (§) it
follows
which
is
still
true for y' =- I.
Thus .z°" is also
O
convex when In summary, for positive values of z , we see that s' is convex when y is
than i , or negative, and concave when If y
r'us large
=- i , the function is "linear", a s we saw above.
4°. | Çn -y- /'z' | is convex
ponrconcave for ab < o,
as long as the binòme soua the radical sign remains positive.
ç°. e' * is a convex function in any interval, while
log z is concave in the interval (o , -}- en).
6º. Differential calculus provides us with a general way of deciding whether
functions that can be differentiated are eonvex or concave. If /'(z) is real and
uniform in a given interval, and has a finite and continuous second derivative, then,
by virtue of Eori.x's theorem, we have
2
.r' being an intermediate number between x and y. So / (z) will be connected as long as
Ç"(z) is positive, concave as long as c(ue {"(z i is nepatiVe. The geometrical
significance of this fact is obvious. lii e8et, if a conveze function p(z i is siisceptihÎe
r1'uno representation ¿-eoinétrique en coor- rlonnees rectangulaires l'équation y - ÿ t z )
cléfinit ii ne courbe, turning its convexity towards negative y.
On convex functions and inequalities entFe mean values.
Remargue.
179
The following three inequalities
(Z) -|- /(ÿ) > 2 ( ÿ 'zy),
'/(z)'/(y) > j/(ÿzy)j\
s and y positive;
z eî y positive, (z) positil;
can be reduced to (i) by simple substitutions, namely, respectivelyment:
log/(s)' P(*)'
/(e") - ( s),
dog (e")
p(s).
z. Generalization o f inequality (i).
Suppose that p(z) is any connected function in a given interval, and that z, ,
z, , z, , . . . are all located in this interval or at its limits. From (i) it follows that
P(-.) + P(-.) + (-.) + P(-.) * 2P(*' *') + 2P(°' t * )
+n
4
and generally, m being a positive integer
u=1
as can easily be seen from t h e complete indnction.If then ri is a positive
integer, and if we choose m such that 2'
ri, we can pose
Then we find
180
J. L. W. V. Jenaen.
which is a generalization o f inequality (i), used to define 1 convex functions.
Clearly, a similar inequality applies to con- cave functions: all you have
to do is reverse the sign o f inequality. For -li- neara- functions, the inequality
simply becomes an equality.
Snpposon8 now that p(z) is an eottfinite and connected function in a certain
interval. We know that such functions exist from the previous examples. Let ri u , -{ri, -[- ... -{- xq, where all nq are positive integers. It follows from (4), by choosing the
z's i n a suitable way,
Let and , o, , . . . eq be any positive numbers whose sum is a, and let 'i grow
indefinitely, but in such a way that
lim
n'
O'
lim -' - "
HO '
.
'
lim -^' - "*'
taD '
which will result in
lim
and by anite, p(z) being continuous by assumption,
which demonstrates the following theorem:
t h e inequality
oîi x , x , ...
repi eseitfenf numbers all within the interval, and yes
a, , a, , .
soul positive numbers, but any numbers.
On oonrex functions and inequalities between average values.
181
For concave functions, the sign of the inequality must be reversed. This
proposition is so general that perhaps a l l known inequalities between mean
values are included as cases.
very special.
V
O
ppliCotions of the formula ($).
3
In the following, the numbers represented by a, h, c, . . . e , $, T, ...
will be assumed positive.
In formula (ç), let's say p(z)
z^, p > i ,
so p(z) is connected, as we' ve seen.
Then we have
or
(6)
where all z are positive. Inequality (6) reduces to a simple identity for z, z , =... z . It also holds for
whereas it must be reversed
for
For e 1
i we
find a result previously given by Mr. R. SIHON.'
By making p -- 2, and replacing a, by a*, z, by , we find
(')
formula by CxUCTiz {loc. ciL, p. 4 Ç), who gives a very different de- monstration.
Posing in (/) e} z} for o, , a, z ' for ô " we have
' Über einige Ongleichungen, Ze its cli ri f t f ü r ôf ath. u. P h ys i - 33 P 37. -8
-
182
J. L. W. V . Jensen.
d'oil follows that the harmonic mean between several positive numbers is
smaller than their algebraic mean.
It is easy to generalize formula i7)
Replace in (6) s:,by
andextract the root p"''° from the deua
members, we will have
which can be written in symmetrical form
z, , r, being positive constants with sum I .
Raising the two members of t h e above inequality to the power z' "°'e where
z' is positive and < I , and multiplying byr
,
we find
which demonstrates t h e following inequality
zi ,
z, , x, being positive constants with sum i.
this way, we demonstrate b y complete induction
Continuing in
(8)
where z are positive constants8 with sum i .
v
In formula ( y), let's put
a, h* instead of a, , and a* b* instead of
in place of fi, where z and y are any real numbers, and for
.
.
.
simplify writing, S(x) - N o, ô( .
We have:
s('j'))"<s,)s(,)
On convex functions and inequalities between mean values.
and log N(z) is therefore
a function
The fundamental relation (ç) then gives:
183
of z in the interval (- en , -J- c'o).
or
S
(9)
2, we have
u
v
o
V
*
For w
*(s) (s(s,))-"(*(si')"' "
or
(* -))"'-*:< (*(-.')"-*(s(-,))"-"Hence
OR POCOIO
1
From (I o) it follows that the function
increases, z , remaining invariant.
(
/)
*' is never decreasing when z
If in (i o') we permute z , and z"
184
J. L. W. V. Jensen.
we must assume z, N z > z, , and conclude that $p(g ') ' " never increases as z decreases.
This demonstrates the following proposition:
The numbers a , , o , , . .. bi , by , . ..
being positive, z being any real
variable, and z
any real constant, the fonetion
is monotonic and never decreases as z increases, either in the interval ( c 'o , z ), or in
the interval (z, , -]- c'o). We have &(z, - e) c 'f(z, -{- e), c being positive.
The last part of the proposal results from the fact that
as mentioned above.
This proposal includes, as special cases, a number of
tions of SciiröàIILCH ':
mustri positive numbers ' x , S, y ... 2, and
S$a- f- Q -\- . . . + I-,
we'll have
V
and
Moreover, BisNAYxE ' has stated, without demonstration, a proposition that
is very close to our above proposal.
Later Mr. R. SIäfON
has published a demonstration o f a special case.
' Über Mitlelgrössen verschiedener Œdnungen, Z ei ta e h r i f t fü r Cf ath em a ti k u n d
Physik,
3. P 3OI, i 8J 8.
' Société philomatique de Paris, Extracts from the minutes of meetings for 1 year 4
p. 6/, Paris
4
' At the 'quoted' location.
On convex functions and entFe lea valenra mean inequalities.
185
SCHLÖMILCH also determines limit values
1
For our function, it is easy to find that
i
and
where b is the largest of the numbers b, , ô , , . .. , ô .
To apply
another application of the fundamental formula (3), let ç (z)
log s, p(z) is concave and we then have
II)
which is a generalized form, due to Mr. L. J. ROGERS', of the classical
proposition about the geometric mean.
Another inequality of a similar nature can be found by noting that z log z
is connected for positive z.
Here it is
(12)
The special case where o, =- o, ... =- a,
i was also given by Mr. Roczns.
These examples should suffice to show just how fruitful formula (ç) can be.
Acta trt4tfiei'irtticn. 30. Imprim ë le 20 décembre 90c.
186
J. L. W. V. Jensen.
Clearly, in the 1e8 preceding formulas, when ja, Za,z" etc., are convergent
series, we can increase ii indefinitely, thus obtaining a n inequality between
certain infinite series.
We can use the formula ( i i ) in series t h e o r y in a different way. let fib , Xb
z . ... Zb" be contiguous series with positive terms, and let al , al , . . . a positive
constants whose sum is i , the series
will also be com urgent.
The proof of this proposition can be deduced immediately from (i i) by
making the b's dependent o n a new index v, posing a, -{- o, -}-i , and summing
in both members for v i , c , ....
ltOOtÎOîtS SttY Îe calcul intégral. There are still other cases where ri
4
can be made to grow indefinitely. Remembering the definition o f an integral defined
as the limit of the values o f a sum, the above gives us a whole range o f
interesting inequalities between integrals.
Suppoaona that a(z) and f(:r) area functions integrablea in the inter-valle
(o , i), and that a(z) is constantly positive. Inequality (ç) gives
where p(z) is assumed to be continuous and connected in the interval (p , , p,),
p, and g being the lower and upper limits of the function /(z) in the interval (o ,
i).
Now, we know that the product of two integrable functions is
an integrable function, and that an integrable function, by substitution in a
continuous function, gives an integrable function.
So, if we allow zt
to grow indefinitely
On eonvex functions and inequalltys between mean values.
It goes without saying that the J
187
in this formula
by corresponding integrals f.
By analogy with the previous formulas,
examples o f the application of formula (ç')
fa(z)dz)
f a (z) b(z)dz
(o(s))*ôz
Iwill cite the following
f a(z)(((z))-dz,
(à(s))*rfs,
/(s) positive, @ I;
b(s) intëgzable and positive;
lea a(z) being positive and integrable, and x, -{- x, -{- . . -]- x, - I ;
fr c) log è{s)d=
O(*)1Og^t=@*
Of these formulas, the second and fourth are generalizations of wellknown formulas. By positing a(z) i in the last one, we find a ScnLöMILca
formula that is particularly interesting because it leads to an important result in
the search for the zéPOB of a Txrron series, as I'll show elsewhere.
3. Further study of conne:ces functions.
After demonstrating the usefulness of the notion of "connected function",
we return to t h e study of such functions in general. In §z, we showed that
formula (4) has li£iu p£lur any connected function
188
If we make n
J. L. W. \ '. Jensen.
this,
we find
Putting - d in place of d, and assuming z -]- ad and .r - nd within t h e given
interval, we find
As a result of the definition,
the inequalities ('x) and (§) can be written as
||V
|V
|
Assuming then that p(z) has a finite upper limit p in the given inter-valle,
we have, taking m
i
from which it follows, if we decrease o to o, that n grows indefinitely, but slowly
enough so that z + u'i does not leave the interval,
km (p(z -}- 'i) - p(z))
o.
This demonstrates the following proposition:
On convex functions and inequalities between mean values.
189
Formula (Ç) therefore applies to any similar function. We could deduce the
following, but it's }a1simpler to start from formula (y).
If we put inthis
iithe pl:ice of à, we find, now assuming o
/'osi/t/
which becomes, by makingconverge
to any positive number, this, smaller
|
V
|
than i
This formula shows that "+never grows when d decreases,
and that this quofient remains constantly greater than
o' is positive, but i n any case arbitrary.
0
- ') or
It follows that
' This proposition also applies to a concave function, by replacing it with
>upper limit" by s lower limit". From the fact that a - linear function s can be considered as a
special case of two clas8ea of functions, it follows:
A "linear function" that has either an upper or a lower limit in a given interval is
continuous.
From this result we can easily conclude the following proposition: a "linear function" having
either an upper or a lower limit in a given interval always has the form a -}- bs dttnB this interval, 'i
and b being constants. This fully justifies the name we have introduced.
190
exists, and we c a n also see that
is also determined.
This theorem is thus demonstrated:
A J'onctimi con "er 'y(x), qiti't in a certain interval a finite upper limit, has a
derivative function tattt ii right, f' (x), than gaucH p?(z); the di,$rrence p (z) - p?(z)
is postfire o" zero.
An analogous proposition applies to concave functions (z), which follows
from the above, since - ( z) is convex.
6. Some suggested functions for /oiicfions.
If, in the interval (p , p'), ÿ (z) is a connected function qni d o e s not
decrease as z increases, and if ¡ r (z) is conveae in an interval, in theqnal takes
place the inequality q K p (z) c ç ', /('r(z)) is also connected.
In eflet, from
it follows
that
/ r("{') <r('*") t'°' ) <j(r(r(-)) + r(r(-))),
which demonstrates the proposition.
The following diagram 8 is similarly demonstrated:
oonveze, growing
eoncave, decroi8aante
related
related
related
ooncave
conveze, decreasing
conoave
convex
concave, growing
conoave
concave
Snr lea functions connexes at les inégalités entra les valeurs moyennes.
' connexe, croissante°
conca e, croissante
connexe,décroissante
concare, dëcroiasante '
.
191
connexe,décroiaaante
concave, décroissante
On a certain. convrxe function.
We saw above that
r, | z - z, | is convex in any interval that includes at least one
points z, , z, , . . . .. This can be used to form a connected function whose right and
left derivatives are di8erent at all points of a real, countable set, given in advance.
Let e8and c , , c , ...
an infinite sequence of positive numbers, z, , z, , ...
an infinite sequence of real and bounded numbers, and suppose the series 2'o, convergent.
It follows from the research into CANTOR's principle of condensation of
singularities that L cook set out in Diff-LfiROTH (T/îeorie der Fu "lfioneii cirer
verändertichen reellen Clrösse, § i o8*) that the function
which is connected
as we saw above, has a unique derivative everywhere,
except at points z, , z, , z, , ....
At these points, the function has
one derivative function on the right and another on the left, and we have ÇJ (z,) -/ (z,)
zc, .
If, for example, for the set (z,), we choose the rational numbers in the
interval (o , i) in a given order, /(z) will be connected in this interval, and will
have a finite derivative at all irrational points, while at rational points the
derivative functions ä right and ä left differ by a positive number.
In closing, I can't resist adding a few remarks. It seems to me that the
notion of "related function" is about as fundamental as these:
positive function",
increasing function". If I'm not mistaken in
this, the notion should find its place in the
elementary expositions of the theory of real functions.
J. L. ÀI". À'. Jensen.
As for the deÔnition o f a connected function rte }i1several variables
the next is the most natural.
The function rce1le p (X) alu roint illlil ÿtiqnc rcel1e N - (z , y, r, ...) is
connected, if rlans a domain Simply connected and coiiveao we always have
<-C
-2
2"
Il is
+
'
'.
'
evirlent that such a tonc tion is always f'inction related to
each of the
xŒia1'1es. L'in i et se ri a pas lieu comme on le voit r-- un
f(° ' '7J
l *'* '
Additi0rt. After iivoir made the conference above ] I noticed that the
fontlamental formula (3) was not entirely new as I thought. I've just found, 'fans une
mémoire tle II. A . l'ltINGSI4EIM,' a quotation f r o m a note by II. 0. HOLIJ Eli '
rtans 1ar{ucl1e is de- shown the formula in question. A la vérite les li1'pot1ieses de Il.
HöLDER are quite different from mine in that he assumes that 'r"(z) exists.
Tva very important formula ( 5') is n o t '
mentioned more than the
most of the -rl"'cations given above.
En mcme temps je veux 'lemontrer une inégalité c1'un autre caractère que
celles tlonnées plus haut. hans une addition1 a se mémoire précitée
II. Then Gsiii iii astonishes an é1é¿-ante demonstration, which he attributes to M. LÜi:oa ir, 'le t h e inequality
r'i the /' are positive and z > I.
' Zmr Theorie der petizeii Ii-anscendenten 4 "unktionen, S i t z u n gs b e r . d. m a t h . p li ya.
G la s s e d. k. ba y er. A k a d. d . \V., t. 3.2, p. i 63 - 9*
Nachtrag . . ibid.
P 295 -5
3
' Über einett. -Mittelicertsalz, fi a c h r. v. cl. k. G e se 11 s c h. d. V'. z u Göt t i ri g e n,
On convex functions and inequalities between mean values.
193
This demonstration is easy to generate. L e t z be a positive variable and Ç(z)
a positive function el increasing with z, and let o, , a, , . .. ô , , b , . . . of8 equally
positive constants, we have
5(b,) < [ b) for r i , 2 , ... it, f' being the sum bl -[- b, -]- ... -]- ô,. By multiplying by a,
the two members of t h e above inequality, and by
Naming for r I , 2 , . . . n, we find
on
This is the inequality I had in mind. For o, b and ÿ (z) z '°',
we find
inequality (e). Using this inequality, we can generalize the conditions under
which formula (8) is valid. Posing in e8and dana this n'q for a,q, we have
which shows that formula (8) is still valid for
only t h e equal signJ is to be discarded.