some applications to inequalities of the method of generalized convexity

SOME APPLICATIONS TO INEQUALITIES

OF THE METHOD OF G E N E R A L I Z E D CONVEXITY*

By SAMUEL KARLIN AND ZVI ZIEGLER

We have always admired the ingenuity of Menahem Max Schiffer in exploiting

existing and developing new inequalities pertaining to variational problems in

complex and real analysis. Inequalities occur naturally in dealing with extremal

problems, in establishing estimates for norms of transformations in L p spaces, in

verifying ordering relations, in evaluating combinatorics, etc.

During the past four years we have been writing a monograph [9] aimed at

constructing a unified framework for deriving an assortment of analytic inequalities

emphasizing the geometry of moment spaces and various concepts of generalized

convexity. This has enabled us to obtain anew inequalities from various sources

previously deduced by ad hoc means and, by understanding better the underlying

structure, to often sharpen and extend them. In this note we offera preview of two subsections of the monograph concerned

with applicat!ons of generalized convexity, coupled with the geometry of moment

space. We confine attention to a number of one-dimensional cases, deferring

multi-dimensional discussions to the book itself. For similar reasons, we shall not

present proofs of some ancillary theorems.

1. Some concepts and definitions of generalized convexity

An important class of "convex functions," apparently identified first by E. Hopf

[7], is the class of convex [unctions o[ order n, [(x), defined on an interval (a, b)

obeying

(1)

1 1 . . . 1

X l X 2 " ~ " X n + l

X ~ - - I xn--I n--I 2 " " " X n + l

f(x,) f(x2) " '" f(x.+,)

->0

for all a < x ~ < x 2 < . . . < x , + l < b .

* Supported in part by Grant NSF MPS71-02905A03.

281

J O U R N A L D ' A N A L Y S E M A T H I ~ M A T 1 Q U E , V o l . 30 (1976)

282 S. KARLIN AND Z. ZIEGLER

Ordinary m o n o t o n e and convex funct ions cor respond to convex functions of

orders 1 and 2, respect ively.

A natural genera l iza t ion occurs by replacing the base funct ions (1, x,- �9 x" - l ) by

(Uo(X), u ~ ( x ) , " ", u,_~(x)), thus defining a funct ion f to be convex with respect to

{Uo,. . . , u,_~} iff

(2)

uo(x, ) . . . Uo(X.+,)

Un__I(XI ) ' ' ' U n _ l ( X n + / )

f ( x l ) "'" f(x, ,- 0

= 0

for a < x ~ < x 2 < - . . < x , + ~ < b.

The set of funct ions satisfying (2) is evident ly a convex cone which we denote

by ~(uo, u , . . . , u , _ O. The more s t ructure we impose on the functions

{Uo, U , . . . , u , _ ~ } , the more proper t ies are endowed to funct ions of

~(Uo, u , . . . , u,,_~). In the case where {Uo, u ~ , . . . , u._~} is an Ex t ended Comple t e

Tchebycheff* system, the cone ~(Uo, u , . . . , u,_~) has been extensively studied (see

[8] and [13]). T h e ex t r eme rays were de t e rmined and a r epresen ta t ion t h e o r e m for

functions of r162 u , . . . , u._ 0 was establ ished.

The set of signed measures d g satisfying

b

f _->0, a

for all f ~ qg(u0, u , . - . , u,_~),

const i tutes the dual cone to r162 u , - - -, u._l) deno ted by r162 u , . . . , u,_~). The

character iza t ion of the measures of the dual cone serves as an essential tool in the

s tudy of inequalit ies, as will be amply exempli f ied in this note .

We start by citing some facts needed later (see [8] for their comple te discussion).

D e f i n i t i o n . A signed measure d/z is said to possess k sign changes on (a, b) if

there exists a par t i t ion of I = (a, b) into k + 1 disjoint cont iguous sets To," �9 ", Tk,

I = U ~0T~, w h e r e / z (T~) ~ 0, ~ mainta ins a single sign as a measure on each T~ and

/z exhibits oppos i t e signs over successive T~, i = 0, 1 , . . - , k.

T h e o r e m A . Let dtz E ~*(Uo, ul, " " ", u.-1). Then the " m o m e n t " relations

* Definition. A system {uo, u. . �9 ,, uk-l} is a Tchebycheff system (Extended) iff each non-trivial linear combination ~-la~u~ can have at most k - 1 zeros (counting multiplicities), A system {uo, ul,-.., u.-L} is an Extended Complete Tehebyeheff system if {uo, u . - . - , u~-l} is an Extended Tchebycheff system for every k _-< n.

GENERALIZED CONVEXITY 283

b

(3) f u, dtz = 0, i = 0, 1 , . . . , n - 1 a

hold. Where {Uo, u~,. �9 u,_~} is a Tchebycheff system, then dl.t changes sign at least n times.

A sufficient condition that dtz belongs to ~*(Uo, ul,'" ", u,-O is that the '$moment conditions" (3) are present and diz exhibits exactly n sign changes having its last sign (+).

Numerous analytic inequalities often reduce to verifying that a specific signed

measure d/~ belongs to ~g*(Uo, ul," �9 ", u,_l). In the next sections we develop a series

of inequalities emanating from this principle.

2. Some classes of dual general ized convex inequalit ies

I. We start by deducing some cases of inverse Jensen and H61der type

inequalities first promulgated by Favard and Berwald. Our methods are, of course,

different and based on the geometry implicit to Theorem A.

Favard [5]. Let f be a non-negative concave function on [0, 1], not identically O, and let ~b(. ) be convex over [0,2f] , where

=/ f f(x)dx. 0

Then

(4) -~fftP(y)dy>= O[f(x)] dx. o

Proof . I t suffices to establish (4) for f continuously differentiable on (a, b) and

strictly concave. We introduce the measure d/x = d/z~(y) defined by

1

0 o

valid for all smooth functions 0(~), s r > 0.

Since f is strictly concave and non-negative, it follows that


d/z:(y) = m (y)dy

where

m(y) = {t*:l~=y~ I f ' (x ) l '

y ~ [fmin, fmax] ,

otherwise.

Hence m (y) is increasing by virtue of the concavity of f. Using these definitions, inequality (4) can be written in the form

2f , f ~

fmin

o r

(5) f ~ ( y ) [ d v ( y ) - m (y)dy] ~ 0 0

where

dv(y) = {~f0f ' 0--< Y --<2/'

otherwise.

Manifestly, we have

f l . [ d r ( y ) - m(y)dy] = 1 - 1= O, f y [ d v ( y ) - m(y)dy] = f - f = O. o 0

Hence, the moment conditions (3), for i = 0, 1 are satisfied. By Theorem A, d r ( y ) - m ( y ) d y must involve at least two sign changes. But m(y) is increasing, implying that there can exist at most two sign changes. Therefore there are exactly two sign changes and the order of the signs is +, - , +. The requirements of Theorem A are fulfilled entailing the inequality (5).

A parallel proof establishes the following generalization of Favard's result.

Berwald 's Inequal i ty [1]. Let f be a non-negative concave function on [0, 1],


not identically O, and let q~ be a strictly increasing positive function on [0, ~). Let ~. be

the unique positive root fulfilling the equation

�9 j ,f -v ~ ( y ) d y = q~[f(x)]dx. Z

0 0

Then

(6) if f -: O(y)dy => qJ[f(x)]dx Z

0 0

holds for all ~ E C(1, ~).

T h e foregoing inequali t ies can be used to obta in an improved version of an

inequal i ty fea tured in [10].

T h e o r e m 1. Let f~, . . ., fi be non-negative concave functions on [0, 1] obeying

the normalization

(7) f f" (x )dx = �89 v = 1, 2 , . . . , n. 0

n - - 1 Let pv > 1, u = 1 , . . . , n, and Ev=~p, = 1. Then

0

< 1 .

P r o o | . Applying Favard 's inequali ty to fi, with ~ ( y ) = yP, and taking account

of (7), we obtain

' i i p. + 1 - y ' d y - [ fv(x)] 'dx , 0 0

v = 1, �9 �9 n

o r

(9)

!

(p~ + 1) f 0

[/, (x)]'- dx _-< 1.

286 s. KARLIN AND Z. ZIEGLER

Because of the condition " -i E w l p , = 1, the left-hand side of (8) is a convex combination of the quantities (9), and (8) immediately ensues. Q.E.D.

II. The following applications of Theorem A were motivated by a recent work of C. Borell [2].

For a > 0 , 1 e t

~:,, = {f; f>~O, f concave, f If(x)l'dx = 1 }

and

~o,~= f ; O = f = M , f convex, [f(x)] 'dx= l 0

With these notations we have:

T h e o r e m 2. Let h(x) be a real function on [0, 1] not identically a constant. Consider ot > 0 and set Ah, a (f) = f~ h (x ) [f(x )]adx.

(a) If h (x) is increasing, then a.i) min{Ah,~(f); f E 3~} is attained for l l ( x ) - ( l + ot)l/~ x). a.ii) max{Ah, a(f); f E 3~} is attained for lo(x)=-(l+a)I/*x.

a.iii) min{Ah,o(f); f E ~O.,u} is attained for q~(x)--(M*+l/(ot + 1))[-~l--LlVl X]+.j

(Here [a]+ = max{a, 0}.)

a.iv) max{Ah,,(f); f E ~3~.M}

(b) If h(x) is decreasing, then

is attained for

[ ~b(x)-- (M*+'/(a + 1)) tx - (1 L

b.i) max{Ah,.(f); f E 3:o} is attained for l,(x). b.ii) min{Ah,a(f); f E 3ra} is attained for lo(x). b.iii) max{A~(f) ; f ~ ~a,~} is attained for ~o(x). b.iv) min{Ah,.(f); f E ~.,M} is attained for O(x).

All the extrema are attained uniquely for the respective functions.

Proof . For each f in the classes under consideration, let c = c(f) be defined by

(lo) c = ] h (x) [f(x)]*dx, 0


so that

(10') | [f(x)] ~ [h(x) - c] ax = O.

0

We restrict ourselves to proving (a), remarking that (b) follows from (a) by taking

/~(x) = h(1 - x).

a.i) Relation (10') implies that h ( x ) - c involves one sign change and it

transpires from - to + (we know that h(x) is monotone). Denote the point of sign

change by yl, and determine the constant A so that [/(x)] = - A [/1(x)]* changes sign at y~. Since [(x) is non-negative and concave we readily find that A is positive

and the sign change is from - to + . Hence, we have

[ h ( x ) - c] {[f(x)] = - A [l,(x)]'}ax >- O, 0

with strict inequality prevailing unless [ ( x ) - l~(x) on (0, 1). By virtue of (10') we conclude that, unless [(x) =- l~(x),

A f [ h ( x ) - c ] [l~(x)]"dx < O, 0

o r

/ i / [h(x)l[l ,(x)l=dx < c [ll(x)l 'dx = c = h(x)[ f (x ) l"dx , o 0 0

establishing this subcase.

a.ii) We specify B[lo(x)]" instead of A [ll(x)]* and observe in this case that the

sign of [/(x)] = -B[ lo (x ) ] " changes from + to - . Paraphrasing the other steps of

the argument, we achieve the proof of a.ii).

a.iii) In view of the definition of q3=,M we have

(11) i ( [ f (x)]* - [ ,p(x) l=)dx = 0 , 0

288 S. K A R L I N A N D Z. Z I E G L E R

so that the function g(x) = [f(x)] a - [<p(x)]* has at least one sign change. By virtue

of the convexity of f (x) , the restriction f(x)<= M and the fact of ~ ( 0 ) = M, it

follows that g(x) exhibits exactly one sign change from - to + . Appealing to

Theorem A we conclude that

{ [ f ( x ) ] ~ - [,p(x )l"}dx <r

Since h(x) is increasing, we thus have

(12) i h(x) { [ f (x ) ] - >- o 0

with equality iff f ( x )=-~(x ) . This completes the proof of case a.iii).

a.iv) This subcase is established in exactly the same way, noting that now the

measure

{[6(x)] a - [ f (x ) l~}dx belongsto cr Q.E.D.

R e m a r k . Observe that where the boundedness condition in the definition of

~.M is deleted, the extremum is not attained.

The next theorem was initially proved by Borell [2] employing more intricate and

special techniques.

T h e o r e m 3. Let h (x) be a real function which does not reduce to a constant,

and assume that h (x ) is first increasing and then decreasing. Let a >= 1. Then, with

the notation of the previous theorem, we have :

(13)

min{Ah.o(f); f ~ , ~ a } isattained (either)for f ( x ) - - ll(X)

or f(x)---- to(x).

R e m a r k . Note that a is now restricted by a _-> 1. The situation with 0 < a < 1

is unresolved.

P r o o t . We define c ( / ) as in (10), (10').

Since [f(x)]a __> 0, it follows that h ( x ) - c changes sign at least once. On the other

hand, by the nature of the form of h(x) we know that h ( x ) - c can involve at most

two sign changes, and if there are two sign changes, the pattern is - , + , - . We

divide the subsequent discussion into two cases:


a) The function h ( x ) - c changes sign twice, at y~ and y2, 0 < y ~ < y 2 < l .

Consider the function

(14) g(x) = [Alg(x)+ BlT(x)l''

where A and B are determined so as to assure the equations g ( y 0 = f ( y l ) ,

g(y2) = f(y2). We claim that A and B are non-negative and they vanish iff f is

linear. In fact, solving the linear equations gives

(15)

f" (y,), (1 - yl)" lyT, f"(y~) (1 + ot)a = f~(y2), ( 1 - y2) ~ = a__ A (1 + ot)B = [ y~' f*(y2) = a___~2

A h ' h A

where

A= Y;' (1- Y')~ I y~, ( 1 - y~)" "

Note that since f is concave and non-negative on [0, 1] the systems (x a, (1 - x)a),

( [ f (x) ]~ , (1- x) a) and ([f(x)]~,x ~) are T-systems on (0, 1), as any non-trivial real

combination of one such pair involves at most one zero inside (0, 1). Thus, every

determinant in (15) maintains a constant sign independent of yl, y2. The signs can

be readily ascertained by specifying yl = e, y2 = 1 - e and subsequently letting

e --> 0, yielding

s i g n h = - l , s i g n a 2 = - l , s i g n a l = - l .

It follows that A, B are positive as claimed. A straightforward computation and

simple analysis verifies that

g(x)={A[lo(x)] ~' + B[l,(x)]"} TM, a > 1,

with A and B positive, is a convex function on (0, 1). Because g(x) is convex and

f (x) is concave, the difference f ( x ) - g ( x ) changes sign exactly twice, at yl and y2.

Its arrangement of sign changes, identical to that of [/(x)] a - [g(x)] a, is ( - , + , - ).

Noting that the sign of h ( x ) - c coincides throughout with that of [f(x)]~ - [g (x)]~,

we conclude that

f [ h ( x ) - c]{[f(x)]" - {A [/o(X)] * + B(l~(x)]"}}dx > O. o


In view of the definition of c it now follows that

1

f [h(x)- c]{A[lo(x)] ~ + B[l,(x)]~ < O, 0

i.e.,

A f h(x)[lo(x)]adx + B i h(x)[l~(x)]*dx 0 0

< c(A + B),

or, equivalently,

1 1 1

A fh ( x ) t l o ( x ) l adX+A_~ fh (x ) t l , ( x ) i~dx<c=f A + B o 0 o

h (x )[f(x )ladx.

This implies that at least one of the integrals on the left side is smaller than

f~oh(x)[f(x)]*dx, and case a) is established.

b) If h ( x ) - c has only one sign change, the arguments paraphrase those used in

the proofs of a.i), b.ii) of Theorem 2.

This concludes the proof of Theorem 3.

The following easy consequence of Theorem 3 is also available.

T h e o r e m 4. Let h(x) be first decreasing and then increasing and let a >- 1. Then

max{A~=(f); f E ~,} isattainedfor f(x) =- lo(x) or

f(x)-- l,(x).

Proof . We need only observe that /~(x)= M - h ( x ) , where M =

max{h (x); x E [0, 1]} is first increasing and then decreasing. Furthermore, in view

of the normalization condition of ~,,

A~,a (f) = M - Ah., (f), for all f E f*,.

Hence, min{Aa,~(f); f E ~ . } = M-max{Ah, a(/); f ~ ~ } and the result follows

from Theorem 3. Q.E.D.

Combining the previous results with Berwald's inequality we obtain some

variations of Theorems 2 and 3, of which the following theorems are typical.


T h e o r e m 5. Let h (x) be specified as in Theorem 3. Let a, /3 be real numbers satisfying 1 <= a <=/3. Then min{Ah,~(f); f E ~;# } is achieved for f ( x ) =- (1 +/3)~/#x or / ( x ) -- (1 +/3)"#(1 - x ) .

P r o o f . Let u (x) be an arbitrary function of ~#. We wish to prove that

(16)

1 1 1

J 0 o 0

where the constant k is chosen so that Lo(x)=-klo(x), Li (x ) - -k l l (x ) fulfill the normalization condition of ~#. Manifestly, k = (1 +/3)1/#/(1 + a ) TM.

Defining

u(x) f(x )=- {[o'[U(X )].dx}"

we see that f E .~. Hence, by Theorem 3,

(17) fo i h ( x ) t u ( x ) l ~ d x

fo ' [ U ( X ) ] ' d x

1

= f h (x) [f(x)]~

! 1

_ _ > min {/h,x,[lo,x,,'dx, f h,x,tll,X,,' x) 0 0

Invoking now Berwald's inequality with q/(x ) -~ x #, ~p(x ) =- x*, f (x ) =- u(x ) yields

1 {( ) #t~ 1+/3 a + 1) [u(x)l"dx >_ u(x)]#dx = 1

0 0

o r

(18) i [ . ( x ) l ' a ~ ---- [(1 + p y ~ ] - = k% t(1 + a ) " J

0

Multiplying (17) by f~ [u(x)]*dx and using (18), we secure (16). Q.E.D.


T h e o r e m 6. Let h (x ) be specified as in Theorem 2 and assume it is increasing.

Let a and fl be two positive real numbers.

(i) I f a < fl then

min {Ah,~ (f); f E ~ } is attained for f ( x ) =- (1 +/3)~/a (1 - x).

(ii) I f / 3 < a then

max{Ah,~(f); f E ~ } is attained for f ( x )=- ( l + /3)V~x.

P r o o L The proof parallels that of Theorem 5.

IIl. We conclude this section with some examples generalizing inequalities

relevant in statistical theory stimulated by work of Van Zwet [12], that do not quite

succumb to a direct application of Theorem A, but where the analysis is

substantially akin to the foregoing methods.

T h e o r e m 7. Let X be a non-degenerate random variable taking values in

[a, b], where - oo < a < b < oo. Assume that ~o(x) is an increasing convex function

on [a,b], r x, satisfying

(19) E I X ] = E [ r = 0

E[X ~] = ~[ ,P (X)I > 0

where E stands for the expected value operator. Then,

b

(20a) f x~i-' [r 2k-2'+' [q~(x)- x ] d F ( x ) > O , 1 <=j < k a

b

I {[~(x)]2ix2k-2i + [~(x)lZk-2'xZJ}[~(x) - x l d F ( x ) > O , 0<-_] <-_ k (20b) a

where F denotes the distribution function corresponding to the random variable X.

P r o o f . In view of (19), q~ is not linear. Hence, by Jensen's inequality

(21) ~(0) = q~(E[X]) < E[~0(X)] = 0.

We note next that ~p(x) - x exhibits at least two sign changes in (a, b). In fact, it has

at least one owing to (19). Assuming that it has precisely one at xo, then the sign

changes from - to + because r is increasing and convex. It follows that


b

f [~(x)-x][x-xoldF(x)>O. a

Strict inequality prevails, since the integrand is strictly positive except at Xo and

F(x) is non-degenerate. Expanding the integrand and using both parts of (19)

results in the inequality

b b b b

f f [f a a a a

which is obviously absurd in view of the Cauchy-Schwartz inequality.

As ~ (x) - x is convex it involves no more than two sign changes. Consequently,

the preceding argument guarantees exactly two sign changes necessarily in the

order + , - , + . In view of (21), the points where the sign changes occur xl, x2 are

situated in the manner x~ < 0 < x2.

We prove now a polynomial lemma which will enable us to establish (20).

L e m m a . Let x , < 0 < x 2 . Define the functions ~(x,y), j = 0 , ' . ",2k, by

(22)

f2j-,(x, y) =

f2j(x, y) =

1 2x, x~ k I

1 2x2 X=K I' l<=j<=k, 1 X -~ y x 2 j - - l y 2k + l - 2 j

1 2xl x~ k

1 2X2 X 2k 2

1 x + y �89 y2/x2k-2']

O<=j<k.

Then

> 0 if (x,y)>(x2, x2) or ( x , y ) < ( x . xO (23) ~(x, y) < 0 if (x., x.) < (x, y) < (x2, x2).

P r o o f . Consider first f2j-l(x,y), l<=j<-k. It is convenient to decompose

fzs-l(x, y) into two parts

[~j-l(X, y) = u(x, y ) + v(x, y)

where


u(x,y)=

1 Xl X~ k

1 x~ x~ k

1 X x2J-ly 2k+1-2/

v(x,y)=

1 Xl X 2k

l X 2 X 2 2k

1 y x 2 j - l y 2k+l-2j

We prove (23) for u(x, y); the proof for v(x, y) is symmetrical (simply change roles

of x and y). The combination yields (23) for ]:2j-~(x, y). Define the polynomials in

one variable

d ( x ) = u ( - x, xO, e ( x ) = u(x,x),h(x)= u(x, x2).

Inspection reveals that e(x) always has a positive leading coefficient. The inequality

x ~ < 0 < x 2 assures that h(x) and d(x) exhibit positive leading coefficients.

Furthermore, all these polynomials share the same negative constant terms

x "x ~ - ' x~ ~-~) 2Xlt z - . Thus, all three polynomials have the form

A x ' + B x + C , s>-2

A x + C , s = l

where A and C are of opposite signs. The Descartes rule of signs implies that each

of these polynomials involves exactly one positive root located at - x~ for d(x), and

at x2 for both e(x) and h (x). Moreover, e(x) is convex, and vanishes at the negative

value x~.

Thus, we have established the following facts:

u(x,x,)=d(_x){~ ~ 0, x~ < x =< 0;

(24) i ' > 0 , x <x~ , x 2 < x ,

u(x , x ) - ~ e ( x )

l < 0 , X~ < X < X2;

f > 0, X2 < X~

u (x, x2) = h (x) t < 0 , O<=x <x2.


Returning now to (23), we consider each region separately.

a) Let (x, y) > (x2, x2). Then, since u (x, y) is increasing in y for x > x2, we have

u(x, y) > u(x, x2) = h(x) > O.

b) Let (x, y) < (x~, xl). Here u(x, y) is decreasing in y for x < x~. We then have

u(x, y)> u(x,x O= d ( - x ) > 0 .

c) Consider finally

(x,, xl) < (x, y) < (x2, x2).

We distinguish three subcases.

c.1) 0=<x<x2. We have

u(x, y)_-< u(x,x~)= h(x) < 0.

c.2) x l < x < 0 , y=>0. Then

u(x, y)_-> u(x,O).

Now u(x,O) is linear, and therefore

u (x, 0) =< max [u (x~, 0), u (0, 0)] < 0.

c.3) x ~ < x < 0 , x ~ < y < 0 . W e h a v e

u(x, y) < u(x, x,) = d ( - x) < O.

The discussion of [2j-l(x, y) is complete. Turning our attention to [2j(x, y), we

paraphrase the previous steps for cases a) and b) where the parity of the power was

irrelevant. The only case that needs additional attention is c) where it remains to prove

(25) f2 j (x ,y )<0 if (x~,xl)<(x,y)<(x2, x2).

We use the fact that, for a fixed x, [2j(x, y) is a convex function of y. Thus, (25) will follow if we prove that

(26) max [/2j(x, xO,]:2j(x, x2)] < 0, Xl < x < x2.


We prove specifically that f2j(x, x l )< 0, the proof of the other part being identical.

Observe that f2j(x, xl) is a convex function of x. Hence, we need to check only the

end points. Manifestly, f2s(x~, x~)= 0 and by direct computation

f~,(x:, x,) = - (x~ - x,)[x~ ~-~'- x~?-~q [x~ j - x~q < o.

Hence (26) is confirmed and the proof of the lemma is complete. We return now to the proof of the theorem. Expanding the determinant in the

following integrand along the last row and taking account of relations (19), we

obtain, for 1 =< j = k,

b

a

1 2x, x~ k

1 2x2 x zk2

1 x +,p(x) x2 ' - ' [~(x)W '-2j

[ ~ ( x ) - x]dF(x) =

b

2(xz - x,) f x zJ-'[ ~o (x)12k-2,+,[ q~ (X) -- x ldF(x). . a

Recalling the definitions (22), we thus have

(27)

b

2(x2- x,) f x2'-'[q~(x)]2k-2J+'[q~(x)- x ldF(x ) a

b

= f f~,_,[x, ,r [ , r x l d F ( x ) . a

In view of the behavior of ~ ( x ) - x and of the inequalities (23), the positivity of (27) ensues, establishing (20a). Inequality (20b) follows in a similar way, relying on

the properties of f2j[x, ~(x)] this time. Q.E.D. The following result of Van Zwet [12] can be obtained as a corollary of

Theorem 7.

C o r o l l a r y . Let X and ~ (x) be specified as in Theorem 7. Then, for all k >- 1, we have

(28) E I X 2j'+'] < E[~o2k+~(X)].


Proof . We rewrite (28) in the form

b b

(29) 0 f = f g(x)Iq (x) - x]dV(x) a a

where

g(x) = [ ~ ( x ) ] ~ + ' - x , p (x ) - x

2 k + l

k k

= ~y . {x2'I,p (x)]~k-~'+ [,p (x)]~,x2k-2,} + Y~ x2'-'f,p (x)] 2~-2'+' . j = O j = l

Relation (29) is obtained by summing the inequalities (20) over j. Q.E.D.

3. S o m e inequal i t i e s der ived wi th the he lp of the g e o m e t r y of

m o m e n t spaces

Some m o m e n t space cones. Let {u,}]' be a prescribed set of real functions defined

on a set T. The moment space M. with respect to {u,}]' consists of all real n-tuples

generated in the manner

~t. = {c_ = (Cl," . . ,c , ) : c, = f u , ( t )do'( t ) , i = l , 2 , " ", n I T

where or(t) traverses the set of all non-negative sigma finite measures defined on T.

The classical moment cones arise from the choice of the u~ as power functions. The

dual cone ~ , to M. can be identified with the collection of all positive "polyno-

mials" (a polynomial in this context refers to a real linear combination E7=1 a~u,). Where {u~}]' constitute a Tchebycheff system of continuous functions on a real

interval [a, b], then the boundary of M. admits a precise characterization. We

review now the fundamental representation theorems of moment points in this

case. We refer to Karlin and Studden [8, chap. 2] for further elaboration.

Every moment point _c = ( c l , . . . , c .) interior to M, admits two special represen-

tations (the upper and lower principal representations) of the following form:

Case 1. (n = 2 m ) .

There exist m + 1 unique points a = _t~ < _t2 < �9 �9 �9 < _t,,+, < b and corresponding

weights

298 S. KARLIN AND Z. Z I EGLER

(30) m+,

_Ak >0 , i = l , ' ' ' , m + l , ~ _ A k = l k=l

such that c admits the representation

b

f m+l (31) c~ = u,(t)d~_(t) = ~ _A~u,(tk), i = 1 , 2 , ' ' . , n. k=l

a

(Lower principal representation.)

Also, there exists {2k}~ +' in the arrangement a < 2, < 2 2 < . . . < 2,..1

corresponding weights {,~k} satisfying

= b and

b

m+l f (32) c , = ~ Aku,(gk) = u,(t)dd'(t), i = 1 , 2 , . . . , n . k=,

a

(Upper principal representation.)

Manifestly, both measures _tr and # concentrate weight at exactly m + 1 points. The

upper principal measure is distinguished from the lower in that it involves the right

end point b.

Case 2. ( n = 2 m + l ) .

The upper principal representation involves m + 2 points {g~}?+2 and the lower principal representation involves the points {_tk}7 '§ satisfying

a = gl <_t, < g2 < _t2 < "-. < gin+, < _t,.§ < 2,,§ b

with the formulas (31) and (32) persisting.

A variety of converse theorems to the Cauchy-Schwartz, H61der and General-

ized Means inequalities is available for functions constrained in some manner (e.g.,

see [4], [11]). The Favard and Berwald inequalities described at the start of the

previous section incorporate one such category of converse inequalities where the

restrictions express some form of convexity constraints. A number of these

converses have natural formulations for classes of self-adjoint operators in a

Hilbert space. We will show here first, via a simple typical example, how the general

problem can be transformed, by using the principal representation of moment

points, into an extremal problem involving a small number of variables which is

subsequently handled by standard calculus methods. The second example of this

type is more recondite.

The following elementary result subsumes some findings of Diaz and Metcalf [4]

and Rennie [11]. It represents an example of a class of inequalities for functions or sequences restricted in range.


T h e o r e m 8. Let F be a distribution function on [m, M], 0 < m < M < oo and let k be a positive number >= 1. Then

M M

(33, (M2---w~_~m)l s/2(f tdF(t))k + (M2k_lm)S/2(f dF_Ft__tt~tt )k<(_nM),2k-w2+(M),2~-w2" r t t r t t

P r o o f . On the interval [re, M] the functions us(t)=-1/t, u2(t)=-1, u3(t)-= t

form a T-system. Consider now the moment space M3 with respect to (us, u2, u3).

The point t? = (cl, c2, c3) whose coordinates are defined by

M M M

r f dF(t,, c2= f dF(t,=l, c3= f tdr(t, r a r a r a

is a moment point in M3. Inequality (33) reduces, therefore, to

1 1,2 + ( M 2 , - S m ) l , 2 c = - -

valid for all points (cl, c2, c3) of M3. Using the existence of a lower principal representation for (cs, c2, c3), we know

that there exist values x, m < x < M and A, 0_--A = 1 , such that cl =

A/m +(1- A)/x, c2 = A �9 1 + ( 1 - A). 1 = 1, c3 = Am + ( 1 - A)x. Hence (34) is equi-

valent to the statement that G(A,x) defined by

satisfies

(m),2' 1,,2 (35) G(A,x)_----- + \ M ] , m < x < M , 0=<A--<_I.

This is a maximization problem of a function of the two variables (A, x), obeying the

restrictions 0_-< A _-- 1, m _--< x _-< M, and is easily manageable. Indeed, direct dif-

ferentiation reveals that G(A, x) is an increasing function of A for each fixed x,

implying

independent of x.

300 s. KARLIN AND Z. Z I EGLER

We close with a more subtle application of the theory of principal representations

in moment spaces coupled to some calculus arguments. The following theorem

substantially generalizes a recent result of W. Gautschi [6], who deals with the

special case a, --- 1/n, i = 1 , . . . , n, using elaborate calculations. The theorem has

interest in statistical contexts and numerical analysis.

Theorem 9. Le t a l , �9 �9 ", a , be positive numbers and F(t) the classical G a m m a

function. Then

(36) ~ [F(t~)] "k -> 1 k = l

for all tk > 0, k = 1,. �9 n satisfying

(37) l~I (tk) ak = 1. E = I

Proof. We start with some preliminaries. Recall the known expansion of

F' ( t ) /F( t ) (see [31),

tF'(t) (t - 1)t (38) F(t) = - l + ( 1 - 3 , ) t + ~ - ~ , , = , ( m + l ) ( m + t ) ' t > 0

where 3, ~ 0.57 is Euler 's constant. A direct computation shows that

tr__r20]" m F(t) ] = 2 =, ( r e + t ) ~ > 0 ' t > 0 ,

so that tF ' ( t ) /F ( t ) is convex on the positive axis. Thus, the functions of

{1, t, tF ' ( t ) /F( t )} constitute an Extended Complete Tchebycheff (ECT)-system for

t > 0. Therefore (1, - 1/t, F ' ( t ) /F( t ) ) also forms an ECT-system, and by integrating

and adjoining the function 1, we deduce easily that

{1, t, - log t, log F(t)}

is also an ECT-system for t > 0. This observation enables us to avail ourselves of

the structure and characterization of the corresponding moment space. Keeping

this in mind, we note that the desired inequality can be stated in the form

(39) 2 at log F(tk) _--> 0, tk > 0 1

subject to

(40)

GENERALIZED CONVEXITY

ak logtk = O. 1

301

As both relations are homogeneous in the ak's we may append the normalization

(41) 2 ak = 1. 1

Since 0 offhand may present technical problems we slightly contract the basic

interval to [e, b], where e is arbitrarily small but fixed, confining all t, inside this

interval.

Consider the moment space ff~3(~, b) constructed with respect to the ECT-system

{1, t, - log t}. Let the point g(s r = (c,(~r cff~:), c3(~r be defined by

(42)

n [c:i:::;i:k:l c3(~) = - ~ ak log tk = 0.

1

We obviously have

(43)

b

2 aklogF(t~) > rain [logF(t).d~(t) 1 o.~v(e(,~)) d

where V(6(~)) comprises the set of measures ~r representing the moment point

6(sr Since {1, t, - l o g t, log F(t)} is an ECT-system it follows that the minimum is

attained for the lower principal representation of the moment point (1, ~r We

thus have

(44)

b b

m i n [ logF(t)&r(t)= mien f logF(t)d~(t), o.~ v(e(,~)) J

where o- has a mass of A at e and a mass of ( l - A ) at the point y, e < y < b ,

satisfying

(45) A loge + ( 1 - A ) l o g y = 0 ,

,~e + (1 - , ~ ) y = ~:.


Note from (45) that y _-> 1. The r ight-hand side of (44) is equal to

where st > 0 is arbitrari ly fixed and A, y are de te rmined by (45). A lower bound for

the left-hand side of (43) will be secured by bounding from below the expression

min~.y G, (A, y ; st) for all positive st and all e close enough to 0. F rom (45) we have

A = logy s t = e l o g y - y l o g e log y - log e ' log y - log e

Substituting these expressions into G~(A, y ;s t ) gives

log y log F(e ) (46) G,(A, y; st) = F~(y) = log y - log e

log e log F(y) . - l o g y - l o g e

Differentiat ing (46) with respect to y and execut ing some obvious simplifications

produces

y [log y - log e 1 d ~ (y) = _ log e [log F(e ) - log F(y) + y F ' ( y ) ] J dy [ l o g y - l o g e F(y) J '

where the factor in f ront of dF,(y)/dy is positive since y _-> 1 _-> e.

From the expansion in (38), we infer that 1 + y F ' ( y ) / F ( y ) is increasing for y _-> 1

and thus its min imum over this region is 1 - Y. Hence

(47) y + Y F ' ( Y ) > o , for y > l . r(y)

We observe next that logF(e)/(- loge) behaves like - er'(e)/F(e) for e small.

Comparing to (38), we conclude that it tends to 1 as e --)0. Thus, there exists an eo

small enough so that

logF(e0)-logF(y) > Y, for 1 <y < b. l o g y - l o g e o = =

Hence , dF, o(y)/dy agrees in sign over the whole region with V + yF ' (y ) /F (y ) , which

min [A log F(e ) + (1 - A ) log F(y)] = min G~ (A, y ; st), A,y A,y

(for e < 1)

G E N E R A L I Z E D CONVEXITY 303

is positive for all y > 1 by (47). Therefore F~o(y) achieves its minimum with respect

to 1 --- y -< b at y = 1 and F~o(1) = 0. We have thus established that 0 is a lower

bound for F~o(y) provided e0 is small enough. Q.E.D.

REFERENCES

1. L. Berwald, Verallgemeinerung eines Mittelwertsatzes yon J. Favard ]:fir Positive Konkave Funktionen, Acta Math. 79 (1947), 17-37.

2. C. Borell, Inverse Hiflder inequalities in one and several dimensions, J. Math. Anal. Appl. 41 (1973), 300-312.

3. P. J. Davis, Gamma functions and related functions, in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun (eds.), N. B. S. Applied Math. Series 55, 1964, pp. 253-293.

4. J. B. Diaz and F. T. Metcalf, Stronger forms of a class of inequalities of G. P61ya-G. Szeg6 and L. V. Kantorovich, Bull. Amer. Math. Soc. 69 (1963), 415-418.

5. J. Favard, Sur les Valeurs Moyennes, Bull. Sci. Math. (2) 57 (1933), 54-64. 6. W, Gautschi, Some mean value inequalities for the Gamma function, SIAM J. Math. Anal. 5

(1974), 282-292. 7. E. Hopf, Ober die Zusammenhiinge Zwischen Gewissen Hi~heren Differenzenquotienten Reeler

Funktionen Eines Reelen Variablen und Deren Differenzierbarkeitseigenschaften, Dissertation, Berlin, 1926.

8. S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience, New York, 1966.

9. S. Karlin and Z. Ziegler, Inequalities, ordering relations and generalized convexity (to be published).

10. Z. Nehari, Inverse H61der inequalities, J. Math. Anal. Appl. 21 (1968), 405-420. 11. B. C. Rennie, On a class of inequalities, J. Austral. Math. Soc. 3 (1963), 442-448. 12. W. R. Van Zwet, Convex transformations of random variables, Mathematical Center Tracts 7,

Amsterdam, 1964. 13. Z. Ziegler, Generalized convexity cones, Pacific J. Math. 17 (1966), 561-580.

DEPARTMENT OF MATHEMATICS THE WEIZMANN INSTITUTE OF SCIENCE

REHOVOT, ISRAEL

AND

DEPARTMENT OF MATHEMATICS STANFORD UNIVERSITY

STANFORD, CALIFORNIA, U.S.A.

DEPARTMENT OF MATHEMATICS THE TECHNION--ISRAEL INSTITUTE OF TECHNOLOGY

HAIFA, ISRAEL

(Received February 13, 1976)

some applications to inequalities of the method of generalized convexity

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