convexity-like inequalities for averages in a convex set

Aequationes Mathematicae 45 (1993) 179-194 000l -9054/93/030179-16 $1.50 + 0.20/0 University of Waterloo © 1993 Birkhfiuser Verlag, Basel

Convexity- l ike inequalities for averages in a convex set

M I H A I D R A G O M I R E S C U A N D C O N S T A N T I N IVAN

Summary. For a function f : X --, R u { + oc }, convex and finite over an algebraically open convex subset C of a linear space X, we provide a simple necessary and sufficient condition for the "convexity inequality"

to be satisfied for every integer n/> 2 and all X l , . . . , xn ~ fl[" and 2~ . . . . . 2, ~ R+ such that ~ 2, = I and

~ 2 , x , e C

(i.e., if just the average ~ 2,x, lies in the set C of convexity, not the points x~ . . . . . x, themselves). Our condition is simply thatfmajorizes a certain convex function, depending only on the restriction

f c o f f to C, namely the smallest convex exlension o f f c , which is given explicitly. We apply this result to enlarge the set of n-tuples ( X l , . . . , x,) ~ X" satisfying (C,), beyond C ~, the

domain ensured by the convexity o f f on C. A number of examples show that such an extension of the domain of validity of (C,,) can be easily obtained when X = R.

1. Introduction

Convexity underlies many important inequalities. Any extension of the basic convexity inequality yields improvements of all its consequences.

This work has been presented at the 14th Int. Syrup. on Math. Programming (Amsterdam, Aug. 5-9, 1991) and belongs to a sequence of papers, starting with [1-3], intended to motivate some of the many convexity-like inequalities which are not justified by the bare convexity of the functions involved.

Throughout this paper we assume that C is a nonempty convex subset of a real linear space (1.s.) X, and that the function f : X ~ R w { + o o } = R is finite and

AMS (1991) subject classification: Primary 26D20, Secondary 26B25.

Manuscript received July 30, 1991 and, in finalform, September 15, 1992.

179

180 MIFIAI DRAGOMIRESCU AND CONSTANTIN IVAN AEQ. MATH.

convex on C, i.e., it satisfies the inequality

f ( (1 - 2)x + 2y) ~<(1 - 2)f(x) + 2f(y) (1)

for all x, y e C and ;t s (0, 1). For the sake of a considerable simplification, we moreover assume here that C is algebraically open.

Then, for every integer n ~> 2, the "convexity inequality"

f ( ~ 2;x;) <~ ~ 2~f(x,) (2)

is satisfied for all x~ . . . . . x, ~ C and all (21, • • •, 2, ) in

A, = { (2~ , . . . , ,~.,) ~ R+ ~ 2 ; = 1}.

(Throughout the paper ~ stands for sum over = 1 . . . . . n. The sign -= stands for "notation", fc denotes the restriction o f f to C.)

Under weak additional conditions on f , we shall provide a set of n-tuples (x~ . . . . . x,) E X" satisfying (2), which may be considerably larger than C".

Specifically, the paper provides a necessary and sufficient condition for f to possess the "convexity-like property" (relative to C)

X 1 . . . . . Xn ~ a'~k'r~ (21 . . . . . 2 . ) ~ A n , Z I~iXi ~ C ~ (2 ) . ( C L ~ )

This property is obviously weaker than convexity o f f (on X), but it implies the (assumed) convexity o f f on C.

We point out that, unlike usual convexity, (CL~) is not implied by the weaker property

x, yeX, - - -2 - -~C~f <~(f (x)+f(y))

whenf i s continuous; this follows from the results in [ 1], where the last property has been investigated in the case X = R.

Obviously (CL~-) =~ (CL~) for every n >~ 2 (taking x2 . . . . . x,). The converse implication is not at all obvious. However, Corollary 1 below will show that properties (CL~.) , ~ 2 are pair-wise equivalent.

Vol. 45, 1993 Convexity-like inequalities for averages in a convex set 181

Section 2 provides the main general results; Theorem 2 gives an intimate (though unexpected) connection between properties (CL~-)n/> 2 and fma jo r i s ing the "smallest convex extension" of f t . Section 3 gives some more detailed results in the (simplest) case X = R, and Section 4 provides a few examples.

2. Main results

THEOREM 1. Let f : X--* R be finite and convex on a convex subset C o f a l.s. X.

I f f >1 g, where g: X --* K is a convex extension o f f c , then f has property ( CL~, ) for every integer n >1 2.

Proof. Let x, . . . . . x, ~ X and (21 . . . . . ~.,) e A, be such that ~ 2~-xe e C. From f ( x i ) >1 g(xi), i = 1 . . . . . n, the convexity of g and the assumption that f ( x ) = g(x) if x e C, we obtain

which proves (2).

Theorem 2 below, the main result of this paper, needs to recall the following definitions.

The algebraic interior (core) of a subset C of a l.s. X, denoted inta C, consists of all points y e C such that, for each h ~ X, there is e > 0 such that y + 2h e C for all 2 e (0, e); C is said to be a-open (algebraically open) if inta C = C [5, p. 7].

I f X is a topological linear space (t.l.s.) and a convex subset C has nonempty interior, then int~ C = int C [5, p. 59, Lemma A], hence C is a-open /f and only i f it is open .

I f C c X is a-open convex and the function f : C--+R is convex then, for every y ~ C and h e X, the directional derivatit, e of f at y (in direction h), defined as

f ' ( y ; h) = lim [ f ( y + 2h) - f ( y ) ] / 2 , (3) 250

exists and is finite, and function f ' ( y ; - ) : X--0 R is sublinear [5, p. 59, Theorem D]. From (1), dividing by 2 and letting 2 + 0, using (3) we obtain

f ( x ) - f ( y ) > ~ f ' ( y ; x - y ) for all x, y e C . (4)

182 MIHAI DRAGOMIRESCU AND CONSTANTIN IVAN AEQ. MATH.

LEMMA 1. If C is an a-open convex subset o f a l.s. X, and f : C ~ R is a convex

function, we have

l i m f ( ( 1 - 2)x + 2y) = f ( x ) f o r all x, y e C. (5) ;.1o

Proof. Let x , y ¢ C and A - { 2 ~ R [ ( I - 2 ) x + A y ~ C } Since C is a-open, 0 E int A. The f u n c t i o n ¢p: A ~ R, ~o(~) = f ( ( 1 - 2 )x + ~y) , is convex, hence continuous at 0 ¢ int A; this impl ies lim~.~ o ~p(,~) = tp(0), wh ich is prec ise ly (5) .

LEMMA 2. /fq~: [0, 1) --* R is a convex function and

lim cp(2) = ~o(0), (6) ;.~o

then

lim [¢p(2) + 2~o'(2; - 1)] = ~o(0). (7) ;.~o

Proof. Since ~o is convex, its left-derivative ~o'_ (2) = - ~ o ' ( 2 ; - 1) exists , is finite and non-decreasing on (0, 1), hence ~o'(. ; - 1) is non-increasing; therefore the limit L -= lira; ~ o ~o'(), ; - l) ~< + ~ exists. I f L is finite then (7) follows directly f rom (6). If L = + ~ then ~o'().; - 1) > 0 for 2 ~ (0, ~); us ing (4) and the p o s i t i v e - h o m o g e n i t y

o f ~o'().; .) we have the inequalities

(p().) < ~p().) + 2g0'(2; -- 1) ~< ~0(0) for all 2 ~ (0, ~),

which, together with the assumption (6), implies (7).

THEOREM 2. Let C be an algebraically-open convex subset o f a l.s. X. A function

f : X--* R w {+ oe} _= R, f inite and convex on C, possesses property ( C L ~ ) f o r some

integer n >1 2 i f and only i f

f >~ E fc (8)

or, equivalently,

f ( x ) >~ E fc (x ) for all x ~ X \ C , (8')

where the function Efc : X ~ R is defined by

E f c ( x ) - sup [ f ( y ) + f ' ( y ; x --y)] , x ~ X. (9) y ~ C

(Efc turns out to be the smallest convex extension o f f c ; see Remark 3.)


Proof. Necessity. Obviously, p roper ty ( C L ~ ) (appl ied with x2 . . . . . x , ) implies (CL~-) . We prove now that ( C L ~ ) implies (8).

Let x ~ X. Since C is a-open and convex, for every y c C there exists ~ ~ (0, 1) such that z ~ - ) . x + ( 1 - 2 ) y e C for all 2 c ( 0 , e). Then (CL~) implies 2f (x) + (1 - 2 ) f ( y ) >~f(zD; hence

f (x ) >~f(y) + [f(z; .) - f (y )] /2 .

for every L e (0, e). Lett ing ;t + 0 and using (3) we obtain

f (x) >~f(y) + f ' ( y ; x - y ) for all y ~ C.

Consequently, using definition (9), f (x) >1 Efc(x). Sufficiency. We prove that the function Efc defined by (9) is a convex extension

o f f c (hence (8) and (8 ') are equivalent), then apply Theorem I with g = E f c . For each y ~ C, f ' (y ; .) is sublinear on X, hence convex. This implies that the

function x ~--~f(y) + f ' ( y ; x - y ) is convex on X. Efc is convex as supremum (for y ~ C) of these convex functions.

It follows f rom (4) and (9) that

f (x ) >~Efc(x) for every x c C. (lo)

We prove now that

f (x ) <~Efc(x) for every x c C, (10')

which proves that actually equality occurs in (10), concluding the p r o o f o f Theorem 2. Let x , y c C. Then, for every 2 ~(0 , 1), z~ - (1 - 2 ) x + 2 y ~ C; hence, by the definition (9), we have

f(z).) +f ' ( z ; . ; x - z;.) ~ Efc(X). (11)

Letting 2 +0 in (11) yields (10'). Indeed, the function cp:[0, 1 ] ~ R defined by

~0(2) =-f(zD is convex, ~0'(2; - 1) = limtso[~O(2 - t) - ~o(2)]/t = limtzo[f(z ~ + t(x - y ) ) - f (z~)]/ t = f ' ( z ~ ; x - y ) , and condi t ion (6) o f L e m m a 2 is satisfied due to conclusion (5) o f L e m m a 1; since x - z ~ = 2(x - y ) , using the posi t ive-homogeni ty of f '(z).; .) and L e m m a 2 we have

f(z~.) + f'(za; x -- z~) = cp(2) + ,~o'(2; - 1) --*f(x) as 2 ~ 0.

184 mtna~ DRA(3OMIRESCU AND COns ' rANTIN ]VAN AEQ. MATH.

REMARK 1. D ~ X being an arbitrary subset, Theorem 2 may be reformulated as follows: the function f : D --* R, convex on an a-open convex subsel C ~ D, possesses ( f o r some n >~ 2) the property

X 1 . . . . . X n ~ D, (2,, . . . , 2,) e A,, ~. )o,x i E C ~ (2) (CL~.D)

i f and only i f

f ( x ) >>- E f t ( x ) f o r all x ~ D \ C . (8")

Proo f Apply Theorem 2 for

f f (x) , i f x ~ D f : X --, R, f ( x ) = ~[. + ~ , if x ~ X \ D "

The consequence of this version is an extension o f the domain o f validity of the convexity inequality (2) when (8") holds for some D larger than C.

REMARK 2. A slightly more general version of Theorem 2 is the following: The funct ion f : X ~ R, f inite on an a-open convex set C ~ X, possesses property

( CL~ ) f o r some integer n >~ 2 i f and only i f

f c is convex and f >~ E f t •

REMARK 3. We have proved (within the proof of sufficiency of Theorem 2) that function Efc: X --, R defined by (9) is a convex extension o f f t . It is noteworthy that E f t is actually the smallest (i.e., minimal) convex extension of re ; this follows from [4, Theorem 1], proved under the somewhat weaker assumptions that into C ~ and f i s (convex and) "algebraically 1.s.c.", (a-l.s.c.), i.e., possesses property (5). The present assumption that C is a-open simplifies considerably the discussion and avoids (due to Lemma 1) the assumption of [4] that f is a-l.s.c.

Since the necessary and sufficient condition for (CL~), given in Theorem 2, does

not depend on n, we have the following result.

COROLLARY 1. Under the assumptions o f Theorem 2, the properties (CL~) and

(CL~-) are equivalent for all integers m, n >1 2.

When (C being an open convex subset of a t.l.s. X) the convex function f c : C --, R is continuous (what happens, e.g., when X is finite dimensional Hausdorff [5, p. 82, Corollary 2], when X is a Banach space and f is lower semi-continuous


[5, p. 197, Ex. 3.50] or, more generally, when f is bounded above on some neighbourhood of some point of C [5, p. 82, Theorem A]), then the function Eft is l.s.c., and is continuous over the interior of its domain of finiteness [4, Theorem 2].

When f is convex and continuous, by Minty's theorem [5, p. 84, Theorem B] f possesses subgradients at every y 6 C, i.e.

gf(y) = {4, ~ x ' I f ( x ) - - f (y ) >~ dp(x --y ) for all x EX} # ~ ,

and by the Moreau-Pshenichnii Theorem [5, p. 86, Theorem D] we have

i f (y; h) = max{~b(h) [4, e gf(y)} for all h ~ X;

consequently the function Efc is alternatively defined by

Eft(x) = sup{f (y) + 4,(x - y)lY ~ c , q~ e Of(y)}. (9')

When f is GS.teaux-differentiable (i.e., for every y ~ C, Of(y) contains a single element Vf(y) ~ X*) then (9') becomes

Efc(x) = sup [ f ( y ) + Vf(y)(x - y)]. (9") y ~ C

For refinements of the representation formulas (9) -(9") when C is algebraically- closed and examples of computing Efc, see [4].

The equivalence between (CL~) and (8) is remarkable and useful, because (CL~), a constrained inequality involving n variables x~ . . . . . x, e X (besides 2~ . . . . . 2, ~ R+ ), is reduced to the unconstrained inequality (8), involving a single variable x e X. The great difficulty consists of computing explicitly (parametrically with respect to x) the supremum defining Eft.

3. The case X = R

To illustrate the kind of improvements of the convexity inequality (2) that Theorems 1 and 2 can yield, we give now two applications of them in the case X = R. In this case computing Eft is very easy due to the following simplest corollary of the results in [4].

186 MIHAI DRA( 3 OM IRE SCU A N D C O N S T A N T I N IVAN AEQ, M A T H .

PROPOSITION 1 [4, Corollary 4]. The smallest convex extension of a continuous convex function f : [a, b] ~ R is the function E f : R ~ R defined by

i f x ~ ( - ~ , a )

i f x ~ [a, b] x ~ ( b , + ~ )

I f (a ) + A(x - a), Ef(x) = ~f(x~,

( f (b ) + B(x b),

where A =--f'(a; 1) =f~_ (a) 1> - ~ and B =- - f ' ( b ; - 1 ) = f ' _ (b) ~< + ~ .

(12)

A' < A and B' > B be such that

f ( x ) >~ f (a ) + (x -- a ) f + (a)

and

f ( x ) >~f(b) + (x -- b )f'_ (b )

for x ~ [A ', a),

for x ~ (b, B'].

(13)

Then inequality (2) is satisfied for every integer n >I 2 and all xl . . . . . x, ~ [A', B'] and (21, • • •, 2,) e A, such that

a ~ 2 2ixi <~ b.

Proof Apply Theorem 2 to the function f defined on R by

~f(x), if x ~ [A', B'] f ( x ) ( + oe, otherwise

with C = (a, b). Being convex on (A, B ) , f i s continuous on [a, b] (and f '+ (a),f'_(b) are both finite), hence Proposition 1 applies. Condition (8') of Theorem 2 is verified by (13); hence the conclusion follows from Theorem 2 /f ~ 2 i x i ~ ( a , b ) . If

2ixi = a, (2) results using a continuity argument: unless xi = a for all i (when (2) is trivial), there exist xj > xk, say x~ > x2; defining 2'~ - 2~ + ~, 2~ = 22 - e, 2~ = 2i for i = 3 . . . . . n, with ~ > 0 small enough, we have ~2~x; E (a, b), which implies f ( ~ ),~xi) ~< ~ 2~f(x,.). Hence, letting e ~ 0, (2) follows since f i s continuous at a. This concludes the proof.

COROLLARY 2. Let f: R ~ R be convex on (A, B). Let [a, b] ~ (A, B), and let

A possibility of easily extending the domain of validity of the convexity inequality (2) (beyond the limits ensured by convexity o f f c ) is described by the following corollary (see Fig. 1).


f

,,. y = fix)\ ~ / / ' - i~ j "

1 ix,,._ ~ ,, ~Et~(.) I t I I

, L_ l [ ~ A" ,4 a b B O'

Figure 1. Geometric interpretation of Corollary 3.

This continuity argument , permit t ing the use of a closed interval C, applies to all examples below.

Another consequence of Theorem 2 is

THEOREM 3. Let f : R ~ R be a continuously differentiable odd function, strictly convex on [0, + ~ ) . Then inequality (2) is satisfied for every integer n >t 2, and all x~ . . . . . xn ~ R and (21 . . . . . 2~) ~ A, such that

~ 2 , x i + K l ' m i n ( x l . . . . . x , )>~0 , (14)

where K! ~ (0, 1] is defined as

K r = sup(-a /x , , ) , (15) a > 0

where, for a > 0, x,, denotes the unique negative solution of the equation

f ( x ) - f ( a ) - ( x --a )f'(a) = 0 . (16)

Proof (i) For a > 0, let Fo(x) denote the left-hand side o f the eq. (16). The strict convexity o f f on R+ implies the following:

(i l) by (4),

F,,(x) > 0 for every x > 0, x # a; (17)

(i2) since f ( 0 ) = 0, using (4) twice,

af'(O) < f ( a ) < a f ' ( a ) for every a > 0 ; (18)

188 MIHAI DRAGOMIRESCU AND CONSTANTIN IVAN AEQ. MATH.

the function f being odd, hence f ' even, it follows that

xf'(O) > f ( x ) > xf ' (x) for every x < 0; (19)

(i3) f ' is increasing on R+ and decreasing on R ; hence the solutions of equation F'~(x) = f ' ( x ) - f ' ( a ) = 0 are only x = __a.

Since (18) and (17) i m p l y / 7 . ( - a ) > 0, F.( + ~ ) > 0, Rolle's sequence

x - ~ - a a +oo

F,(x) + 0 +

proves that eq. (I 6) has an unique negative solution x~ < - a ; hence 0 < - a / x , < 1, implying Kt ~(0, 1], and we have F,(x) >>. O, i.e.,

f ( x ) >~f(a) + (x -- a)f '(a) for every x /> x,. (20)

(ii) Now we prove that, for every x < 0, there exists a > 0 such that xa = x. Let x < 0, and let Gx(a) denote the left-hand side o feq . (16). Since the function

a ~ Gx(a) is continuous, and by (19) we have G x ( 0 ) = f ( x ) - x f ' ( 0 ) < 0 and G x ( - x ) = 2 ( f (x ) - xf ' (x)) > 0, there exists a e (0, - x) satisfying ( 16); hence X a = X .

(iii) For arbitrary a > 0 , we prove that inequality (2) is satisfied if

(21 . . . . . 2,) ~ A, and

xl . . . . . x~/>x~ and ~2ixi>~a. (21)

This follows applying Theorem 2 to f : R ~ / ~ defined by

i f (x ) , i f x ~ > x , f(x)

=-(+Go, ~ if x < x , , '

convex on C-= (a, + oo). Since by Proposition 1 we have

Ef t (x ) = f ( a ) + (x - a)f '(a) for x < a,

condition (8') is satisfied by (20). By Theorem 2, inequality (2) is verified if 2,-xi > a; if ~ 2ix i = a, (2) follows by continuity, as in the proof of Corollary 2. (iv) We now conclude the proof of Theorem 3. Let x~ . . . . . x, e R. I f m

-= min(xj . . . . . x , ) ~> 0, then inequality (2) follows from the convexity o f f on R+.

Vol. 45. 1993 Convexity-like inequalilies for averages in a convex set 189

I f m < 0, by (ii) above there exists a > 0 such that x, = m. Then the first condit ion (21) is trivially satisfied, while the second results f rom (14), K t > ~ - a / x , and - m > 0 as follows: ~. 2,x~ ~> - K r m >~ (a/x,,)m = a. Then inequality (2) is satisfied due to (iii) above.

REMARK 4. Under the assumpt ions of Theorem 3, if(x~ . . . . . x,,) e R'+ then (2) follows f rom the convexity of f on R+ (and condition (14) is superfluous). Theorem 3 provides an extension of the domain of validity of this convexity inequality: for fixed (21 . . . . . )~,) e A, , the set of points (xj . . . . . xn) e R" satisfying (14) is obviously larger than R" " actually this set is a convex polyhedral cone, including R n (but included in the set o f all (x~ . . . . , x~) e R" satisfying (2)).

COROLLARY 3. Under the assumptions o f Theorem 3, the inequality

f ( Z 2~t~)>~ 2~f(t,) (2 ')

is verified for all tl . . . . . in e R and ('~1, . • •, 2,) e A, such that

ZAit~ + K / • max(tj . . . . . t , ) ~<0, (22)

where K r e (0, 1] is defined by (15).

Proof It follows f rom (22) that xi = - t ; , i = 1 . . . . . n, satisfy (14) and hence (2). Tak ing xi = - t~ , (2) yields (2') because f i s odd.

REMARK 5. In fact, Theorem 3 and Corol lary 3 are valid if f is cont inuously differentiable on R\{0}, and f ( 0 ) = 0 ( implied e.g. by continuity at 0). The differentiability of f at 0 is not necessary, because condit ions ( 1 8 ) - ( 1 9 ) are obviously verified even when f ' ( 0 ) = - o o (note that, since f i s convex on [0, + or) and odd, f ' ( 0 ) exists and is > ~ - oo).

4. Examples

In this section we note a few "ex tended" convexity inequalities, which follow as s t ra ightforward applicat ions o f Theorems 2, 3 or Corollaries 2, 3. All o f them extend the domain of validity of known convexity inequalities. N o n e of them is trivial; the reader should imagine the difficulties o f proving them directly.

190 MIHAI DRAGOMIRESCU AND CONSTANTIN IVAN AEQ, MATH.

E X A M P L E I . The inequality

( ~ 2;xi)3 ~< ~ 2ix 3 (23)

is satisfied for every integer n >1 2 and all Xl . . . . . x . ~ R and (21 . . . . . A..) ~ A. such that

1 2ixi + ~ min(xl . . . . . x~)/> 0. (24)

Proof. We apply Theorem 3 to f (x ) = x 3. For a > 0, the negative solution of eq. (16) is x. = - 2 a , hence - a / x ~ = 1/2 = Kf.

REMARK 6. For xj . . . . . xn/> 0, (23) (as well as (25) or (30)) is the well-known inequality between the weighted averages of orders 1 and 3 (respectively pfq), and condition (24) (respectively (26)) is superfluous.

EXAMPLE 2. I f p, q are odd integers and p > q >1 1, the inequality

( ~ 2ix~ )P:q <~ ~ i.ix~/# (25)

is satisfied for all Xl . . . . . xn e R (n >1 2) and (2~ . . . . . 2.) ~ An such that

2~x~ + K" min(xi . . . . . x , ) /> 0, (26)

where K e (0, 1) is the unique positive solution of the equation

(p _ q)Kp/q +pKp:q - l _ q = 0. (27)

The converse of inequality (25) holds i f

2,x,. + K . max(xl . . . . . x,) ~< 0. (26')

Proof. We apply Theorem 3 to f ( x ) = x p/u. Equation (16) becomes

q x p / q __ pxaP/U - 1 + (p _ q)aP/# = O,

hence K = - a / x satisfies (27), The last statement follows from Corollary 3, (Such a statement could be also given for each of the next examples.)


EXAMPLE 3. The inequality

(2 ,~ixi) l /3 ~ Z ~ix]/3 (28)

is satisfied for all Xl . . . . . x, ~ R (n >1 2) and (21 . . . . , ,I.,) ~ A, such that

1 2ixi + ~ m i n ( x l . . . . . x , ) ~> 0. (29)

Proof We app ly T h e o r e m 3 to f ( x ) = - x 1/~ (see R e m a r k 5). F o r a > 0, the

negat ive so lu t ion o f eq. (16) is xa = - 8 a ; hence - a / x , = 1/8 = Ky. Exam ple 3 can

be genera l ized as follows.

EXAMPLE 4. I f p, q are odd integers and q > p >~ 1, we have

(Z,t,xi /> Z ,xf/q (30) P/q

for all x~ . . . . . x, ~ R and (2L,. • . , 2 , ) E A , satisfying (26).

Proof We app ly T h e o r e m 3 to f ( x ) = -xP/q; see Exam ple 2.

EXAMPLE 5. I f f : R -~ R, f (x) - - x( 1 + x 2) - l/2, inequality (2) is satisfied for all xl . . . . . xn ~ R and ()~l . . . . . 2 , ) ~ An satisfying (24).

Proof F o r a > 0 , eq. (16) has the so lu t ion x , = - a g ( a ) where g(a) = 1 + a 2 -t- ( 1 + a 2 -t- a 4) 1/2; hence K r = sup~ > 0 g(a) - ~ = [ in f , > o g(a)] - ~ = 1/2.


( Z 2ixi) ln ~ Rix~ <~ ~,~ixilnlxi[ (31)

is satisfied for all xl . . . . . x , e R and (21 . . . . . 2 , ) c A n satisfying (14), where Ky "-, 0 .27846454 is the unique solution of the equation

1 + K + I n K = 0 . (32)

192 M I H A I D R A G O M I R E S C U A.ND C O N S T A N T I N I V A N A E Q . M A T H .

(Remark. For x~ . . . . . x , > 0, (31) reduces ~ to the classical inequality implying the

well-known property that the "weighted average o f order x " o f the positive numbers a~ . . . . , a~, i.e. the function x ~-* ( ~ 2ia~) ~/x, is nondecreasing on R.)

Proo f We apply Theorem 3 to f : R ~ R , f ( x ) = x ln[x] if x ¢ 0, and f ( 0 ) = 0. Equat ion (16) becomes x lnlx I - x(1 + In a) - a = 0; hence K =- - a / x verifies eq.

(32).


is satisfied for every n >~ 2 and all x , . . . . . x , ~ [ - ~, rr] and (21, . • . , Z,~) ~ A,, such that

2ixi] <~ re~4.

Proo f We apply Corol lary 2 to f defined by f ( x ) = - c o s x if x c [ - ~ , ~] = [A', B'] (and +o~ otherwise), with [a, b] = [ - r e /4 , zc/4]. Since f and

f c are even functions, we only need to prove the second inequality (13); for

x e (re/4, n] we have E f t ( x ) = (x - 7t/4 - 1 ) / x ~ < f ( x ) (because E f t ( x ) < 1).

EXAMPLE 8. For f : R --, R defined by f (x) - x e ~, inequality (2) is satisfied for all

x I . . . . . x , ~ R and (21 . . . . . )~) ~ A , such that

~, 2,xi >~ - 1 .

Proo f We apply Corol lary 2 with (A, B) = ( - 2 , + ~ ) (where

f " ( x ) = ( x + 2 ) e x > 0 ) and a = - l , A ' = - ~ , b = B ' = + G o . S i n c e f ' ( - 1 ) = 0 and f ( - I) = min f (R) , the first condit ion ( 1 3 ) , f ( x ) ~>f( - 1) for x ~ ( - or, - 1), is

trivially satisfied.

EXAMPLE 9. For f : R ~ R, f ( x ) ==-IxI3/(1 -j-x2), inequality (2) is satisfied f o r all

Xl . . . . . x , ~ R and (2t . . . . , 2 , ) ~ A, such that

),iX, <~ I.

Vol. 45, 1 9 9 3 Convexity-like inequalities for averages in a convex set 193

Proof. We apply Corollary 2 with (A, B ) = ( - w / 3 , x/~) (where f " ( x ) = 2 t x l ( 3 - x 2 ) ( l + x 2)-3>~0) and C = [a, b] = [ - l , 1 ] . Since f and E f t are even functions, we only need to prove that f ( x ) >~ E f t ( x ) for x e (1, + oo). This is easily seen to hold, either on the graph of f, or as follows: for x/> 1, f ' ( x ) =

1 + (x 2 - 1)(l + x 2 ) - 2 > 1, and by Lagrange's theorem there exists c e (1, x) such that f ( x ) = f ( 1 ) + f ' ( c ) ( x - 1) > f ( 1 ) + x - 1 = E f t (x).

One easily obtains p-dimensional extensions of all previous examples. For instance:

EXAMPLE 10. (A p-dimensional extension of Example 1). For f:RP-->R, f ( x ) = ~P= l x 3, inequality (2) is satisfied for every n >1 2 and all xl . . . . . x , ~ R p and

(21 . . . . ,2 , ) ~ An satisfying (24), where rain(x1 . . . . . x , ) =- (min(xl/ . . . . . xn/))/= 1...: ( f x, = (x,j)j = i,...: for i = 1 . . . . . n.

Proof. For every j e {1 . . . . ,p}, by (24) we have

1 Y 2ix;, + ~ min(xlj . . . . . x,,) >i 0.

It follows from Example 1 that ( ~ 2;x,;) 3 ~ y. 2,x~:,j = 1 . . . . . p. Summing up these inequalities yields (2).

EXAMPLE 11. Let p>~3 be an odd integer, and let f :R2--~R be such that

f i x , y) = x p + yP if(x, y) ~ R2+ =- C. Then f possesses property ( CL~c ) for every n >1 2 i f and only i f

f ( x , y) ~ (X+) 3 q- (y+)3

where x+ =-max(x, 0).

for all (x, y) e R 2 \ R %

Proo f We apply Theorem 2, using the fact that supx>o[pax p 1 _ (p _ 1)x p] = (a+)P; hence Ef t (a , b) = (a+) p + (b+) p.

EXAMPLE 12. Let X be a Hilbert space, C {x Xlllx-cll r} (where

c ~ X, r > 0), and let f : X ~ R be such that f ( x ) = ]Ix H 2 for x ~ C. Then fposses ses property ( C L ~ ) f o r every n >~ 2 i f and only i f

f(x) ~ [ I x l T - ( I I x - c l I - r ) 2 for every x E X \ C .

Proo f We apply Theorem 2, using the fact that E f t ( x ) = Ilxll 2 - ( l lx - eli - 0 2 [4, Example 2].

t94 MIHAI DRAGOMIRESCU AND CONSTANTIN IVAN AEQ. MATH.

References

[1] DRAGOMIRESCU, M. and IVAN, C., Extensions of Jensen's inequality (H). Rev. Roumaine Math. Pures Appl. 35 (1990), 669-682.

[2] DRAGOMIRESCU, M., Extensions of Jensen's inequalities (111): On the inequality f (x) + f ( y ) +f(z) >I 3f((x + y + z)/3). Stud. Cerc. Mat. 44 (1992), 337-354.

[3] DRAGOMtRESCU, M., Extensions of Jensen's fnequality f ( x 0 + . . . + f ( x , ) >t nf((x I + . . . + xn)/n ). Rev. Roumaine Math. Pures Appl. 37 (1992), 847-871.

[4] DRAGOMIRESCU, M. and IVAN, C., The smallest convex extensfon of a convex function. Optimization 24 (1992), 193-206.

[5] HOLMES, R. B., Geometric functional analysis and its applications, Springer-Verlag, New York, 1975.

Romanian Academy, Centre of Mathematical Statistics, Bd. Magheru 22, RO- 70158 ,Bucharest 22, Romania.

FH Niederrhein, FB Elektrotechnik, Reinarzstr. 49, D-W-4150 Krefeld, Germany.

convexity-like inequalities for averages in a convex set

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