On the preservation of periodic monotonicity

Constr. Approx. (1992) 8:129-140 CONSTRUCTIVE

APPROXIMATION �9 1992 Springer-Verlag New York Inc.

On the Preservation of Periodic Monotonicity

Stephan Ruscheweyh and Luis C. Salinas

Dedicated to the memory o f Professor I. J. Schoenberg

Abstract. A 2n-periodic continuous real function f is said to be periodically monotone if it has the following property: there exist numbers t 1 < t 2 < t#= tl + 2z~ such that f i s nonincreasing for t 1 < t z and nondecreasing in t 2 < t < t 3. For any 2n-periodic, integrable real function g with S~=[g(t)l dt < oo we define

1 t 2 ~ ( f , g)(x):= 2~ & f(t)o(x - t) dr.

g is said to be periodic monotonicity preserving (g ~ PMP) if f * g is periodically monotone whenever f is periodically monotone. This class of functions was introduced by I. J. Schoenberg in 1959. In the present paper we give an explicit description of the members in PMP. It turns out that an old necessary condition due to Loewner is (essentially) also sufficient. Our result extends to noncontinuous periodically monotone functions, solves Schoenberg's problem about the preservation of convex curves, and even improves on the present knowledge concern- ing properties of cyclic yariation diminishing transforms.

I. Introduction

For any finite sequence {xl . . . . . x,} f rom 1i let v ( x l , . . . , x.) denote the number of sign changes in that sequence (after deleting all zero terms). Fo r a 2~-periodic function f : R --* R we define its cyclic variation by

vc(f) := sup v ( f ( x 0 . . . . , f(xm), f(xt)) ,

where the s u p r e m u m is extended over all finite selections

X 1 < X 2 < "'" < Xra < X 1 -I- 2re, m E N.

We call f periodically monotone ( f ~ P M ) if

vc(f - a) < 2 for all a e R.

Such a function, roughly speaking, increases in an interval (x 1, x2) and decreases in (x2, x~ + 2ir). P M b denotes the subset of bounded functions in P M .

Date received: December 7, 1989. Date revised: October 5, 1990. Communicated by J. Milne Anderson. A M S classification: 42A45, 42A85. Key words and phrases: Periodic monotoxicity, Variation diminishing transforms, Convex curves.

129

130 S. Ruscheweyh and L. C. Salinas

Let L be the class of functions g: R ~ R, 2n-periodic with

fo Z'~] g(x)l < o% dx

where the integral is in the sense of Lebesgue. The following problem has been posed and studied by Schoenberg [9]: characterize the functions g ~ L defined by the property

f * g ~ P M for all f ~ P M b, (1)

where

(2) ( f * g)(x):= f ( y )g ( x - y) dy, x ~ R.

Such a function g is said to be periodic monotonicity preserving (g ~ PMP) . Note that under our general assumptions f �9 g is continuous in R.

The purpose of the present paper is to establish the following complete solution for Schoenberg's problem.

Theorem. Let g ~ L. Then g E P M P if and only if g = 0 a.e., where ~ ~ P M b satisfies the following conditions:

(i) 0 is continuous except for at most two points in a period. I f SUpR 0 = S, infR 0 = t, and 0 not continuous at Xo, then [0(Xo + 0) - O(Xo - 0)1 = s - t.

(ii) 0 is continuously differentiable in each interval in which 0 neither assumes nor approaches s or t. Furthermore, log]0'[ is concave in those intervals.

We mention the following immediate consequences of the theorem.

Corol lary 1. Let g ~ P M P , and 0 be as above. Then the function

~(O'(x + o) + O'(x - 0)), x ~ R,

is well defined, has at most four discontinuities in a period, and is in P M b.

Corollary 2. Let g E P M P , and M, N ~ R, M > N. Then

t M, M < g(x), gM,n(x) := g(x), N <_ g(x) < M, x ~ R,

( N , g(x) < N,

is in P M P .

Corol lary 3. Let g ~ P M ~ C 3. Then g ~ P M P if and only if

g'(x)g"(x) < (g"(x)) z, x ~ R.

An application of Corollary 3 yields easily that, for instance, the Poisson kernel

1 -- r 2

Pr(x) := 1 + r 2 - - 2r cos(x)' x ~ R,

On the Preservation of Periodic Monotonicity 131

is in P M P for 0 < r < x/2 - 1, but for no larger r. This result has been conjectured by Schoenberg [9].

Schoenberg [9] also observed that the P M P problem is identical with the characterization of integral kernels which preserve "convexity" of curves in C. A bounded 2n-periodic function a: R ~ C is said to be a "convex curve" (a ~ CC) if

(3) a R e a + b l m a ~ P M for all a , b ~ R .

If (the restriction to a period of) a is a Jordan curve in C, then (3) obviously holds if and only if its interior domain is a convex set. This observation justifies the name "convex curves" for these functions. It is now clear that g ~ P M P if and only if

a * g .'= [(Re a) * g] + i[(Im ~) �9 g] ~ CC for all a ~ CC.

The geometrical imagination drawn from this concept is a major tool in our proof of the theorem.

The P M P problem is not an isolated one but has a bearing on the theory of cyclic variation diminishing transformations, introduced and studied by Schoen- berg [9], P61ya and Schoenberg [7], and Mairhuber, Schoenberg, and Williamson [5]. This theory, together with his own major contributions, is discussed by Karlin in his book [4]. We briefly review the relevant results. For simplicity, we restrict our attention to functions h in the set A, defined through the following properties:

(i) h is 2n-periodic, piecewise continuous, and of bounded variation over a period;

(ii) 2h(x) = h(x + O) + h(x - 0) holds for all x ~ R.

If, in addition,

(iii) h is nonnegative with S 2~ h(x) dx = 2n;

then we say that h ~ A 1. A function g E A1 is said to be cyclic variation diminishing of order 2r (g E CVD2r),

r ~ N , if

vc(f * g) < vc(f) for all f ~ A, vc(f) <_ 2r.

Note that g ~ CVDo~ diminishes the cyclic variation of any f e A. A function g ~ AI is said to be a cyclic P61ya frequency function of order 2r + 1

(g ~ CPF2 ,+ 1), r E N, if

det [(g(xi - Yj))(Zm+ 1,2,,+ 1)] > 0 for all choices

X 1 < X 2 < " ' " < X 2 m + l < X 1 + 27r) Yl < Y2 < "'" < Y2m+l < Yl -k- 2 ~ J ' y m = 0, 1, . . . , r.

Karlin [4, p. 465] has shown that C V D z r = C P f . z r + l holds for r c N . It is obvious that CVD 2 c P M P n A I (compare Lemma 3 below), and hence CPF 3 c P M P n A 1. However, we do not have equality here, although this has essentially been stated in [4, p. 479]. For instance, a numerical test shows clearly

132 s. Ruscheweyh and L. C. Salinas

that the Poisson kernel Pr is not in C P F 3 for r > 0.16 while, as we have seen above, the corresponding range in P M P is different. In fact, P M P (and P M P c~ At ) is a much richer class than C P F 3. An interesting point in this connection is that the smoothness properties of the functions in P M P as stated in the theorem are apparently considerably stronger than what has been known so far for C P F 3

(compare [-4, p. 492]), and even for the much smaller class CVDoo. Our present work is in continuation of the previous paper [8], where a

characterization of P M P c~ C ~ has been given (the actual purpose of that paper was a quite different one). Corollary 3 has already been established there. We shall make use of the results in [8], see Lemma 5 below.

2. Proof of the Theorem

A basic tool in our deduction are the de la Vall6e-Poussin means

V~(x) := (2nn)-1 (2 [x'~'~ 2" cost )), The following lemma is a celebrated result by P61ya and Schoenberg 171.

Lemma 1. For f E L, n E N, we have

v<(V. * f ) <_ Vc(f).

In particular, V, ~ P M P .

Furthermore, since V~, n ~ ~ , is an even approximate identity (see Butzer and Nessel [1, p. 61]), we can state the following fact.

Lemma 2. Let f ~ L, and assume that for some x o ~ R the limits f ( x o - O) , f (x o + O) exist in R. Then

lim (V,,. f)(Xo) = ~(f(Xo + O) + f (Xo - 0)).

Lemma 3. Let g E L. Then g ~ P M P if and only i f

(4) f . g ~ P M f o r a l l f ~ P m c ~ c 1.

Hence (1) can be replaced by the weaker condition (4). To prove Lemma 3 we first observe that with f ~ P M b also fo ~ PMb, where

fo(x):= ~ f ( x + O) + f ( x - 0)), x ~ R,

and we have f = fo a.e. Furthermore, if {f,} is a sequence from P M which converges pointwise to some function f , then f ~ P M . Thus, if f ~ PMb we have (Lemma 1) f , .'= f * V. ~ P M b ~ C t, n s N, and f , --+ fo by Lemma 2. Now, using (4), we see that f , * g ~ P M , and the result follows using dominated convergence.


Lemma 4. For g, h ~ P M P we have g �9 h ~ P M P . Furthermore, g ~ L is in P M P if and only if V, * g ~ P M P for n ~ N.

Proof. For f ~ P M b we have f , g ~ P M b and hence

f , ( g , h ) = ( f , g ) , h e P M .

In particular, by Lemma 1, V, �9 g e P M P . N o w let f e P M ~ C ~ and F = f , g. Then obviously

F -- lim F , V, -- lim f , ( V , , g ) . n---> oo n--* oo

If V, * g ~ P M P , n e N, then we deduce now that F is the pointwise limit of functions in P M and therefore in P M itself. The result follows from Lemma 3. �9

Lemma 5. Let g E L ~ C 1. Then g 6 P M P if and only generally, g ~ L is in P M P if and only if

(5) ( V , . g ) + i (V ,*g ) ' 6CC, n ~ N .

if g + ig' E CC. More

Proof. The "if" par t is a consequence of [8, Theorem 5] in combinat ion with Lemma 3 (note that (4) was the defining proper ty for P M P in [8]). The "only if" par t is due to Loewner (see [9]). However, we prefer to give a somewhat simpler a rgument for this direction. First we observe that V, + iV', ~ CC. To see this it suffices to prove that aV', + bV[ has at most two zeros of odd multiplicity in a period, a, b ~ R. But

tl a V ' . + b V , ~ - ( 2 n ] (1 + c o s ( x ) ) n - l [ b ( n - 1 ) - a s i n ( x ) - b n c o s ( x ) ] ,

\ n /

and x = n is a zero of even multiplicity while the expression in the brackets has at mos t two zeros of odd (simple) multiplicity. N o w let g ~ P M P c~ C a. Then

( V . + i V ' . ) * g = ( V . * g ) + i ( V ' . . g ) = ( V . . g ) + i ( V . . g ' ) e C C , h e N ,

and these converge pointwise to g + ig' which is therefore also in CC. For general g s L the second statement in Lemma 5 is a consequence of the first one, and of Lemma 4 (note that if f , f �9 g e C 1, g s L, we have f ' �9 g = ( f �9 g)'). �9

Al though the whole structure of g ~ P M P can be extracted from the condit ion (5), such a direct approach seems to be a tedious one since our general assumption about g, namely g ~ L, is a weak one. To simplify the reasoning we take a shortcut using univalent function theory. Fo r 0 < r < 1 we have o', c CC, where

1 + re ix

a~(x) . - 1 -- re ix' x ~ R.

Hence, if g ~ P M P , g ~ const, a.e., we have z , : = a r , g ~ CC, 0 < r < 1. However, it is easily seen that in this part icular case there exists a single analytic function


G in the unit disk D so that

(6) zr(x) = G(rei~), x ~ R, 0 < r < 1.

By construction, G(reiX), r fixed, 0 < x < 2re, is a Jordan curve with a convex interior domain, and this implies that G is a univalent function in D with G(D) a convex domain. About this kind of function the following facts are well known.

Lemma 6 (Suffridge [10]). Let G be univalent in D with G(D) convex. Then

(7) G(eiX):= lim G(re ix) r ~ l - 0

exists in C for every x e R . I f G is extended this way to D, then G is continuous in f ) with at most two exceptions. I f z o e ~D is such an exception, then

lim G(z) = ~ . z-~ zo

zeD

I I G has two discontinuities, then G(D) is a strip domain bounded by two parallel lines.

Lemma 7 (Eenigenburg and Keogh [2]).

A (8) G(z) - + 8 ,

1 - ez

or [GI EL.

Let G, G be as above. Then either

A , B , e e C , lel = 1,

It is easily seen that the function G as defined in (6) cannot be of the form (8). Hence, O(x):= Re G(e ~x) is in L, and using the Schwarz-Poisson formula we get

G(re ix) = ~ ~rr(y)O(x - y) dy, 0 < r < 1, x e R.

Comparing the two representations for G(re~X), we see that we must have g = a.e., because all of their moments are equal. Properties of 9, however, can now be read from Lemma 6.

Lemma 8. Let g e P M P . Then g = ~ a.e., where in every period either:

(i) ~ is continuous (with possible values +_ ~ in at most two points); or (ii) ~ is continuous with exactly one exception (mod 2n) which corresponds to a

jump from SUpR ~ to infR ~ or vice versa; or (iii) ~ assumes only two (finite) values and has exactly two jumps between these

values.

Furthermore, there is no interval in which ~ is constant, except possibly if that

constant equals supl~ ~ or infR ~.

Proof. Case (i) is clear. It contains the case that G(D) is a doubly infinite strip not parallel to the imaginary axis.


Case (ii) deals with the situation that G is not bounded, but is not a strip, and that ~ has a discontinuity: The conclusion about the size of the jump in the discontinuity comes from the geometry of the curve G(e ix) which has an infinite imaginary part at the jump and bounds a convex set.

Case (iii) corresponds to the case that G(D) is a doubly infinite strip parallel to the imaginary axis.

The last statement comes again from the geometry of G(e ~) and the well-known properties of analytic functions on the boundary of their domains. �9

From now on, talking about 9 ~ P M P , we always (including in the statements of subsequent lemmas) assume that 9 has already been replaced by a corresponding function ~. We don't care about the values of 9 in the possible discontinuities, except from the general assumption that

(9) min{o(x - 0), 9(x + 0)}< g(x) <<_ max{9(x - 0), g(x + 0)}.

Lemma 9. Let g e P M P . Then g e P M and we can f ind points x o < x t < x o + 21r so that g is convex in (x o, xa) and concave in (xl, Xo + 2~z). One o f these intervals can be empty. Furthermore, Xo, x I are the only points where g may have a discontinuity and/or assume infinite values.

Proofi We have V, �9 g e PM, n e N, and the continuity properties stated in Lemma 8 show that we have pointwise convergence to g (except, possibly, in the discontinuities). However, (9) ensures that g e P M . From (5) we deduce that (V, �9 0)' e PM, and therefore we can find intervals (X{o "), x~")), (x~ "), X(o ") + 2n) in which (V,, O)' is increasing and decreasing, respectively. Hence V,, g is convex (respectively, concave) in those intervals. We choose a subsequence so that Xto"k~ ~ xo, x~ "k) --* xi. Clearly, x o < x 1 < Xo + 2re. Assume x o < x 1 and choose YI, Y2 e (xo, xl) so that neither of the points ya, Yz, (Yl + y2)/2 corresponds to a discontinuity of 9 or to a point where g assumes an infinite value. We then have (Lemma 2)

lim (V. * 9)(x) = g(x), x e {Yl, Yz, (Yl + y2)/2}. n--~ oo

Since, by assumption, for large n

we conclude that

g ( ~ Z - - ~ ) < {(g(Yt) + g(Yz)).

It is clear that this condition will be violated for some suitably chosen Yl, Y2 if we assume that one or more of the mentioned exceptions (discontinuities and/or infinite values of g) occur in (Xo, xO. Hence g is convex and finite in the whole of (Xo, xl) (and concave and finite in (x~, x o + 2r0).


Lemma 10. Let g ~ P M P and let x 0, x 1 be as in L e m m a 9. Then for x ~ Xo, xa (rood 2n) the symmetric derivative

g(x + 8) - g (x - 8) O(x) := lim

~-~o 28

exists and is finite.

This is a general property of convex/concave functions (see Hewitt and Stromberg [3, p. 271]).

Lemma 11 (Schoenberg [9]). For 0 < 8 < n let

~n/8, --8 < x < 8,

k~(x) := (0, e < x < 2n - 8.

Then k~ ~ P M P .

In fact, Lemma 11 we note that

1 ~x+~ (g * k~)(x) = ~ ~ x - , g(y) dy ~ P M P

In view of Lemma 10 we see that

lim (g �9 k~)(x) = g(x), ~-+ 0

lira (g �9 k f f (x) = O(x), e---~ 0

holds for all x ~ Xo, Xl (mod 2n). We also shall need the following two simple technical lemmas.

Schoenberg proved (4) for g = k~. As an important consequence of

for all g ~ P M P .

Lemma 12. Le t h ~ Cl(a, b), h' > O, satisfy the equation h'(x) = H(h(x)), x ~ (a, b). Then H is concave in (h(a), h(b)) i f and only i f log(h') is concave in (a, b).

Proof. It is readily checked that

lim log h'(x + e) - log h'(x) = lira H ( y + e) -- H(y) r=h(x) ~ - ~ 0 + 0 8 e ~ 0 + 0 8

holds for every single x ~ (a, b) if at least one of the two limits exists, and a corresponding observation is true for the left-hand derivatives. If, for instance, H is concave it has all left-hand and right-hand derivatives, and they are decreasing with x. The same is then true for log h'(x) and proves its concavity (note that, for instance, log h' is the indefinite integral of its left-hand derivatives). �9

Lemma 13. Le t h: (a, b) ~ R be such that h exists in (a, b) and is continuous. Then h ~ Cl(a, b), and h' = h in (a, b).


Proof. Let

r(x) .'= h(x) - J l x h(t) dt,

so that ~ - 0 in (a, b). Set s(x):= ~ r(t) dt. Then (see N a t a n s o n [6, p. 337])

s(x + ~) + s(x -- ~) -- 2s(x) r(x + ~) -- r(x -- ~) lim = lim = P(x) = 0 ~ o ~2 ~-~o 2~

in (a, b). By a theorem of H. A. Schwarz (see Na t anson [6, p. 337]) we see that s is a linear function and therefore r = const. The conclusion follows immediately.

Lemma 14. Let 0 be as in Lemma 8(iii). Then g ~ P M P .

Proof. It is readily seen that 0 can be writ ten as

O(x) = AkE(x + t) + B, A, B, t ~ R,

which by L e m m a 11 (and obvious invariance propert ies of P M P ) is in P M P . �9

Note that L e m m a s 8 and 14 establish our theorem as far as functions 0 with two (or more) discontinuities in a per iod are concerned. F r o m now on we shall restrict our a t tent ion to the cases (i), (ii) of L e m m a 8.

To simplify further discussion we introduce the following notat ion. Let g ~ P M be cont inuous except for at mos t one discontinuity in a period, where it j umps f rom its s u p r e m u m s down to its inf imum t (or vice versa). An interval (a, b) is said to be maximal for g if

{s, t} (~ g((a, b)) = f2J, {s, t} c g((a, b)).

If g has a discontinuity then it has exactly one maximal interval (mod 2n), otherwise it has exactly two or is constant.

Proof of the Theorem ("/f"-part) . Condi t ion (ii) of the theorem says that log lg'[ is concave in each maximal interval, hence g' ~ 0 there. I t also implies that

~g '(x) x in maximal interval,

g*(x) := (0, elsewhere,

is a bounded 2n-periodic function (note also that g' = g* a.e.). We m a y also assume that g is, say, cont inuous f rom the left everywhere.

Assume that g' > 0 in a maximal interval 11. Then, f rom L e m m a 12, we deduce that g'(g-1) is concave in 9(11) (clearly we assume here g - l : g(i1) ~ I1). This shows that

al (x) := O(x) + ig*(x), x ~-I 1,

is a Jo rdan arc in the upper half-plane so that

U 1 := {u + iv: 3x ~ 11 with u = g(x), v <_ g*(x)}


i#"

~ ~ 2 j

9 -5

Case 1. g continuous. Case 2.9 has one discontinuity (mod 27@

Fig. 1

is a convex set. If there is a second maximal interval (mod 270, 12, then we must have g' < 0 there; Gz(X):.=g(x ) q-ig*(x), XEI2, is a Jo rdan arc in the lower half-plane with a similar cor responding convex set U2. See Fig. 1 for typical curves g-(x):= g ( x ) + ig*(x), 0 _< x < 2r~. It is now obvious that a ~ CC, and therefore

~ r , V . = ( V ~ , g ) + i ( V , , g * ) = ( V , , g ) + i ( V , , g ) ' ~ C C , n o N .

L e m m a 5 then gives the desired conclusion: g c P M P . �9

For the p roof of the other direction of the theorem we shall need some more lemmas.

L e m m a 15. Let g ~ P M P be a nonconstant trigonometric polynomial. Then log lg'I is concave in each maximal interval o f g. In particular, g' has precisely two zeros in a period, and the interval between any two consecutive zeros is maximal.

Proof. Let I = (a, b) be maximal for g. Then g is strictly m o n o t o n e (increasing, say) in 1. Hence, g - 1: g(i) ___, I is strictly increasing, and from (3) and L e m m a 5 we deduce that for A, B e R the equat ion

g,(g- l(y)) = Ay + B, y E g(I),

has at mos t two solutions. This clearly implies that g'(g-1(y)) is either convex or concave in the whole of g(I). But this function is nonnegat ive in g(I) with g'(a) = g'(b) = 0. Hence g'(g-1) must be concave in g(I). In part icular, g' > 0 in I, since otherwise we would have g' - 0, which is absurd. The result now follows f rom L e m m a 12 (if g is decreasing in I we replace g by - g ) . �9

L e m m a 16. Let g E P M P be strictly monotone and convex (or concave) in-some open interval I (I is not assumed to be maximal). Then g is continuously differentiable in I with log]g'] concave in I.


Proof. Let [Yl, Y2] c I, and choose e > 0 so small that [Yl - ~, Y2 + e] c I. We define

h~.~ := V . * k ~ * g , n ~ N .

Then h.,~ ~ P M P and is a t r igonometr ic polynomial . Assume there are sequences n k and x.~ ~ [y~, Yz] with x,, k -~ x o so that h'.~.~(x.~) = 0. By a uniform convergence a rgument we conclude tha t

g(xo + e)-g(Xo ~) lim h'.~, ~(Xo) =- (k~ * g)'(Xo) = = O,

k ~ o o 2e

a contradic t ion to the assumpt ion that g is strictly m o n o t o n e in I. In view of L e m m a 15 we now see that (Yl, Y2) is contained in a max imal interval of h,,~ for all n large enough. Thus log I ~ , ( k s �9 0)'t is concave in (Yl , Y2), a p roper ty which is inherited by its cont inuous limit

log[ g(xo + e) -- g(xo -- e) logl(k~ g)'(x)] , x ~ (YD Yg- I

For e ~ 0 we find, using L e m m a 10, that log[O[ is concave in that same interval, a p roper ty which obviously extends to the whole of I. This implies, in part icular, that 0 is cont inuous in I, and the assert ion follows f rom L e m m a 13. �9

L e m m a 17. Let g ~ P M P . Then g is bounded.

Proof. If not, we can assume to find x a with

(10) lim g ( x ) = ~ . x ~ X ! - -0

Then there mus t be an interval (x*, xl) in which g is convex and strictly increasing (compare L e m m a 9). T o that interval, however, we can apply L e m m a 16 and conclude that loglg 'I is concave, which clearly rules out the possibility that g' approaches ~ , a necessity if g satisfies (10). �9

Proof of the Theorem ( " o n l y / f " - p a r t ) . It will be sufficient to show that g satisfies the conclusions in condi t ion (ii) of the theorem for any maximal interval I = (a, bj. g is strictly m o n o t o n e in I (compare the last s ta tement in L e m m a 8). Hence, if g is convex (or concave) in the whole of I, then our assertion follows immediate ly f rom L e m m a 16.

Next we note that g switches f rom convexity to concavi ty (or vice versa) in at mos t one point x o ~ I. This is obvious if g has a discontinuity (Lemma 9). If g is cont inuous in R and [x o, x , ] c I, where Xo, x l are as in L e m m a 9, then it must be convex (or concave) in the whole of [x 1, x o + 2hi which is impossible by our assumpt ions abou t I.

Assume now that such an Xo exists, and, without- loss of generality, that g increases in I. Then, f rom L e m m a 17 and the shape of the approx imat ing functions V. �9 g in I it is clear that g must be convex in (a, Xo) and concave in (xo, b). F r o m L e m m a 16 we find that loglg'[ is concave in bo th intervals, which shows, in


particular, that g' cannot approach infinite values for x ~ x o +_ 0. This implies that 0 exists and is finite in x o, moreover

~(Xo) = ~(O(Xo - 0) + 0(Xo + 0)) ~ 0.

Hence (V. *g)'(Xo) ~ 0 for n large, and as in the proof of Lemma 15, we see that loglg'l is concave in some neighborhood of x o. This, together with the earlier conclusions about the intervals (a, Xo) and (Xo, b), completes the proof of the theorem. �9

Acknowledgments. Research supported by the Fondo Nacional de Desarrollo Cientifico y Tecnol6gico (FONDECYT, Grant 263/89), by the Universidad F. Santa Maria (Grant 89.12.06), and by the German Academic Exchange Service (DAAD).

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S. Ruscheweyh Mathematisches Institut Universit~it Wtirzburg W-8700 Wiirzburg F.R.G.

L. C. Salinas Departamento de MatemMica Universidad T+cnica F. Santa Maria Valparaiso Chile

On the preservation of periodic monotonicity

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