A SUBSET OF METRIC PRESERVING FUNCTIONS

Internat. J. Math. & Math. Si.

VOL. 21 NO. 2 (1998) 409-410

409

RESEARCH NOTES

A SUBSET OF METRIC PRESERVING FUNCTIONS

ROBERTW.VALLIN Departmentof Mathematics Slippery Rock University of Pennsylvania Slippery Rock, Pennsylvania 16057,U.S.A.

(ReceivedApril 15, 1996andin revisedformJune17,

1996)

ABSTRACT. Inthispaperwe definea subset ofmetricpreserving functions and give someexamples and acharacterizationofthissubset.

KEY WORDS AND PHRASES: Metrics,metricpreservingfunctions, derivatives.

1991AMSSUBJECT CLASSIFICATIONCODE: 54E35.

1. INTRODUCTION

Wecallafunction

f ^R

^{+ -,}

^R

^{+ ametricpreserving}

function

^ifandonlyif

f

^(p)

^{M M}

^--,

^R

^{+ is a}

metricfor everymetricp

M M -- ^{R +,}

^where

^(M,

^p)isan arbitrarymetricspaceand

R

⁺denotes the nonnegative reals. Wewilldenote the collection ofmetricpreservingfunctionsbyAd. There aremany papers out there which deal withthese functions (see the

references).

Of particular interest is the derivative ofmetric preserving functions. In

[1]

J. Borik and J. Dobo show that if

f

^{6 A,[ is}

differentiablethen

If’ ^(x)l -< ^{f’ (0).}

^{J.Dobo[andZ.Piotrowskiin}

^[2]

^constructtwoexamples concerning differentiation and metric preserving functions. The first

f ^.M

is continuous and nowhere differentiable. Theother is metric preserving, differentiable and the derivative is infinite exactly on

{0}

^U2

-n,

n 1,2,3,.... In

[9]

^thisauthor answersaquestionof Dobo andPiotrowskibyshowing how for any measure zero,

6

^{set in}

[0, oo)

^thereis a continuous metricpreserving function whose derivativeis infiniteonthatsetunionzero.

Thesubsetofmetricpreserving functionswewishtoconsider is definedbelow.

DEFINITION. Let

f

E

.M

bedifferentiable on

(0, oo).

Defineg(x)as

f’(x)

z

e (0, oo)

(1.1)

0 x=0

Wesay

f

^{2)ifandonlyiff,g j.}

Thepurposeofthispaperistogiveexamples ofthese types of functions andtocharacterizethetype of

f

^{which canbein29.}

2.MAIN RESULTS

Wenotehere that theset29 is nonempty. It iseasy^toseethat 29 contains all functions of the form

f(x)

^=/cx,/c

>

^{0. A}natural questiontothen askis if it ispossiblethat there are functions

f

^{suchthat g}

defined aboveiscontinuousatthe origin(whichis notthatcase for

f(x)

lcx). ^{The answer}isnoandis givenin thefollowingtheorem.

THEOREM1. If

f

^isdifferentiable on

[0, oo)

and metric preserving

f’(x)

is not a metricpreserving function.

PROOF. If

f’ ^M

^then

^f’(0)

^{would havetobezeroand}

f’ ^>

^{0 on}

^{(0, oo)}

^{impliesthere mustbe}

some

[0, )

^where

f

^mustbe strictly convex. Then

f

^{.A4 from}

^Prop.

^10in[1].

Norcanwegoin theoppositedirectionand assumethat if g is metricpreservingitsintegralwillalso be metricpreserving.

EXAMPLE.

There exists ametric preservingfunction g whose integral,

f0 ^g(),

^{dr, is not also}

metric preserving.

PROOF. Let

g(x)

1 e

-x.

Then

f0

¹

^e-tdt

is strictly convex^{in a}neighborhood of the origin.

410 R.W.VALLI

Note

that

g(x)

2xwould also serve in theexampleabove. Whilebotharecontinuous,1 e has theadded strength ofbeingbounded. Wenowcanlookatsome properties of these functionsin23.

THEOREM2. If

f

^E23,

f

is nondecreasing.

PROOF. This is aconsequenceof the fact that the functiong(x)mustbegreaterthanzerosince g is metdcpreserving.

LEMMA. Let

f

^{E ,Mandlimsup_,0/}

^f(x)

^{a. Thenfor allx E}

[0, co), f(z) >_ a/2.

PROOF. This is aproperty of

f

^beingmetricpreserving. SeeCorollary in ].

THEOREM3. Let

f (x)

^x

.

^Only

^f

^E

^D

^{if andonlyifk 1.}

PROOF.

If k

>

1then

f

^,Adsince

f

would be strictlyconvexaround the origin.

If kE

(0, 1)

then gviolatesthe lemma above.

If k 0then g ^J/[since gwouldbe identicallyzero.

If k

<

0then

f

^violatesthelemma above.

Inordertocharacterizefunctionsinthe set 23we needthenotionofatriangle triplet. The3-tuple

(a,

b,

c)

E

(R +)3

^iscalledatriangle triplet ifa

_<

b

+

^{c, b}

^_<

^a

+

^{c, andc}

^_<

^a

+

^{b. Thisis}another way todetermine ifa function ismetricpreserving

(see

F.

Terpe [8]).

Afunction

f

^{isa rnetficpreserving}

functionifand onlyif

f(0)

0and

(f(a), f(b), f(c))

^{is a}triangle triplet whenever

(a,

b,

c)

is one. This givesus away^todescribe these functions in 23.

,THEOREM

^{4. Let}

^{g(x) R}

^+-,

^R

^{+beafunctionsatisfying}

Va

>

⁰

g(x)dx >_

g(x)dx where c b a. (2.1) If thereexists an

A >

0such that

A <_ N + Mg(z) <_

2A,

(2.2)

N+Mg(x)

x>0

then both

G(x)

_{0 x 0} ^and

F(x) G(t)dt

are in3/[.

PROOF. Thecondition

(2.2)

^{gives us}

G(x)

is metricpreserving (Proposition3 in

[1]).

Condition (2.1) assures that

F(x)

will satisfy the triangle triplet condition. Assume a

<

^b

<

c. Then

F(a) <_ F(b)

^4-

F(c)

^and

F(b) <_ F(a)

4-

F(c)

are automatic. Lastly,

F(c) F(b) + G(t)dt < F(b) + G(t)dt F(a) + F(b). (2.3)

Thisdescribessuchexamplesin23 using l+e

-x,

3+

cos(l/x),

and 3

+

^{e cosxforN+Mg(x).}

To

closewe notethatthisgives anotherwaytocreatemetricpreserving

functio,

^ns.

COROLLARY. If

g(x)

^meetscondition

(2.1)

^and0

< g(x)

almosteverywhere theng(x)^neednot beinA4,but

f0 ^g(t)dA

^{is inA4where}

,

denotesLebesguemeasure.

REFERENCES

[1]

BORIK,

J. and

DOBOg,

_J.,Onmetricpreservingfunctions, Real AnalysisExchange,13(1987-88), 285-293.

[2]

IX)BOg,

J. and PIOTROWSKI, Z., Some remarks on metric preserving functions, Real Analysis Exchange,19

(1993-94),

^317-320.

[3]

DOBOg,

J. andPIOTROWSKI, J., Anoteonmetricpreserving functions,

Internat.

J.Math.andMath.

Sci.,19

(1996),

^199-200.

[4]JUZA, M., Anoteoncompletemetricspaces,Matematicko-fyzikdlny

sopis

Save,6(1956), No.3, 143-148.

[5]

POKORNS r,

I., Some remarks on metric preserving functions, Tatra^{MountainsMath.}

Publ.,

²

(1993),

^66-68.

[6]SHIRAI,T., Onthe relations between thesetand its distances,Mere. Coll. Sci.

Kyoto Imp. Umv.

Serv.,

22

(1939),

^369-275.

[7] SREENISAVA,T.K., Someproperties ofdistancefunctions,J.IndianMath.

Soc.,

11(1947),^38-43.

[8] TERPE, F.,Metricpreserving functions,Proc.

Conf.

^{TopologyandMeasure}

^IV,

G-reifswald(1984), 189-197.

[9] VALLIN,R.W., Onmetricpreserving functions and infinitederivatives(submitted).

[10] WILSON, W.A., Oncertain typesofcontinuous transformations of metric spaces,Amer.

J.

Math,57

(1935),

^62-68.