A SUBSET OF METRIC PRESERVING FUNCTIONS

In this paper we define a subset of metric preserving functions and give some examples and a characterization ofthis subset.


INTRODUCTION
We call a function f R + -, R + a metric preserving function if and only if f (p) M M --, R + is a metric for every metric p M M --R+, where (M, p) is an arbitrary metric space and R + denotes the nonnegative reals.We will denote the collection of metric preserving functions by Ad.There are many papers out there which deal with these functions (see the references).Of particular interest is the derivative of metric preserving functions.In [1] J. Borik and J. Dobo show that if f 6 A,[ is differentiable then If' (x)l -< f' (0).J. Dobo[ and Z. Piotrowski in [2] construct two examples concerning differentiation and metric preserving functions.The first f .M is continuous and nowhere differentiable.The other is metric preserving, differentiable and the derivative is infinite exactly on {0} U2-n, n 1,2,3,....In [9] this author answers a question of Dobo and Piotrowski by showing how for any measure zero, 6 set in [0, oo) there is a continuous metric preserving function whose derivative is infinite on that set union zero.
The subset of metric preserving functions we wish to consider is defined below.
DEFINITION.Let f E .M be differentiable on (0, oo).Define g(x) as f'(x) z e (0, oo) We say f 2) if and only if f, g j.
The purpose of this paper is to give examples of these types of functions and to characterize the type of f which can be in 29.

MAIN RESULTS
We note here that the set 29 is nonempty.It is easy to see that 29 contains all functions of the form f(x) =/cx,/c > 0. A natural question to then ask is if it is possible that there are functions f such that g defined above is continuous at the origin (which is not that case for f(x) lcx).The answer is no and is given in the following theorem.
THEOREM 1.If f is differentiable on [0, oo) and metric preserving f'(x) is not a metric preserving function.
PROOF.If f' M then f'(0) would have to be zero and f' > 0 on (0, oo) implies there must be some [0, ) where f must be strictly convex.Then f .A4 from Prop. 10 in [1].
Nor can we go in the opposite direction and assume that if g is metric preserving its integral will also be metric preserving.
EXAMPLE.There exists a metric preserving function 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 ofthe origin.
Note that g(x) 2x would also serve in the example above.While both are continuous, 1 e has the added strength ofbeing bounded.We now can look at some properties ofthese functions in 23.PROOF.This is a consequence of the fact that the function g(x) must be greater than zero since g is metdc preserving.
LEMMA.Let f E ,M and limsup_,0/ f(x) a. Then for all x E [0, co), f(z) >_ a/2.PROOF.This is a property of f being metric preserving.See Corollary in ].THEOREM 3. Let f (x) x.Only f E D if and only if k 1.

PROOF.
If k > 1 then f ,Ad since f would be strictly convex around the origin.
If k E (0, 1) then g violates the lemma above.
If k 0 then g J/[ since g would be identically zero.
If k < 0 then f violates the lemma above.
In order to characterize functions in the set 23 we need the notion of a triangle triplet.The 3-tuple (a, b, c) E (R+)3 is called a triangle triplet if a _< b + c, b _< a + c, and c _< a + b.This is another way to determine if a function is metric preserving (see F. Terpe [8]).A function f is a rnetfic preserving function if and only if f(0) 0 and (f(a), f(b), f(c)) is a triangle triplet whenever (a, b, c) is one.This gives us a way to describe these functions in 23.
,THEOREM 4. Let g(x) R + -, R + be a function satisfying Va > 0 g(x)dx >_ g(x)dx where c b a.This describes such examples in 23 using l+e-x, 3+ cos(l/x), and 3 + e cosx for N+Mg(x).
To close we note that this gives another way to create metric preserving functio, ns.COROLLARY.If g(x) meets condition (2.1) and 0 < g(x) almost everywhere then g(x) need not be in A4, but f0 g(t)dA is in A4 where , denotes Lebesgue measure.

THEOREM 2 .
If f E 23, f is nondecreasing.