TRANSFORMATIONS WHICH PRESERVE CONVEXITY

Let C be the class of convex nondecreasing functions f: (0,) (0,) which satisfy f(0) O. Marshall and Proschan (I) determine the one-to-one and onto functions 4: (0,) (0,=) such that g f 4 -1 belongs to C whenever f belongs to C. We study several natural models for multivariate extension of the Marshall-Proschan result. We show that these result in essentially a restatement of the original Marshall-Proschan characterization.

Let C denote the class of convex, nondecreasing functions f: [0,=) [0, ) which satisfy f(O) O.A known and useful result is that if p i, then 1 g(x) fP (xp) belongs to C whenever f belongs to C. There is an interesting geometrical inter- pretation of the relationship between f and g.Beginning with the equation of the graph of g, 1 y fP(xP), one obtains that 1 1 yP f (x p) and, thus, that the graph of f is obtained from the graph of g (and vice versa) by applying the same transformation to both of the coordinate axes.
In [I], A. W. Marshall and  : [0, ) [0, ) such that g V belongs to C whenever f belongs to 6.The answer to this question is of interest in several applications.Karlin ([2], pp. 368, 369) uses the answer to the question in obtaining bounds on the survival function (x) in terms of an exponential survival function.Barlow and Proschan   (F3] pp.Ii0, III) also use the result in obtaining bounds on the survival function in terms of , where C IF is convex.Marshall and Proschan show that if is continuous at some point, then g belongs to C whenever f belongs to C if and only if (x) cx p for some c 0 and p i.
In this note we study several natural models for multivariate extension of the Marshall-Proschan result.We show that these result in essentially a restatement of the original Marshall-Proschan characterization.

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We begin by establishing some notation.Let R denote the nonnegative orthant n o n-dimensional Euclidean space equipped with coordinatewise ordering, multiplication, and addition.If _x and y R, let _x + y and xy denote the sum and the product, computed coordinatewise, of _x and _y.Let _0 and _e denote the vectors all of -i whose entries are 0 and I, respectively.If x O, let x denote the multi- n plicative inverse of _x.Finally, let _x denote the nth power of _x.
We now examine several possible extensions of the Marshall-Proschan result.Let (Xl n for _x (Xl,...,Xn) in P -, + , n does g belong to C? The following example shows that the answer is "No".
EXAMPLE i.Let n -> 2 and p i.For x_ in , let f(x_) x I + x 2. Then is not convex since the partial derivative of g with respect to x 2 is decreasing as a function of x2, for fixed x I. Thus, for fixed Xl, g is not a convex function of x2([],Thm.12B ).Hence g does not belong to C.
Example i, the theorem of Marshall and Proschan, and the fact that the present class C contains a copy of the class C of interest to Marshall and Proschan show   that, given the present choice of C, there is no point in considering real-valued functions (x) which operate on each coordinate separately.If we wish to look at such functions (x), we must choose C differently.For example, we might let C R + be the class of functions f: [0, ) which are convex and nondecreasing in each n variable separately and which satisfy f(0) O; however, Example 1 shows that this choice is not suitable.Alternatively, we might let C consist of those functions f: R +-[0, ) which satisfy f(0) 0, are nondecreasing in each argument, and whose n restrictions to rays through the origin are convex functions of one variable.In this TRANSFORMATIONS WHICH PRESERVE CONVEXITY In one way, the result of Example i is not surprising: there is an obvious difference between the dimension of the transforming function x p and the argument x of the functions to be transformed.Thus we now consider the case where both the convex functions and the transfor,,,ation function map R + into It 4.
We say that n n (or bounded in a neighborhood of some point; the answer is the same) and have the property that -i f C implies that @ f .C. (2.1) In Example 2 (below) we shall find a necessary condition that (2.1) is satisfied by a function of a certain type.To aid us, we will use the following remark: REMARK I. Let al,a2,...,an be real numbers.Let n aj h(x) x.
x 0. j=l Suppose that a.
O and a k O for some integers and k such that # k.We claim that h is no__t convex.To see this, suppose, without loss of generality, that i, k 2. The first two principal minors of the Hessian matrix, the matrix of second order partial derivatives of h, must be nonnegative if h is convex ([],Thm. 42F).Thus the inequalities al(l a I) <_ 0 and ala2(l a I a 2) -> 0 must simultaneously hold if h is convex.These inequalities cannot both hold" thus h is not convex.
EXAMPLE 2. Let A be a real n n matrix.Let f be the R+-valued function with n domain x_ 0, each of whose component functions is of the type given in Remark such that the exponents in the kth component function are, in order, the elements of the kth row of A, k l,...,n.Represent f as follows: f(x) x A for x O.
It is easy and interesting to see that if _x where BA is the usual matrix product of B and A. Also f is invertible if and only if A is invertible and, in this case, -i

A -I f (x)
x Let us call a matrix simple if it is invertible and each row contains exactly one nonzero entry.A permutation matrix is a simple matrix such that the nonzero entry in each row is I.
We will now present a result about non-preservation of convexity.Let A be an n n non-simple invertible matrix and let : R + R + be one-to-one and onto and n n also satisfy O. (2.2) An easy argument, which we omit, shows that there exists an n n diagonal matrix P, all of whose diagonal entries are greater than or equal to one, such that the matrix -i Q APA has a row with two (strictly) positive entries.Choose such a P and let It is clear that f belongs to C. On the other hand, by Remark i, the does not belong to C. Thus _ does not satisfy (2.1).
Suppose that we now consider an arbitrary simple matrix A. Using "test" functions in C of the form f(x)--(g(x I) g(x 2) g(x )) n where g: [0, ] [0, =] is convex, nondecreasing, and satisfies g(O) 0 and using the Marshall-Proschan result, it is easy to see that if satisfies (2.1) and (2.2), then A must be a permutation matrix.
THEOREM.Suppose that n e 2 and : R + R + is one-to-one, onto, and continuous at n n some point. ]'hen satisfies (2.1) ,if and only if for x in R + some vector c 0, and some permutation matrix B n' PROOF.If _ satisfles(3.1)forsome vector c__ O and some permutation matrix A, then 4' clearly satisfies (2.1).
Suppose that _ satisfies (2.1).We shall derive a functional equation which is satisfied by _.Motivated by the proof in [i] and consideration of invertible linear functions in C, we ask the following question: If g is one-to-one, onto, and g and -i both belong to C, what must be true of g?It is easy to see that the equations g(x + y) g(x) + K(Y) for some nonnegative simple matrix A. -i For any a_ O, let f(x_) a__x and let g f__ Since satisfies -i -i (2.1) and both f and f belong to C, both g and g also belong to C. Th.
there is a nonnegative simple matrix A such that f -l(z zA, z R+. Substituting _z (x) and f(x) a__x, we obtain R + (ax) (x)A, x n' where A depends on a.
We will now show that A is a diagonal matrix.First, note that since A is simple, there is some vector b 0 and a linear transformation , depending on a, which permeates coordinates, such that R + for all a_ 0 and some c > 0 which depends on a.
Our next step is to express c in (3.4) in terms of .We claim that if _z + O, then _(z_) O. Suppose that z i 0 for some i, i -< i _< n.Let a_ O and b 0 be such that a i bi and aj bj, j i.By (3.4)there exist _c _0 for all _x in R + .n Suppose that _@(z)_ _0-Since __az __bz and _(_z) has a multiplicative inverse, it follows from (3.5) that c_ d.Using (3.5) again with _x e__, we obtain (a__) _(b), which contradicts the fact that is one-to-one.
Thus our claim is established Furthermore, since ,,.-IX also satisfies (3.4) with F. Proschan, motivated by the special case (x) xp, pose and solve the following problem: Determine those one-to-one and onto functions R. A. FONTENOT AND F. PROSCHAN -i hold for all I 0 and all x_,x_ It then follows that g(x) xA, x in n n'