Internat. J. Math. & Math. Sci.
(1985) 49-55
49
Vol. 8 No.
TRANSFORMATIONS WHICH PRESERVE CONVEXITY
ROBERT A. FONTENOT
Department of Mathematics
Whitman College
Walla Walla, Washington
99362
FRANK PROSCHAN
Department of Statistics
Florida State University
Tallahassee, Florida
32306
(Received January 7, 1983)
ABSTRACT.
Let
which satisfy
C
f(0)
4: [0,)
functions
belongs to
C.
be the class of convex nondecreasing functions
[0,)
f:
[0,)
Marshall and Proschan [I] determine the one-to-one and onto
-1
f
[0,=) such that g
belongs to C whenever f
4
O.
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.
KEY WORDS AND PHRASES. Convexity, preservation of convexity, tnansformations
multivariate, permutation matrices.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. AS Subject classification: 26B25
i.
INTRODUCTION.
Let
C
denote the class of convex, nondecreasing functions
which satisfy
f(O)
A known and useful result is that if
O.
f:
p
[0,=)
i,
[0, )
then
1
fP(x p)
g(x)
belongs to
C
whenever
f
C.
belongs to
pretation of the relationship between
f
There is an interesting geometrical interand
g.
Beginning with the equation of the
graph of g,
1
y
fP(xP),
one obtains that
1
yP
1
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 F. Proschan, motivated by the special case (x)
x
p,
pose and solve the following problem:
Determine those one-to-one and onto functions
R. A. FONTENOT AND F. PROSCHAN
50
:
[0, )
[0, )
such that
g
-i
V
belongs to
C
([2],
Karlin
The answer to this question is of interest in several applications.
6.
belongs to
f
whenever
pp. 368, 369) uses the answer to the question in obtaining bounds on the survival
Barlow and Proschan
in terms of an exponential survival function.
function (x)
,
(F3] pp. Ii0, III) also use the result in obtaining bounds on the survival function
in terms of
(x)
only if
cx
p
IF
C
where
Marshall and Proschan show that if
is convex.
then
continuous at some point,
g
for some
c
belongs to
C
0
i.
and
p
whenever
belongs to
f
C
is
if and
In this note we study several natural models for multivariate extension of the
We show that these result in essentially a restatement of
Marshall-Proschan result.
the original Marshall-Proschan characterization.
EXTENSIONS OF IARSHALL-PROSCHN RESULT
2.
+
Let
We begin by establishing some notation.
denote the nonnegative orthant
R
n
n-dimensional Euclidean space equipped with coordinatewise ordering, multiplication,
o
_x
If
and addition.
and
R,
y
_x
computed coordinatewise, of
0
whose entries are
plicative inverse of
_x + y
let
_y.
and
denote the sum and the product,
and xy
Let
and
_0
I, respectively.
n
denote the
let _x
If
and
x
_e denote the vectors all of
-i
denote the multiO, let x
nth power of
_x. Finally,
_x.
Let
We now examine several possible extensions of the Marshall-Proschan result.
C
denote the set of those convex functions
f
belongs to
C, and
1
(Xl,...,Xn)
_x
does
n
-> 2
and
x_
For
i.
p
is not convex since the partial derivative of
as a function of
for fixed
x2,
x2([],Thm. 12B ).
x
let
f(x_)
g
x
with respect to
Xl,
does not belong to
g
and
I,
p
I
+
x
2.
Then
x )p
Thus, for fixed
I.
Hence
If
0
1
+
function of
,
in
1
f(O)
The following example shows that
belong to C?
g
"No".
Let
EXAMPLE i.
P-,+,n
in
such that
1
x p)
n
fP (XlP
g(x)
the answer is
[0, )
n
The following question is natural:
is nondecreasing in each argument.
for
R+
f:
g
x
2
is decreasing
is not a convex
C.
Example i, the theorem of Marshall and Proschan, and the fact that the present
class
C
functions
(x)
such functions
C, there
(x),
C
we must choose
f:
R+
n
choice is not suitable.
[0, )
f(0)
f(0)
O;
however, Example 1 shows that this
C
consist of those functions
0, are nondecreasing in each argument, and whose
restrictions to rays through the origin are convex functions of one variable.
case, the C-preserving functions are the
In this
same as those in the one-dimensional case;
the proof is a trivial and, hence, uninteresting application of the theorem of
Marshall and Proschan.
C
which are convex and nondecreasing in each
Alternatively, we might let
which satisfy
If we wish to look at
For example, we might let
differently.
[0, )
variable separately and which satisfy
+n
is no point in considering real-valued
which operate on each coordinate separately.
be the class of functions
f: R
of interest to Marshall and Proschan show
C
contains a copy of the class
that, given the present choice of
TRANSFORMATIONS WHICH PRESERVE CONVEXITY
In one way, the result of Example i is not surprising:
difference between the dimension of the transforming function
x
Thus we now consider the case where both the
of the functions to be transformed.
convex functions and the transfor,,,ation function
f:
R+
+
R
R
map
+
into
n
4.
It
n
We say that
is convex if the inequality
_f(Ix +
holds for all
R+
x, y
n
if and only if each of the
Let
sense.
there is an obvious
p
x
and the argument
C
l_f(_x) +
>)2)
(I
O, I].
and all
>,)_f(y)
(i
It is immediate that f
is convex in the usual
coordinate functions of
n
f
+
f: R
f(O)
those one-to-one and onto functions
O.
;,: Rn+
R
n
+
which are continuous at some point
n
the answer is the same) and have the
(or bounded in a neighborhood of some point;
property that
C implies that @
f
R+
which are nondecreasing
n
lTe now pose the following problem: Determine
denote the set of convex functions
in each coordinate and satisfy
is convex
f
.
-i
(2.1)
C.
In Example 2 (below) we shall find a necessary condition that (2.1) is satisfied
by a function
REMARK I.
To aid us, we will use the following remark:
of a certain type.
Let
Let
be real numbers.
al,a2,...,an
n
h(x)
x.aj
0.
x
j=l
Suppose that
claim that
2.
i, k
O
a.
h
is
and
O
a
k
no__t convex.
for some integers
k
# k.
such that
We
The first two principal minors of the Hessian matrix, the matrix of
second order partial derivatives of
42F).
and
To see this, suppose, without loss of generality, that
h, must be nonnegative if
h
is convex
([],Thm.
Thus the inequalities
al(l
a
<_ 0
I)
and
ala2(l
must simultaneously hold if
h
h
a
a
I
is convex.
-> 0
2)
These inequalities
cannot both hold"
thus
is not convex.
EXAMPLE 2.
domain
x_
Let
A
be a real
n
n
Let
matrix.
f
be the
R+-valued
n
function with
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(x)
as follows:
f
x
A
for
x
It is easy and interesting to see that if
_g(_x)
for
x
0
and for some real
n
n
g
B
x_
matrix
(x)
B, then
BA
_x
O.
R.A. FONTENOT AND F. PROSCHAN
52
A
only if
B
is the usual matrix product of
BA
where
is invertible
and,
f
-i
A.
and
Also
is invertible if and
f
case,
in this
(x)
x
A -I
Let us call a matrix simple if it is invertible and each row contains exactly one
A permutation matrix is a simple matrix such that the nonzero entry in
nonzero entry.
I.
each row is
:
non-simple invertible matrix and let
n
n
+
R
n
+ be one-to-one
R
n
A
Let
We will now present a result about non-preservation of convexity.
be an
and onto and
also satisfy
A
(x)
_
(2.2)
O.
x
x
_
An easy argument, which we omit, shows that there exists an
n
n
P,
diagonal matrix
all of whose diagonal entries are greater than or equal to one, such that the matrix
-i
APA
has a row with two (strictly) positive entries. Choose such a P and let
Q
f(x)
It is clear that
-i
f
x
P
+
R
x
n
belongs to
C.
On the other hand, by Remark i, the function
does not belong to
C.
Thus
f
does not satisfy (2.1).
Suppose that we now consider an arbitrary simple matrix
in
C
f(x)-- (g(x I)
[0, ]
where g:
A
then
[0, =]
some point.
x
in
PROOF.
If
__
is convex, nondecreasing, and satisfies
]’hen
+
R
n e 2
functions
g(O)
0
and using
satisfies (2.1) and
(2.2),
functions in
+
+
:
R
R
is one-to-one, onto, and continuous at
n
n
satisfies (2.1) ,if and only if
and
(x)
n’ some vector
cx
_.
B
(3.1)
0, and some permutation
c
satisfles(3.1)for some vector
4’ clearly satisfies (2.1).
Suppose that
satisfies (2.1).
satisfied by
-i
"test"
must be a permutation matrix.
3. MAIN RESULTS.
THEOREM. Suppose that
then
Using
g(xn ))
g(x 2)
the Marshall-Proschan result, it is easy to see that if
for
A.
of the form
c__
O
matrix B
and some permutation matrix
A,
We shall derive a functional equation which is
Motivated by the proof in [i] and consideration of invertible linear
C, we ask the following question: If g is one-to-one, onto, and g and
C, what must be true of g? It is easy to see that the equations
both belong to
g(x + y)
-i
I
g-i (x_) +g(Z)
(x+y)
g
must hold for all
g(x) + K(Y)
0 and all
x_,x_
for some nonnegative simple matrix
A.
+
R
n
It then follows that
g(x)
xA, x
in
R
+
n’
TRANSFORMATIONS WHICH PRESERVE CONVEXITY
O, let
a_
For any
f(x_)
-i
(2.1) and both f and f
a__x
and let
A
there is a nonnegative simple matrix
f(x)
(x) and
_z
a__x,
A
depends on
and
g
satisfies
Since
also belong to
C.
Th.
+.
R
zA, z
we obtain
(ax)
where
g
-i
such that
-l(z
f
Substituting
C, both
belong to
-i
f__
g
53
+
(x)A, x
R
n’
a.
We will now show that
A
simple, there is some vector
is a diagonal matrix.
b
0
,
First, note that since
and a linear transformation
A
is
depending on
a,
which permeates coordinates, such that
(ax)
.q
Note that
b(,(x))
+
(3.2)
is multiplicative, that is,
n(x_l)
Using (3.2) and
n(x)n(y)
0
c
x
ci
which depends on
a.
m
R+n"
Z
(3.3), for any positive integer
$(amx)
for some
R
x
m, we obtain that
(,(x)), x
Take
(] ])
R
+
n.’.
m
Using elementary group theory
and the fact that the linear transformations on
of order
of permutation type form a group
R
n
is the identity transformation.
Thus
Nm
m, we have that
@(amx)
m
a_
Replacing
by
a_,
c@(x)
a_
0
and some
c > 0
+ O,
then
_(z_)
c,(x), x
R
+
n
O.
and
b
0
be such that
and
d
0
such that
in (3.4)
c
Suppose that
a
i
+
bi
z
and
0
i
aj
(3.4)
n
which depends on
Our next step is to express
_z
R
we may write
q(ax)
for all
x
a.
in terms of
for some
bj,
@(ax)
c$(x)
(bx)
d_(_x)
j
i.
.
i, i -< i
We claim that if
_< n.
Let
By (3.4)there exist
O
a_
_c
_0
and
for all
_x
in
+.n
R
(3.5)
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 ,,.-I
also satisfies (3.4) with
X
R. A. FONTENOT AND F. PROSCHAN
54
and
a
(e)
interchanged, it follows that if
-1
c
0
Let
((e))
and
(x)((e)) -I.
(x)
_0,
z
(z)
then
In particular,
_0-
exists.
It is clear that (3.4) is equivalent to the functional
e qua t ion
(a)(x)
(ax)
O,
a
+
(3.6)
R
x
(3.6) for
and
0
a
and using exponential and logarithmic functions
0
x
O. Considering
(0)
Using (3.6) and the result in the previous paragraph, we obtain
coordinatewise as appropriate, we transform (3.6) into a functional equation of the
type
B(y) + 8(z)
8(y + z)
Note that
8(z)
C.
for some real matrix
n
The solution of this equation [5] is
R
is bounded on some open set in
R
y, z
n
Rn,
zC, z
A
Transforming and letting
denote the transpose of
C,
we get
A
(x)
Using Example 2 and the fact that
x__
(3.7)
O.
x
_
satisfies
matrix.
(2.1),
To finish the proof, we must show that (3.7) holds for all
(x)
Note that
x
A
-I
x
transformaticn, that
is a linear
A
we have that
x in
is a permutation
+.
Let
R
n
R
N
satisfies
(2.1),
that (O)
and that
(x)
x,
o.
x
(3.8)
To complete the argument, we require the following result, whose proof is left to the
reader:
If
g: R
at every point
all
x
n, and
_x,
x
# O implies
n
+
+
R
n
n
is convex and nondecreasing then
__
that is, for every sequence
x,
f(x)
g(X_n)
O;
g(x).
(_n)
f
Choose
for example, take
g
is continuous from above
o f points such that
x
C
in
f(x)
x_B
where
n matrix all of whose entries are positive.
_
B
x
--n
x, for
is one-to-one and
such that f
is an invertible
Using (3.8) and the result about continuity from above, we obtain that
f
By the choice of
_(x)
x
(-i (x))
f, _f_
holds for all
x
in
-i
(__x)
f(x)
R
+.
n
x
for all
holds for
x
x
# O.
R+n
Thus
-I
(x)
x
This completes the proof of the theorem
and hence
O,
TRANSFORMATIONS WHICH PRESERVE CONVEXITY
ACKNOWLEDGEMENT.
Department
Re,a
55
Research was performed whi;e the first author was visiting the
f Statistics, Florida State University during the academic year 1981-82.
h was performed under the support of the Air Force Office of Scientl
Research under contract
AFOSR-82-K-O007.
REFERENCES
2,
3o
MARSHALL, A. W., and PROSCHAN, F., Convexity preserving scale transformations,
Ine_Rual_ities Vo!. III, Academic Press, New York, 1972, pp. 225-234.
KARLIN, S,, Total Positivity_,. Vol__.__Io., Stanford University Press, Stanford,
Calif., 1968.
BARLOW, R. E., and PROSCHAN, F., Statistical Theory of Reliability and Life
Testing: Probability_problems, Second Printing, To Begin Uith, Silver Spring,
MD, 1981.
ROBERTS, A. W., and VARBERG, D. E., Convex Functions, Academic Press, New York, 1973.
ACZEL, J., Lectures on Functional Equations and their Applications Academic Press,
New York, 1966.