The shape of the Borwein-Affleck-Girgensohn
function generated by completely monotone and
Bernstein functions
Hristo S. Sendov and Ričardas Zitikis
Department of Statistical and Actuarial Sciences, University of Western Ontario,
London, Ontario N6A 5B7, Canada
(E-mails: hssendov@stats.uwo.ca, zitikis@stats.uwo.ca)
Abstract
Borwein, Affleck, and Girgensohn (2000) posed a problem concerning the shape (that
is, convexity, log-convexity, reciprocal concavity) of a certain function of several arguments
that had manifested in a number of contexts concerned with optimization problems. In
this paper we further explore the shape of the Borwein-Affleck-Girgensohn function and
especially its extensions generated by completely monotone and Bernstein functions.
AMS 2010 Classification:
26A48 Monotonic functions, generalizations
26A51 Convexity, generalizations
26B25 Convexity, generalizations
47N10 Applications in optimization, convex analysis, mathematical programming, economics
52A41 Convex functions and convex programs
Keywords and phrases: Borwein-Affleck-Girgensohn function; completely monotone function; Bernstein function; harmonically convex function; convex measure; harmonically
convex measure; Laplace transform; quasi-arithmetic mean; Kolmogorov-Nagumo mean.
1
Contents
1 Introduction
2
2 Three classes of functions
3
2.1
Completely monotone functions . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
Bernstein functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.3
Harmonically convex functions . . . . . . . . . . . . . . . . . . . . . . . . .
8
3 Convex and harmonically-convex measures
9
4 The shape of the BAG function Fn [f ]
11
4.1
Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2
Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5 The shape of the transformed BAG function g ◦ Fn [f ]
18
5.1
Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2
Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A Appendix: The determinant of the Hessian of Mg
1
24
Introduction
Borwein, Affleck, and Girgensohn (2000) posed a problem concerning the shape of the
function
(x1 , . . . , xn ) 7→
X 1
X
X
1
1
−
+
x
x + xj 1≤i<j<k≤n xi + xj + xk
1≤i≤n i
1≤i<j≤n i
− · · · + (−1)n−1
1
(1.1)
x1 + · · · + xn
defined on (0, ∞)n . We call this function the Borwein-Affleck-Girgensohn (BAG) function.
In particular, these authors conjectured that, for every integer n ≥ 1, function (1.1) is
(1) convex,
(2) logarithmically convex (i.e., the logarithm of function (1.1) is convex), and
(3) reciprocally concave (i.e., the reciprocal of function (1.1) is concave).
2
Affleck (2000b) proved that the BAG function is convex. Borwein and Hijab (2000) applied
a probabilistic reformulation of the function and established all statements (1)–(3).
The BAG function has manifested in a number of contexts. For example, Borwein,
Affleck, and Girgensohn (2000) explain how the function arises as the objective function in
a probabilistic network optimization problem. Affleck (2000a,b) notes a connection of the
BAG function with the coupon collector’s problem (see Nath, 1973, for details) and also
utilizes the function to explore the expected time to broadcast in unreliable arborescence
networks.
A natural generalization of function (1.1) arises (cf. Borwein, Affleck, and Girgensohn,
2000) in the following form: Given a function f on (0, ∞) and an integer n ≥ 1, let Fn [f ]
be the function on (0, ∞)n defined by
Fn [f ](x1 , . . . , xn ) =
X
1≤i≤n
f (xi ) −
X
f (xi + xj ) +
1≤i<j≤n
X
f (xi + xj + xk )
1≤i<j<k≤n
− · · · + (−1)n−1 f (x1 + · · · + xn ). (1.2)
We call it the f -BAG function, with f being a ‘generating’ function. Note that function
(1.1) is equal to Fn [f1 ] with the generating function f1 (x) = 1/x, which is convex and
even completely monotone.
A number of natural problems concerning the f -BAG function Fn [f ] arise, such as
establishing its convexity or concavity, logarithmic convexity, reciprocal concavity, or
other shape-type properties. Naturally, certain assumptions on the generating function
f need to be imposed and, as a consequence, completely monotone, Bernstein, and other
functions arise (Section 2) accompanied with certain classes of measures such as convex
and harmonically convex (Section 3). The shape of the function Fn [f ] will be investigated
in Section 4. A number of non-trivial extensions of shape-type results to the ‘transformed’
BAG function g ◦ Fn [f ] for certain classes of ‘transforming’ functions g will be derived in
Section 5.
2
Three classes of functions
In this paper, our research is based on three classes of functions: completely monotone,
Bernstein, and harmonically convex. Hence, we devote this section to introducing these
3
functions, their notations and definitions, as well as basic facts that we shall need later
in the paper.
Function
Notation
Measure
Completely monotone (generic)
C
γ
Completely monotone (example)
fα (x) = 1/xα
γα
Bernstein (generic)
B
β
Bernstein (example)
fα (x) = 1/xα
βα
H
η
Harmonically convex (generic)
Notes
α>0
α ∈ (−1, 0)
Table 2.1: Functions, their notations, associated measures, and illustrative examples
Our ‘canonical’ illustrative example throughout the current paper is the function
fα (x) =
1
,
xα
which is completely monotone when α > 0 and Bernstein when α ∈ (−1, 0). To get an
initial flavour of our results, we note (details in Section 4 below) that the function Fn [fα ]
is convex when α = 1/2 (a completely monotone case) and concave when α = −1/2 (a
Bernstein case). From these shapes, one in turn trivially concludes (with the help of, e.g.,
Theorem 7.2.1 of Kuczma, 2009) the shapes of the function exp ◦Fn [fα ] when α = 1/2
as well as of the functions log ◦Fn [fα ] and 1/Fn [fα ] when α = −1/2, with ‘◦’ denoting
the composition of two functions. The shapes of the remaining three functions, which
are log ◦Fn [fα ] and 1/Fn [fα ] when α = 1/2, and exp ◦Fn [fα ] when α = −1/2, will be
explored in Section 5, where we develop a general technique for establishing the shape
of the transformed function g ◦ Fn [f ] for completely monotone and Bernstein generating
functions f .
In the following three sub-sections we discuss completely monotone, Bernstein, and
harmonically convex functions, but only at the depth needed for our following considerations in Sections 3–5.
2.1
Completely monotone functions
Definition 2.1 A non-negative function C : (0, ∞) → [0, ∞) is called completely monotone if it is infinitely differentiable and
(−1)n C (n) (x) ≥ 0 for all x > 0
4
for every integer n ≥ 1.
For historical comments and references on completely monotone functions, we refer to
Comments on pages 9-10 of Schilling, Song, and Vondraček (2010), where we learn, for
example, that the concept of complete monotonicity originates from the work of Bernstein
(1914).
By definition, every completely monotone function is non-increasing and convex. For
example, the function fα is completely monotone for every α > 0. For numerous examples of completely monotone functions, we refer to Schilling, Song, and Vondraček (2010).
Whenever possible in the current paper, we illustrate our main results with simple examples such as fα .
The Bernstein theorem says (cf., e.g., Theorem 1.4 on p. 3 in Schilling, Song, and
Vondraček, 2010) that completely monotone functions can be viewed as the Laplace transforms of certain measures. As a prelude to this general theorem, we note as an example
that, for every α > 0,
1
1
=
α
x
Γ(α)
Z
∞
−xt α−1
e
t
Z
∞
e−xt γα (dt),
dt =
0
0
where the measure γα is given by
1
γα [0, x] =
Γ(α)
x
Z
tα−1 dt =
0
xα
.
αΓ(α)
Now, the general Bernstein theorem follows.
Theorem 2.1 (Integral representation of completely monotone functions) Let C :
(0, ∞) → [0, ∞) be a completely monotone function. Then it is the Laplace transform of
a unique measure γ on [0, ∞), that is,
Z
e−xt γ(dt)
C(x) =
(2.1)
[0,∞)
for all x > 0. Conversely, if γ is a measure on [0, ∞) such that integral (2.1) is finite for
every x > 0, then the integral defines a completely monotone function C.
Theorem 2.1 implies that the derivative of any order of any completely monotone
function C can be calculated using the formula
Z
(k)
k
C (x) = (−1)
tk e−xt γ(dt).
[0,∞)
5
(2.2)
Hence, for example, (−1)k C (k) is a completely monotone function with the measure γ̄
defined by γ̄(dt) = tk γ(dt), for every integer k ≥ 1.
We see from Theorem 2.1 that for every completely monotone function C, the corresponding measure γ is uniquely determined. Given C, this measure can be recovered using
the following theorem (cf., e.g., Proposition 1.2 on p. 2 in Schilling, Song, and Vondraček,
2010).
Theorem 2.2 (Inversion of the Laplace transform) Given a completely monotone
function C : (0, ∞) → R, we have that
k
X
x
k (k)
lim
(−1) C (x)
= γ[0, t]
x→∞
k!
k≤ x t
(2.3)
at every point t ∈ [0, ∞) where the function t 7→ γ[0, t] is continuous.
2.2
Bernstein functions
Definition 2.2 A non-negative function B : (0, ∞) → [0, ∞) is called Bernstein if it is
infinitely differentiable and
(−1)n B (n) (x) ≤ 0 for all x > 0
for every integer n ≥ 1.
For the evolution of the notion and terminology related to Bernstein functions, we
refer to Schilling, Song, and Vondraček (2010), where the authors note in particular that
the term ‘Bernstein function’ originates in the work by Faraut (1965/1966).
By definition, every Bernstein function is non-decreasing and concave. Note also that
the first derivative B 0 of the Bernstein function B is completely monotone. However, not
every completely monotone function has a Bernstein primitive.
An elementary but useful example of the Bernstein function is fα for any α ∈ (−1, 0),
and we shall use it to illustrate our general results. The function has the integral representation
1
α
=−
α
x
Γ(1 + α)
Z
∞
−xt
(1 − e
α−1
)t
0
Z
∞
dt =
(1 − e−xt )βα (dt),
0
where the measure βα is defined by
α
βα (x, ∞) = −
Γ(1 + α)
Z
x
6
∞
tα−1 dt =
xα
.
Γ(1 + α)
For many examples of Bernstein functions, we refer to Schilling, Song, and Vondraček
(2010). We note in passing that, for any α ∈ (−1, 0), the function fα is actually a
complete Bernstein function, because the measure βα is absolutely continuous with a
completely monotone density.
The following Lévy–Khintchine theorem characterizes any Bernstein function (cf., e.g.,
Theorem 3.2 on p. 15 in Schilling Schilling, Song, and Vondraček, 2010) in terms of an
integral representation.
Theorem 2.3 (Integral representation of Bernstein functions) The function B :
(0, ∞) → [0, ∞) is a Bernstein function if and only if it admits the representation
Z
(1 − e−xt ) β(dt)
(2.4)
B(x) = a + bx +
(0,∞)
for some constants a, b ≥ 0 and a measure β on (0, ∞) such that
Z
min{1, t} β(dt) < ∞.
(2.5)
(0,∞)
The triplet (a, b, β) uniquely determines the function B, and vice versa. The measure β
associated with the Bernstein function B is usually called the Lévy measure.
As we shall see from the next Proposition 2.1, condition (2.5), which guarantees the
finiteness of the integral in (2.4) for every x > 0 (cf., e.g., p. 16 in Schilling Schilling,
Song, and Vondraček, 2010) can be reformulated in terms of the measure µ defined by
µ(dt) = tβ(dt). This measure will play an important role later in this paper: it is
associated with the first derivative B 0 of the Bernstein function B.
Proposition 2.1 With the measure µ defined by µ(dt) = tβ(dt), we have that
Z
Z
min{1, t} β(dt) =
µ(0, 1/s] ds.
(0,∞)
Proof. We start with the equation
Z
Z
min{1, t} β(dt) =
(0,∞)
(2.6)
(0,1)
Z
µ(0, 1] ds +
(0,1)
7
(1,∞)
1
µ(dt),
t
(2.7)
and then show that the right-most integral is equal to
R
(0,1)
µ(1, 1/s] ds. This we do as
follows:
Z
(1,∞)
1
µ(dt) =
t
Z
Z
1(1,∞) (t)1(0,1/s] (t) µ(dt) ds
(0,∞)
Z
(0,∞)
Z
=
1(1,1/s] (t) µ(dt) ds
(0,1)
(0,∞)
Z
µ(1, 1/s] ds.
=
(2.8)
(0,1)
Combining equations (2.8) and (2.7), we conclude the proof of Proposition 2.1.
Hence, Proposition 2.1 reformulates condition (2.5) in terms of the integrability of the
function x 7→ µ[0, 1/s] over the interval (0, 1) and allows us to look at Proposition 3.4 of
Schilling, Song, and Vondraček (2010) from a somewhat different angle, namely, in terms
of the ‘shape’ of the function s 7→ µ[0, 1/s].
Corollary 2.1 Let C(x) = b +
R
(0,∞)
(1 − e−xt ) µ(dt) be a completely monotone function
with some measure µ. Then the function C has a Bernstein primitive if and only if the
function s 7→ µ[0, 1/s] is integrable in a neighbourhood of 0 with respect to the Lebesgue
measure.
Proof. The proof is analogous to that of Proposition 3.4 of Schilling, Song, and Vondraček
(2010), using the bounds t/(1 + t) ≤ min{1, t} ≤ 2t/(1 + t) for t ≥ 0.
The shape of the function x 7→ µ(0, 1/s] will play a considerable role in our subsequent
considerations, starting in Section 3. For example, those measures µ for which the function
x 7→ µ[0, 1/x] is convex will be called harmonically convex. In anticipation of these
development, we next discuss the class of harmonically convex functions.
2.3
Harmonically convex functions
Definition 2.3 A function H : (0, ∞) → R is called harmonically convex if the function
x 7→ H(1/x) is convex on (0, ∞).
For example, the logarithmic function x 7→ log(x) is harmonically convex (cf., e.g., a
note on p. 211 of Merkle, 2004) even though it is concave from the classical point of view.
8
In general, the appropriateness of the modifier ‘harmonically’ follows from the fact
that the function H is harmonically convex if and only if
2
H(x1 ) + H(x2 )
H
≤
1/x1 + 1/x2
2
(2.9)
for every x1 , x2 ∈ (0, ∞). Note that 2/(1/x1 + 1/x2 ) is the harmonic mean of x1 and
x2 . Hence, harmonically convex functions relate the harmonic mean of two points to the
arithmetic mean of the function values at the two points. For properties and further
examples of harmonically convex functions, we refer to Merkle (2004). We note that
Merkle (2004) calls a function h reciprocally convex if it is concave and the function
x 7→ h(1/x) is convex. Hence, in general, there is a difference between reciprocally
convex functions of Merkle (2004) and the above introduced class of harmonically convex
functions.
Due to the relationship 2/(1/x1 +1/x2 ) ≤ (x1 +x2 )/2 that holds for all x1 , x2 ∈ (0, ∞),
every non-decreasing convex function H : (0, ∞) → R is harmonically convex on (0, ∞).
The following lemma (cf., e.g., Lemma 2.2 on p. 211 of Merkle, 2004) further clarifies the
relationship between convex and harmonically convex functions.
Lemma 2.1 A function H : (0, ∞) → R is convex if and only if x 7→ xH(1/x) is convex.
3
Convex and harmonically-convex measures
Definition 3.1 A measure µ on [0, ∞) is called (harmonically) convex if the function
x 7→ µ[0, x] is (harmonically) convex on (0, ∞).
A few observations follow. Lemma 2.1 and notes preceding the lemma immediately
imply that
• every measure µ is convex if and only if the function x 7→ xµ[0, 1/x] is convex, and
• every convex measure µ is harmonically convex.
We shall next investigate the shape of measures corresponding to completely monotone
functions. To illustrate, the function fα is convex for every α > 0. Its corresponding
measure γα is harmonically convex because γα [0, x] = xα /(αΓ(α)), which implies the
equation γα [0, 1/x] = 1/(xα αΓ(α)) whose right-hand side is a convex function of x for
every α > 0. Note, however, that the measure γα is convex only for α ≥ 1.
9
Theorem 3.1 Let C be a completely monotone function with measure γ. If γ is (harmonically) convex, then, for every integer k ≥ 1, the measure corresponding to the function
(−1)k C (k) is also (harmonically) convex.
Proof. By formula (2.2), the measure corresponding to the completely monotone function
(−1)k C (k) is γ̄(dt) = tk γ(dt). Thus
Z t
Z
k−1
ks ds γ(dt)
γ̄[0, x] =
[0,x)
0
Z
Z
k−1
=k
s
1[0,x] (t)1[s,∞) (t) γ(dt) ds
[0,∞)
[0,∞)
Z
sk−1 γ[0, x] − γ[0, s] + ds.
=k
(3.1)
[0,∞)
If the measure γ is convex, or harmonically convex, then such is the function x 7→ (γ[0, x]−
γ[0, s])+ because the positive part x 7→ x+ is an increasing and convex function; and so is
the integral with respect to s. In summary, if γ is convex, or harmonically convex, then
such is the measure γ̄. This completes the proof of Theorem 3.1.
The next theorem highlights a close relationship between the shapes of completely
monotone functions and their measures.
Theorem 3.2 Let C be a completely monotone function with measure γ, and let Tn be a
function defined by
Tn (x) =
n X
k
(−1) C
k=0
(k)
k
x
.
(x)
k!
The measure γ is convex if and only if the function Tn is harmonically convex for every
integer n ≥ 0. Furthermore, the measure γ is harmonically convex if and only if the
function Tn is convex for every integer n ≥ 0.
Proof. Since T0 (x) = C(x), we have that
Z
Z
−λx
T0 (x) =
1{λ ≥ t}xe dλ γ(dt)
[0,∞)
[0,∞)
Z
=
γ[0, λ]xe−λx dλ.
[0,∞)
Upon noticing that, for every integer k ≥ 1,
k
d
xe−λx = (−1)k−1 kλk−1 e−λx + (−1)k λk xe−λx ,
dx
10
(3.2)
and using the mathematical induction, we obtain from equation (3.2) that, for every
integer n ≥ 0,
Z
1
Tn (x) =
γ[0, λ]xn+1 λn e−λx dλ
n! [0,∞)
Z
1
=
γ[0, λ/x]λn e−λ dλ.
n! [0,∞)
(3.3)
Hence, if the measure γ is harmonically convex, then Tn is convex, and replacing x everywhere by 1/x, we have that if γ if convex, then Tn is harmonically convex, and these
statements hold for every integer n ≥ 0.
To show that the two convexity properties of the function Tn imply the corresponding
convexity properties of the measure γ, we utilize Theorem 2.2. In its statement (2.3), we
replace x by x/t and obtain
x k 1
X
k (k) x
= γ[0, t].
lim
(−1) C
x→∞
t
t
k!
k≤ x
(3.4)
When the function Tn is harmonically convex for every integer n ≥ 0, then the function
x k 1
X
k (k) x
t 7→
(−1) C
t
t k!
k≤ x
is convex, as so is the limiting function t 7→ γ[0, t]. Replacing t by 1/t on both sides
of equation (3.4), we conclude that when the function Tn is convex for every integer
n ≥ 0, then the limiting function t 7→ γ[0, 1/t] is also convex, and so the measure γ is
harmonically convex. This concludes the proof of Theorem 3.2.
Corollary 3.1 Let C be a completely monotone function with measure γ. When γ is
convex, then the function C is harmonically convex.
Corollary 3.1 is a simple consequence of Theorem 3.2 when n = 0, but we mention it
separately here as it implies the shape of the Laplace transform (cf. Theorem 2.1).
4
The shape of the BAG function Fn[f ]
Here we are mainly concerned with determining when the function Fn [f ] is convex or
concave.
Throughout the section and beyond, we use Z1 , . . . , Zn to denote independent and
identically distributed exponential random variables on a probability space (Ω, F, P).
Hence, P[Zi ≤ z] = 1 − e−z for all z ≥ 0.
11
4.1
Convexity
Theorem 4.1 Let C be a completely monotone function with measure γ. Then the function Fn [C] is non-negative. Furthermore, if γ is harmonically convex, then Fn [C] is convex
on (0, ∞)n for every integer n ≥ 1.
Proof. Utilizing Bernstein’s integral representation (2.1), we have
Z
n
Y
1−
P Zi /xi < t γ(dt)
Fn [C](x1 , . . . , xn ) =
[0,∞)
i=1
Z
=
P
h
[0,∞)
i
max {Zi /xi } ≥ t γ(dt).
i=1,...,n
(4.1)
The non-negativity of Fn [C] follows. Manipulating with the probability measure just like
with any other measure so far, we have from equation (4.1) that
Z
Z
Fn [C](x1 , . . . , xn ) =
1[t,∞) max {Zi (ω)/xi } P(dω) γ(dt)
i=1,...,n
[0,∞) Ω
Z Z
(t) γ(dt) P(dω)
=
1
0, max {Zi (ω)/xi }
Ω [0,∞)
i=1,...,n
Z h
i
γ 0, max {Zi (ω)/xi } P(dω)
=
i=1,...,n
ZΩ
=
max γ[0, Zi (ω)/xi ] P(dω).
Ω i=1,...,n
(4.2)
Since the measure γ is harmonically convex by assumption, the function x 7→ γ[0, 1/x] is
convex and so is the function x 7→ γ[0, Zi (ω)/x] for every i = 1, . . . , n and ω ∈ Ω. Hence,
the function (x1 , . . . , xn ) 7→ max γ[0, Zi (ω)/xi ] is convex, and so is the integral on the
i=1,...,n
right-hand side of equation (4.2). This concludes the proof of Theorem 4.1.
To illustrate Theorem 4.1, recall that fα is a completely monotone function for every
α > 0. The corresponding measure γα is harmonically convex and given by the formula
γα [0, x] = xα /(αΓ(α)). Hence, we have the following corollary to Theorem 4.1.
Corollary 4.1 For every α > 0, the BAG function Fn [fα ] is non-negative and convex on
(0, ∞)n for every integer n ≥ 1.
The following corollary to Theorem 4.1 tells us how to create a variety of functions
such that the their generated BAG functions are non-negative and convex.
Corollary 4.2 The following three statements hold for every integer n ≥ 1 and k ≥ 0:
12
a) If C1 and C2 are completely monotone functions with harmonically convex measures,
then, for all real a, b ≥ 0, the function Fn [(−1)k (aC1 + bC2 )(k) ] is non-negative and
convex on (0, ∞)n .
b) If C1 and C2 are completely monotone functions and one of them has a convex measure, then the function Fn [(−1)k (C1 C2 )(k) ] is non-negative and convex on (0, ∞)n .
c) If h : [0, ∞) → [0, ∞) is such that x 7→ (1/x)2 h(1/x) is non-increasing, then the
R
Laplace transform Lh (x) = [0,∞) e−xt h(t) dt is convex on (0, ∞). Moreover, the
(k)
function Fn [(−1)k Lh ] is convex on (0, ∞)n .
Proof. To establish part a), let C be a completely monotone function with measure γ.
Since (−1)k C (k) is completely monotone with the measure γ̄(dt) = tk γ(dt), Theorem 3.1
says that γ̄ is harmonically convex. Furthermore, the measure corresponding to the
function aC1 + bC2 is harmonically convex. This concludes the proof of part a).
To prove part b), we first note that the measure corresponding to the function C1 C2
is the convolution of the measures corresponding to C1 and C2 (cf., e.g., Corollary 1.6
of Schilling, Song, and Vondraček, 2010). Hence, we only need to establish the following
result, formulated in a slightly more generally form than currently needed: If µ and ν
are two measures on [0, ∞) and one of them is convex, then their convolution µ ∗ ν is
convex, and hence harmonically convex. To prove this statement, suppose for the sake
of concreteness that µ is convex. We need to show that the function x 7→ (µ ∗ ν)[0, x] is
convex. We have
Z
(µ ∗ ν)[0, x] =
1[0,x] (s + t) µ(ds) ν(dt)
[0,∞)2
Z
=
1[0,x−t] (s)1[0,x] (t) µ(ds) ν(dt)
[0,∞)2
Z
µ[0, x − t]1[0,x] (t) ν(dt)
=
[0,∞)
Z
=
µ[0, φt (x)] ν(dt),
(4.3)
[0,∞)
where φt (x) = max{x − t, 0}. The functions φt and x 7→ µ[0, x] are convex and nondecreasing, and so is their composition x 7→ µ[0, φt (x)] for every fixed t ≥ 0. Hence, the
integral on the right-hand side of equation (4.3) is a convex function of x. This establishes
the above statement about the shape of the convolution µ ∗ ν and concludes the proof of
part b).
13
Rx
e−xt ν(dt), where ν[0, x] = 0 h(t) dt. The
R 1/x
measure ν is harmonically convex because the convexity of x 7→ 0 h(t) dt follows from
To prove part c), we write Lh (x) =
R
[0,∞)
the assumption that x 7→ (1/x)2 h(1/x) is a non-increasing function.
The next corollary is inspired by Remark 2 of Borwein and Hijab (2000). Note that
the corollary does not require any assumption on the measures γ and β corresponding to
completely monotone and Bernstein functions.
Corollary 4.3 If the function f is completely monotone or Bernstein, then, for every
integer n ≥ 1, the function
(x1 , . . . , xn ) 7→
Fn [f ](x1 , . . . , xn )
x 1 · · · xn
(4.4)
is non-negative and convex on (0, ∞)n , and also decreasing in each argument.
Proof. Since the random variables Zi are independent and exponentially distributed, the
random variables Yi = Zi /xi are independent and have the densities y 7→ xi e−xi y . Hence,
when f is completely monotone, from equation (4.2) we have that
Z
Fn [f ](x1 , . . . , xn ) = x1 · · · xn
e−(x1 y1 +···+xn yn ) max γ[0, yi ] dy1 . . . dyn .
i=1,...,n
[0,∞)n
The function (x1 , . . . , xn ) 7→ e−(x1 y1 +···+xn yn ) is convex, and so is function (4.4).
Similarly, when f is a Bernstein function, we use equation (4.6) and have
Z
Fn [f ](x1 , . . . , xn ) = x1 · · · xn
e−(x1 y1 +···+xn yn ) min β[yi , ∞) dy1 . . . dyn .
i=1,...,n
[0,∞)n
The function (x1 , . . . , xn ) 7→ e−(x1 y1 +···+xn yn ) is convex and decreasing, and so is function
(4.4). This concludes the proof of Corollary 4.3.
4.2
Concavity
We need an additional definition.
Definition 4.1 A measure ν on (0, ∞) has a harmonically concave tail if the function
x 7→ ν(1/x, ∞) is concave on (0, ∞).
Theorem 4.2 Let B be a Bernstein function with measure β. Then the function Fn [B] is
non-negative. Furthermore, if β has a harmonically concave tail, then the function Fn [B]
is concave on (0, ∞)n for every integer n ≥ 1.
14
Proof. Utilizing the Lévy–Khintchine integral representation of the function B, we have
n
Y
P Zi /xi < t β(dt)
Z
Fn [B](x1 , . . . , xn ) =
[0,∞) i=1
Z
h
i
P max {Zi /xi } < t β(dt).
=
i=1,...,n
[0,∞)
(4.5)
The non-negativity of Fn [B] follows. We proceed with equation (4.5) and have that
Z
Z
Fn [B](x1 , . . . , xn ) =
1[0,t) max {Zi (ω)/xi } P(dω) β(dt)
i=1,...,n
[0,∞) Ω
Z Z
(t) β(dt) P(dω)
=
1
max
{Z
(ω)/x
},
∞
i
i
Ω [0,∞)
i=1,...,n
Z β max {Zi (ω)/xi }, ∞ P(dω)
=
i=1,...,n
ZΩ
min β Zi (ω)/xi , ∞ P(dω).
(4.6)
=
Ω i=1,...,n
Since the measure β has a harmonically concave tail, the function x 7→ β(1/x, ∞) is
concave, and so is the function x 7→ β(Zi (ω)/x, ∞) for every i = 1, . . . , n and ω ∈ Ω.
Consequently, the function (x1 , . . . , xn ) 7→ min β(Zi (ω)/xi , ∞) is concave, and so is the
i=1,...,n
integral on the right-hand side of equation (4.6). This concludes the proof of Theorem
4.2.
The following corollary to Theorem 4.2 extends an example of Borwein and Hijab
(2000).
Corollary 4.4 For every α ∈ (−1, 0), the BAG function Fn [fα ] is concave on (0, ∞)n
for every integer n ≥ 1.
Proof. Recall that the function fα is Bernstein for every α ∈ (−1, 0) with the integral
R∞
representation 0 (1 − e−xt )βα (dt), where βα (x, ∞) = xα /Γ(1 + α). Hence, βα (1/x, ∞) =
x−α /Γ(1 + α), which is a concave function for every α ∈ (−1, 0). Consequently, the
measure βα has a harmonically concave tail. This concludes the proof of Corollary 4.4.
The two end-points α ∈ {0, −1} not covered by Corollary 4.4 produce the BAG
function

 1
Fn [fα ](x1 , . . . , xn ) =
 0
15
if α = 0,
if α = −1.
Hence, when α = 0 and α = −1, the BAG function Fn [fα ] is both concave and convex.
This observation together with Corollaries 4.1 and 4.4 establish the shape of the BAG
function Fn [fα ] for every α ≥ −1.
In a sense, we may view the case α = 0 as a ‘boundary’ between the classes of
completely monotone and Bernstein functions, but one may very well argue that a better
‘boundary’ between the two classes would be the logarithmic function
flog (x) = log(1 + x),
which is a Bernstein function with the Lévy–Khintchine representation
Z ∞
e−t
log(1 + x) =
(1 − e−xt )
dt.
t
0
Note, however, that the corresponding measure
Z
β(x, ∞) = −
x
∞
e−t
dt
t
does not have a harmonically concave tail, which is seen from the formula
d
−1 −1/x
β(1/x, ∞) =
e
.
dx
x
Since the latter is not a decreasing function of x on (0, ∞), we cannot infer from Theorem
4.2 the concavity of the function Fn [flog ]. This, however, is natural because even though
the function is concave when n = 1, it is neither concave nor convex when n = 2. A proof
of the latter fact follows from the formulas of the two (main) diagonal terms
−1
1
+
,
2
(1 + x1 )
(1 + x1 + x2 )2
−1
1
+
,
2
(1 + x2 )
(1 + x1 + x2 )2
(4.7)
and the determinant
−1 + 2x1 x2
(1 + x1 ) (1 + x2 )2 (1 + x1 + x2 )2
2
(4.8)
of the Hessian of (x1 , x2 ) 7→ F2 [flog ](x1 , x2 ). Both diagonal terms (4.7) are negative, and
thus the BAG function (x1 , x2 ) 7→ F2 [flog ](x1 , x2 ) is concave along each of the individual
x1 and x2 axes. However, the BAG function is not concave or convex jointly in both
x1 and x2 , because determinant (4.8) is not of the same sign everywhere on (0, ∞)2 . To
illustrate, in Figure 4.2 we have depicted the diagonal BAG function z 7→ F2 [flog ](z, z)
and its second derivative, demonstrating that the BAG function (x1 , x2 ) 7→ F2 [flog ](x1 , x2 )
can be neither concave nor convex on (0, ∞)2 .
16
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.5
1
1.5
2
0.5
-0.1
1
1.5
2
-0.1
Figure 4.1: The function z 7→ F2 [flog ](z, z) (left) and its second derivative (right).
We next establish a technical lemma to be used later when proving a result concerning
Bernstein functions whose first derivative – as we have already noted above – is associated
with the measure µ defined by the equation µ(dt) = tβ(dt).
Lemma 4.1 If the measure ν on (0, ∞) has a harmonically concave tail, then the measure
µ defined by the equation µ(dt) = tν(dt) is harmonically convex on (0, ∞).
Proof. We start the proof with elementary equations:
Z
1[0,x] (t)tν(dt)
µ[0, x] =
[0,∞)
Z
Z
1[0,x] (t)1(s,∞) (t)ν(dt)ds
=
[0,∞) [0,∞)
Z
ν(s, ∞) − ν(x, ∞) ds.
=
+
[0,∞)
Consequently,
Z
µ[0, 1/x] =
ν(s, ∞) − ν(1/x, ∞)
ds.
(4.9)
+
[0,∞)
Since ν has a harmonically concave tail, the function x 7→ −ν(1/x, ∞) is convex, and so
is the integral on the right-hand side of equation (4.9) as a function of x. This implies
that the measure µ is harmonically convex and concludes the proof of Lemma 4.1.
Corollary 4.5 Let B be a Bernstein function with measure β, and let k ≥ 1 be any
integer. If β has a harmonically concave tail, then Fn [(−1)k B (k) ] is non-positive and
concave on (0, ∞)n for every integer n ≥ 1.
Proof. Since the first derivative B 0 of the Bernstein function B is completely monotone,
Theorem 2.1 gives us a measure γB such that
Z
0
B (x) =
e−xt γB (dt).
[0,∞)
17
(4.10)
On the other hand, we have from equation (2.4) that
Z
0
B (x) = b +
e−xt tβ(dt).
(4.11)
(0,∞)
Equations (4.10) and (4.11) imply that γB (dt) = bδ0 (dt) + tβ(dt), where δ0 is the Dirac
measure, that is, δ0 (B) = 1 if 0 ∈ B and δ0 (B) = 0 otherwise, for every Borel set
B ⊆ [0, ∞). Hence, we have that γB [0, x] = b + µ(0, x], and so the harmonic convexity of
γB follows from the harmonic convexity of the measure µ defined by µ(dt) = tβ(dt) (cf.
Lemma 4.1). Part (a) of Corollary 4.2 concludes the proof of Corollary 4.5.
5
The shape of the transformed BAG function g◦Fn[f ]
In this section we further explore the shape of the function Fn [f ] by investigating the
shape of its transformations such as log(Fn [f ]), 1/Fn [f ] and, generally, g ◦ Fn [f ]. The
obtained results generalize those of Borwein and Hijab (2000), who proved in particular
that in the case of the generating function f (x) = 1/x, the logarithmically and reciprocally
transformed BAG functions log(Fn [f ]) and 1/Fn [f ] are convex and concave, respectively.
To clarify our following research, we note that for some functions g, the shape of
g ◦ Fn [f ] follows immediately from the shape of the original BAG function Fn [f ], which
was explored in the previous sections. For example, we know from Corollary 4.1 that
the function Fn [fα ] is convex for every real α > 0 and every integer n ≥ 1, and so it
immediately follows (see, e.g., Theorem 7.2.1 in Kuczma, 2009) that
(i) if g is convex and increasing, then g ◦ Fn [fα ] is convex, and
(ii) if g is concave and decreasing, then g ◦ Fn [fα ] is concave.
We can use analogous elementary arguments to derive the shape of g ◦ Fn [fα ] for transforming functions g such as g(x) = ex and g(x) = xρ for any ρ > 1.
Note, however, that neither g(x) = log(x) nor g(x) = 1/x falls into any of the two
classes (i)-(ii) of functions g.
Furthermore, we know from Corollary 4.4 that the function Fn [fα ] is concave for every
real α ∈ (−1, 0) and every integer n ≥ 1. Hence (cf., e.g., Theorem 7.2.1 in Kuczma,
2009), we have that
(iii) if g is convex and decreasing, then g ◦ Fn [fα ] is convex, and
18
(iv) if g is concave and increasing, then g ◦ Fn [fα ] is concave.
Both g(x) = log(x) and g(x) = 1/x fall into one of the two classes of functions g, and
we can therefore conclude that, for every real α ∈ (−1, 0) and every integer n ≥ 1, the
functions log(Fn [fα ]) and 1/Fn [fα ] are concave and convex, respectively.
However, such elementary arguments do not imply the shape of g ◦ Fn [fα ] for the
functions g(x) = ex and g(x) = xρ for any ρ > 1, as they do not fall into any of the two
classes (iii)-(iv) of functions g.
Given the above, a more elaborate analysis is needed, and we shall pursue it in this
section. To proceed, we need another definition.
Definition 5.1 Let g be an increasing one-to-one function. Then for any pair x and y of
real numbers, the quasi-arithmetic (qa) g-mean Mg (x, y) of x and y is defined (cf., e.g.,
Aczél, 2006; Kuczma, 2009) by the equation
−1 g(x) + g(y)
Mg (x, y) = g
.
2
We say that an increasing one-to-one function g is qa-concave if Mg is concave. Likewise,
we say that the function g is qa-convex if Mg is convex.
The quasi-arithmetic mean has a long history, going back to at least the works of
Kolmogorov (1930), Nagumo (1930), de Finetti (1931), Jessen (1931), Kitagawa (1934).
Not surprisingly, therefore, the quasi-arithmetic mean has frequently been called the Kolmogorov mean, or the Kolmogorov-Nagumo mean. Since then, numerous works dealing
with such means and their generalizations have appeared, especially in context related to
information and decision theories. For recent applications of quasi-arithmetic means, as
well as for references, we refer to, e.g., Porcu, Mateu and Christakos (2009), and Chapter
4 (in particular, Section 4.5) of Denuit, Dhaene, Goovaerts, and Kaas (2005).
5.1
Convexity
The main result of this section will establish the convexity of the function g ◦ Fn [f ] and
will be based on increasing, one-to-one, and qa-concave functions g. Examples of such
functions are:
• g(x) = x with Mg (x, y) = (x + y)/2
19
• g(x) = xρ with Mg (x, y) = ((xρ + y ρ )/2)1/ρ for any ρ ∈ (0, 1)
• g(x) = −1/x with Mg (x, y) = 2/(1/x + 1/y)
• g(x) = log(x) with Mg (x, y) =
√
xy
• g(x) = log(1 + x) with Mg (x, y) =
p
(1 + x)(1 + y) − 1
• g(x) = −e−x with Mg (x, y) = log(2/(e−x + e−y ))
The verification that all of these functions are qa-concave is equivalent to checking that
the corresponding functions (x, y) 7→ Mg (x, y) are concave, which is easily done by calculating the Hessian and noticing that, for every function g above, the two (main) diagonal
members as well as the determinant of the Hessian are non-positive. In fact, the determinant is equal to 0 in all these cases, which we have found an interesting phenomenon,
but we defer additional notes on it until Appendix A.
Theorem 5.1 Let C be a completely monotone function with measure γ. If the transforming function g is increasing, one-to-one, and qa-concave, and if the function x 7→ g(γ[0, x))
is harmonically convex, then the function g ◦ Fn [C] is non-negative and convex on (0, ∞)n
for every integer n ≥ 1.
Proof. In the proof of Theorem 4.1, we have established the equation
Fn [C](x) = E
h
i
max γ[0, Zi /xi ] .
i=1,...,n
(5.1)
In what follows, we use the notation x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ). Denote a
random field X : (0, ∞)n → R by the formula
X(x) = g
max γ[0, Zi /xi ]
i=1,...,n
= max g γ[0, Zi /xi ] ,
i=1,...,n
where the last equation holds because the function g is increasing. Since x 7→ g(γ[0, x])
is harmonically convex, the random field X is convex. Hence, our task is to check that
x 7→ g E[(g −1 ◦ X)(x)]
20
(5.2)
is convex. Since X is convex and both g and g −1 are increasing functions, we have that
g E[(g −1 ◦ X)((x + y)/2)] ≤ g E[(g −1 ((X(x) + X(y)/2)]
= g E[Mg ((g −1 ◦ X)(x), (g −1 ◦ X)(y))] .
(5.3)
Since Mg is concave, an application of the two-dimensional Jensen inequality (e.g., Schaefer, 1976) gives the bound
g E[Mg ((g −1 ◦ X)(x), (g −1 ◦ X)(y))] ≤ (g ◦ Mg )(E[(g −1 ◦ X)(x)], E[(g −1 ◦ X)(y)])
= g(E[(g −1 ◦ X)(x)]) + g(E[(g −1 ◦ X)(y)]) /2,
which together with bound (5.3) establish the convexity of function (5.2). This concludes
the proof of Theorem 5.1.
As an example, with γα [0, 1/x] = 1/(αΓ(α)xα ) for any α > 0, we have that the
function x 7→ g(γα [0, x)] is harmonically convex if and only if the function
1
x 7→ g
αΓ(α)xα
(5.4)
is convex. We easily check that the following functions g from the list at the beginning
of this subsection result in convex functions (5.4):
• g(x) = x
• g(x) = xρ for any ρ ∈ (0, 1)
• g(x) = −1/x, provided that α ∈ (0, 1]
• g(x) = log(x)
• g(x) = log(1 + x)
Theorem 5.1 initiates a number of extensions, including the following corollary, which
is a consequence of the earlier noted fact (cf.
Theorem 3.1 and its proof) that, for
every integer n ≥ 1, the function (−1)k C (k) is completely monotone with the measure
γ̄(dt) = tk γ(dt). Since the case k = 0 is covered by Theorem 5.1, in the following
corollary we consider the case k ≥ 1.
21
Corollary 5.1 Let C be a completely monotone function with measure γ, and let k ≥ 1
be any integer. If the transforming function g is increasing, one-to-one, and qa-concave,
and if x 7→ g(γ̄[0, x]) is harmonically convex, then the function g ◦ Fn [(−1)k C (k) ] is nonnegative and convex on (0, ∞)n for every integer n ≥ 1.
To verify harmonic convexity of the function x 7→ g(γ̄[0, x)), we note that
Z
k−1
γ[0, x] − γ[0, s] + ds
g(γ̄[0, x]) = g k
s
[0,∞)
Z
k
k−1
= g x γ[0, x] − k
s γ[0, s] ds ,
(5.5)
[0,∞)
which is a consequence of equation (3.1).
Note that the measure γ may or may not be harmonically convex, yet the function
x 7→ g(γ̄[0, x]) can be harmonically convex. Hence, one would need to verify the harmonic
convexity of the right-hand side of equation (5.5) with respect to x. Assume for the sake of
argument that the measure γ is harmonically convex. Since g is increasing by assumption,
and if we also assume that g is convex, then equation (5.5) immediately implies that
x 7→ g(γ̄[0, x]) is harmonically convex.
To illustrate the above arguments, consider the case when γ = γα with γα [0, 1/x] =
1/(αΓ(α)xα ) for any α > 0, which is an example discussed at the beginning of Section 3.
In this case, equation (5.5) simplifies to
1
g(γ̄[0, 1/x]) = g
.
(k + α)Γ(α)xk+α
(5.6)
We easily check that the following functions g from the list at the beginning of this
subsection result in convex functions x 7→ g(γ̄[0, 1/x]):
• g(x) = x
• g(x) = xρ for any ρ ∈ (0, 1)
• g(x) = log(x)
• g(x) = log(1 + x)
Hence, Corollary 5.1 implies that for these functions g and for any completely monotone
function C, the function g ◦ Fn [(−1)k C (k) ] is non-negative and convex on (0, ∞)n for all
integers n ≥ 1 and k ≥ 1. Likewise, we conclude from Theorem 5.1 that the function
g ◦ Fn [C] is non-negative and convex on (0, ∞)n for every integer n ≥ 1.
22
5.2
Concavity
The main result of this section, which establishes the concavity of the function g ◦ Fn [f ],
requires increasing, one-to-one, and qa-convex functions g. Examples of such functions
are:
• g(x) = x with Mg (x, y) = (x + y)/2
• g(x) = xρ with Mg (x, y) = ((xρ + y ρ )/2)1/ρ for any ρ > 1
• g(x) = ex with Mg (x, y) = log((ex + ey )/2)
The verification that all of them are qa-convex can easily done by calculating the Hessian
of (x, y) 7→ Mg (x, y) and then noticing that, for every function g above, the two (main)
diagonal members as well as the determinant of the Hessian are non-negative. In fact,
the determinants are equal to 0 in all the cases, and we shall give an insight into this
phenomenon in Appendix A.
Theorem 5.2 Let B be a Bernstein function with measure β. If the transforming function g is increasing, one-to-one, and qa-convex, and if the function x 7→ g(β(1/x, ∞))
is concave, then the function g ◦ Fn [B] is non-negative and concave on (0, ∞)n for every
integer n ≥ 1.
Proof. In the proof of Theorem 4.2, we have established the equation
Fn [B](x) = E
h
i
min β(Zi /xi , ∞) .
i=1,...,n
(5.7)
Denote
X(x) = min g β(Zi /xi , ∞) .
i=1,...,n
Since x 7→ g(β(1/x, ∞)) is concave by assumption, the random field X is concave. Our
task is to check that
x 7→ g E[(g −1 ◦ X)(x)]
(5.8)
is concave. Since X is concave and both g and g −1 are increasing functions, we have that
g E[(g −1 ◦ X)((x + y)/2)] ≥ g E[(g −1 ((X(x) + X(z)/2)]
= g E[Mg ((g −1 ◦ X)(x), (g −1 ◦ X)(y))] .
23
(5.9)
Since Mg is a convex function, an application of the two-dimensional Jensen’s inequality
on the right-hand side of equation (5.9) gives the bound
g E[Mg ((g −1 ◦ X)(x), (g −1 ◦ X)(y))] ≥ (g ◦ Mg )(E[(g −1 ◦ X)(x)], E[(g −1 ◦ X)(y)])
= g(E[(g −1 ◦ X)(x)]) + g(E[(g −1 ◦ X)(y)]) /2,
which together with bound (5.9) establish the concavity of function (5.8). This concludes
the proof of Theorem 5.2.
In the proof of Corollary 4.4 we have noted that the function fα is Bernstein for every
α ∈ (−1, 0) with the measure βα such that βα (1/x, ∞) = x−α /Γ(1 + α). We easily check
that the following functions g from the list at the beginning of this subsection result in
concave functions x 7→ g(βα (1/x, ∞)):
• g(x) = x
• g(x) = xρ , provided that 1 < ρ ≤ (−α)−1
• g(x) = ex
Hence, Theorem 5.2 implies that for these transforming functions g, the function g ◦Fn [B]
is non-negative and concave on (0, ∞)n for every integer n ≥ 1.
We conclude this subsection with a note that there is no need for deriving an analogue
of Corollary 5.1 for Bernstein functions, because for every such function B, the derivative
B 0 is completely monotone and thus Theorem 5.1 and Corollary 5.1 imply desired shapetype results for g ◦ Fn [(−1)k B (k) ] for every integer k ≥ 1.
A
Appendix: The determinant of the Hessian of Mg
The determinant of the Hessian of Mg (x, y) is equal to 0 in the case of every transforming
function g that we used to illustrate our theory in Section 5. We find this an interesting
and puzzling phenomenon. To get an insight, let g be any transforming function. The
determinant of the Hessian of Mg (x, y) is equal to
2 g 0 (Mg (x, y))2 g 00 (x) g 00 (y) − g 0 (y)2 g 00 (x) + g 0 (x)2 g 00 (y) g 00 (Mg (x, y))
8 g 0 (Mg (x, y))4
24
.
(1.1)
Assuming that the second derivative g 00 is not 0, which is the case for all the illustrative
functions g except when g(x) = x, determinant (1.1) is equal to 0 if and only if
!
g 0 (Mg (x, y))2
1 g 0 (x)2 g 0 (y)2
=
+ 00
.
g 00 (Mg (x, y))
2 g 00 (x)
g (y)
With the notation
(1.2)
g 0 (z)2
,
g 00 (z)
Gg (z) =
we rewrite equation (1.2) as follows:
Gg (Mg (x, y)) =
Gg (x) + Gg (y)
.
2
(1.3)
Below is the list of the aforementioned illustrative functions g and their corresponding
functions Gg :
• If g(x) = xρ , then Gg (x) = xρ ρ/(ρ − 1) for any ρ ∈ (0, 1)
• If g(x) = −1/x, then Gg (x) = −1/(2x)
• If g(x) = log(x), then Gg (x) = −1
• If g(x) = log(1 + x), then Gg (x) = −1
• If g(x) = −e−x , then Gg (x) = −e−x
• If g(x) = xρ , then Gg (x) = xρ ρ/(ρ − 1) for any ρ > 1
• If g(x) = ex , then Gg (x) = ex
We easily check that the above seven examples satisfy equation (1.3).
Acknowledgments
The research was partially supported by the Natural Sciences and Engineering Research
Council (NSERC) of Canada.
25
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