Journal of Inequalities in Pure and
Applied Mathematics
A PICK FUNCTION RELATED TO AN INEQUALITY
FOR THE ENTROPY FUNCTION
volume 2, issue 2, article 26,
2001.
CHRISTIAN BERG
Department of Mathematics,
University of Copenhagen,
Universitetsparken 5
DK-2100 Copenhagen, DENMARK.
EMail: berg@math.ku.dk
URL: http://www.math.ku.dk/ berg/
Received 6 November, 2000;
accepted 6 March, 2001.
Communicated by: F. Hansen
Abstract
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Abstract
The function ψ(z) = 2/(1 + z) + 1/(Log (1 − z)/2), holomorphic in the cut plane
C\[1, ∞[, is shown to be a Pick function. This leads to an integral
P representation
n
of the coefficients in the power series expansion ψ(z) = ∞
n=0 βn z , |z| < 1.
The representation shows that (βn ) decreases to zero as conjectured by F.
Topsøe. Furthermore, (βn ) is completely monotone.
A Pick Function Related to an
Inequality for the Entropy
Function
2000 Mathematics Subject Classification: 30E20, 44A60.
Key words: Pick functions, completely monotone sequences.
Christian Berg
Contents
1
Introduction and Statement of Results . . . . . . . . . . . . . . . . . . .
2
Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
5
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J. Ineq. Pure and Appl. Math. 2(2) Art. 26, 2001
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1.
Introduction and Statement of Results
In the paper [2] about bounds for entropy Topsøe considered the function
(1.1)
ψ(x) =
2
1
+ 1−x ,
1 + x ln 2
−1 < x < 1
with the power series expansion
(1.2)
ψ(x) =
∞
X
βn xn
n=0
and conjectured from numerical evidence that (βn ) decreases to zero.
The purpose of this note is to prove the conjecture by establishing the integral
representation
Z ∞
dt
, n ≥ 0.
(1.3)
βn =
tn+1 (π 2 + ln2 t−1
)
1
2
This formula clearly shows β0 > β1 > · · · > βn → 0. Furthermore, by a
change of variable we find
Z 1
ds
βn =
sn
, n ≥ 0,
s(π 2 + ln2 1−s
)
0
2s
which shows that (βn ) is a completely monotone sequence, cf. [3].
The representation (1.3) follows from the observation that ψ is the restriction
of a Pick function with the following integral representation
Z ∞
dt
(1.4)
ψ(z) =
, z ∈ C \ [1, ∞[ .
(t − z)(π 2 + ln2 t−1
)
1
2
A Pick Function Related to an
Inequality for the Entropy
Function
Christian Berg
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J. Ineq. Pure and Appl. Math. 2(2) Art. 26, 2001
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From (1.4) we immediately get (1.3) since βn = ψ (n) (0)/n!.
A Pick Function Related to an
Inequality for the Entropy
Function
Christian Berg
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J. Ineq. Pure and Appl. Math. 2(2) Art. 26, 2001
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2.
Proofs
A holomorphic function f : H → C in the upper half-plane is called a Pick
function, cf. [1], if Im f (z) ≥ 0 for all z ∈ H. Pick functions are also called
Nevanlinna functions or Herglotz functions. They have the integral representation
Z ∞
1
t
(2.1)
f (z) = az + b +
−
dµ(t) ,
t − z 1 + t2
−∞
where a ≥ 0, b ∈ R and µ is a non-negative Borel measure on R satisfying
Z
dµ(t)
< ∞.
1 + t2
a = lim f (iy)/iy , b = Re f (i) , µ = lim+
y→∞
y→0
1
Im f (t + iy)dt ,
π
where the limit refers to the vague topology. Finally f has a holomorphic extension to C \ [1, ∞[ if and only if supp (µ) ⊆ [1, ∞[.
Let Log z = ln |z| + i Arg z denote the principal logarithm in the cut plane
C\] − ∞, 0], with Arg z ∈ ]−π, π[. Hence Log 1−z
is holomorphic in C \ [1, ∞[
2
with z = −1 as a simple zero. It is easily seen that
(2.3)
ψ(z) =
Christian Berg
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It is known that
(2.2)
A Pick Function Related to an
Inequality for the Entropy
Function
2
1
+
,
1 + z Log 1−z
2
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z ∈ C \ [1, ∞[
J. Ineq. Pure and Appl. Math. 2(2) Art. 26, 2001
http://jipam.vu.edu.au
is a holomorphic extension of (1.1) with a removable singularity for z = −1
where ψ(−1) = 1/2. To see that V (z) = Im ψ(z) ≥ 0 for z ∈ H it suffices by
the boundary minimum principle for harmonic functions to verify
lim inf z→x V (z) ≥ 0 for x ∈ R and lim inf |z|→∞ V (z) ≥ 0, where in both cases
z ∈ H.
We find

x ≤ 1 (with ψ(1) = 1)
 ψ(x) ,
lim ψ(z) =
z→x
 2 +
1
,
x>1
1+x
ln x−1 −iπ
2
hence
lim V (z) =
z→x

π
π 2 +ln2
Christian Berg
x≤1

 0,
x−1
2
,
A Pick Function Related to an
Inequality for the Entropy
Function
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x > 1,
whereas lim|z|→∞ ψ(z) = 0. This shows that ψ is a Pick function, and from
(2.2) we see that a = 0 and µ has the following continuous density with respect
to Lebesgue measure

x≤1
 0,
d(x) =
 2
1 π + ln2 x−1
,
x > 1.
2
Therefore
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Z
ψ(z) = b +
1
∞
t
1
−
t − z 1 + t2
dt
π 2 + ln2
t−1
2
.
J. Ineq. Pure and Appl. Math. 2(2) Art. 26, 2001
http://jipam.vu.edu.au
In this case we can integrate term by term, and since limx→−∞ ψ(x) = 0, we
find
Z ∞
dt
ψ(z) =
2
(t − z)(π + ln2 t−1
)
1
2
and
8 ln 2
b = Re ψ(i) = 1 − 2
=
π + 4 ln2 2
which establishes (1.4).
Z
1
∞
(1 +
tdt
+ ln2
t2 )(π 2
t−1
)
2
,
A Pick Function Related to an
Inequality for the Entropy
Function
Christian Berg
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References
[1] W.F. DONOGHUE, Monotone Matrix Functions and Analytic Continuation, Berlin, Heidelberg, New York, 1974.
[2] F. TOPSØE, Bounds for entropy and divergence for distributions over a
two-element set, J. Ineq. Pure And Appl. Math., 2(2) (2001), Article 25.
http://jipam.vu.edu.au/v2n2/044_00.html
[3] D.V. WIDDER, The Laplace Transform, Princeton 1941.
A Pick Function Related to an
Inequality for the Entropy
Function
Christian Berg
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J. Ineq. Pure and Appl. Math. 2(2) Art. 26, 2001
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