# Complete characterization of symmetric Kubo-Ando operator means satisfying
Molnár’s weak associativity

Graeme W. Milton111Email<EMAIL_ADDRESS>
University of Utah
Department of Mathematics
Salt Lake City, UT, USA 84112

Aaron Welters222Email<EMAIL_ADDRESS>
Florida Institute of Technology
Department of Mathematics and Systems Engineering
Melbourne, FL, USA 32901

###### Abstract

We provide a complete characterization of a subclass of means of positive
operators in the class of symmetric Kubo-Ando means that was first introduced
and studied in L. Molnár, Characterizations of certain means of positive
operators, Linear Algebra Appl. 567 (2019) 143-166. In Theorem 6 of that
paper, he gives a characterization of this subclass (which we call the Molnár
class of means) in terms of the operator monotone functions representing the
means, which includes the geometric mean. Furthermore, he leaves open the
problem to determine if the geometric mean is the only such mean in that
subclass. Here we give an alternative characterization of the Molnár class of
means in terms of the boundary-values of bounded harmonic functions on certain
rectangles which completely characterizes this class of means. Moreover, we
use this to construct an explicit example of a mean in the subclass that is
not the geometric means thereby solving the open problem of L. Molnár.

Keywords: Arithmetic-harmonic-geometric means, Kubo-Ando means, operator
monotone functions, Herglotz-Nevanlinna functions, Molnár class of means,
bounded harmonic functions, harmonic analysis
2020 MSC: primary 47A64, 47B90; secondary 26E60

##  1 Introduction

In this paper, following [Mol19], we adopt the following notations:

  * •

$H$ always denotes a complex Hilbert space.

  * •

$B(H)$ \- the set of all bounded linear operators on $H$.

  * •

$B(H)^{+}$ \- the set of all positive semidefinite operators in $B(H)$.

  * •

$B(H)^{++}$ \- the set of all invertible operators in $B(H)^{+}$.

  * •

$I$ denotes the identity operator on $H$.

  * •

The order relation $A\leq B$ for $A,B\in B(H)^{+}$ means $B-A\in B(H)^{+}$
[i.e., $\leq$ is the Loewner order on $B(H)^{+}$].

  * •

We write $A_{n}\downarrow A$ if $(A_{n})$ is a monotonically decreasing
sequence in $B(H)^{+}$, i.e., $A_{1}\geq A_{2}\geq\ldots$, and $A_{n}$
converges strongly to $A$ in $B(H)$.

Following [KA80] (see also [Hia10] and [Sim19, Chap. 36 and 37]), a binary
operation

$\displaystyle\sigma:B(H)^{+}\times B(H)^{+}\rightarrow
B(H)^{+},\;(A,B)\mapsto\sigma(A,B)=:A\sigma B$

is called a connection on $B(H)^{+}$ if the following requirements are
fulfilled:

  * (I)

(Joint Monotonicity) $A\leq C$ and $B\leq D$ imply $A\sigma B\leq C\sigma D$;

  * (II)

(Transformer Inequality) $C(A\sigma B)C\leq(CAC)\sigma(CBC)$;

  * (III)

(Upper Semicontinuity) $A_{n}\downarrow A$ and $B_{n}\downarrow B$ imply
$A_{n}\sigma B_{n}\downarrow A\sigma B$.

Moreover, a connection $\sigma$ is called a mean (or, more precisely, a Kubo-
Ando mean) or symmetric, respectively, if

  * (IV)

(Normalization) $I\sigma I=I$;

  * (V)

(Permutation Symmetry) $A\sigma B=B\sigma A,\;\forall A,B\in B(H)^{+}$.

In particular, a symmetric Kubo-Ando mean is any binary operation $\sigma$ on
$B(H)^{+}$ satisfying (I)-(V).

Three important examples of symmetric Kubo-Ando means (which motivated the
axiomatic approach of [KA80] to positive operator means) are the arithmetic
mean $\nabla$, harmonic mean $!$, and geometric mean $\\#$ (see also [PW75,
LL01, ALM04] and the recent survey [LL24]) which, for all $A,B\in B(H)^{++}$,
are given by:

$\displaystyle A\nabla B=\frac{1}{2}(A+B);$ (1.1) $\displaystyle
A!B=2\left(A^{-1}+B^{-1}\right)^{-1};$ (1.2) $\displaystyle
A\\#B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}.$ (1.3)

The values of these means at non-invertible elements of $B(H)^{+}$ can be
determined from (III) by passing to the limit of monotonically decreasing
sequences from $B(H)^{++}$. For instance, if $A,B\in B(H)^{+}$ then the
formula for $A\nabla B$ is still (1.1), whereas if $A\in B(H)^{++},B\in
B(H)^{+}$ then the formula for $A\\#B$ is still (1.3). But in general, if
$A,B\in B(H)^{+}$ then (1.2) and (1.3) become

$\displaystyle
A!B=2\lim_{n\rightarrow\infty}\left(A_{n}^{-1}+B_{n}^{-1}\right)^{-1},$ (1.4)
$\displaystyle
A\\#B=\lim_{n\rightarrow\infty}A_{n}^{1/2}(A_{n}^{-1/2}B_{n}A_{n}^{-1/2})^{1/2}A_{n}^{1/2},$

for any two sequences $(A_{n}),(B_{n})$ in $B(H)^{++}$ such that
$A_{n}\downarrow A$ and $B_{n}\downarrow B$.

Aside from the factor of $2$, the right hand side of (1.4) has been given the
name ‘parallel sum.’ More precisely, the parallel sum is denoted by $:$ and
gives another well-known example of a connection:

$\displaystyle A:B=\frac{1}{2}A!B,\text{ for }A,B\in B(H)^{+}.$ (1.5)

The concept of the parallel sum was introduced for positive semidefinite
matrices in [AD69] and extended to bounded (positive semidefinite) linear
operators in [AT75] (see also [KA80, NA76]).

As shown in [KA80], there is an important relationship between connections,
Loewner’s theory on operator-monotone functions, and the properties of a
special class of analytic functions called Herglotz functions [GT00, Ges+01,
Tes14] (also called Nevanlinna [ADL09, Lug15], Herglotz-Nevanlinna [ABT11,
LN19, LO22, OL22], Pick [AD56, AD64, Don74, Ber08], or $R$-functions [Wig51,
Wig52, WN54, KK74], [KN77, Appendix]). Let us briefly elaborate on this.

First, a function $f$ is called a Herglotz function if
$f:\mathbb{C}^{+}\rightarrow\mathbb{C}^{+}\cup\mathbb{R}$ is analytic, where
$\mathbb{C}^{+}=\\{z\in\mathbb{C}:\operatorname{Im}z>0\\}$ denotes the open
upper half-plane. Next, let us introduce the following notation.

######  Notation 1.

Let $OM_{+}$ denote the class of all analytic functions
$f:\mathbb{C}\setminus(-\infty,0]\rightarrow\mathbb{C}$ satisfying $f(x)\geq
0$ if $x\in\mathbb{R}$ with $x>0$ and $\operatorname{Im}f(z)\geq 0$ if
$z\in\mathbb{C}$ with $\operatorname{Im}z>0$.

In particular, if $f\in OM_{+}$ then its restriction
$f|_{\mathbb{C}^{+}}:\mathbb{C}^{+}\rightarrow\mathbb{C}^{+}\cup\mathbb{R}$ to
$\mathbb{C}^{+}$ is a Herglotz function. Next, a function

$\displaystyle f:(0,\infty)\rightarrow[0,\infty)$

is called a (positive) operator monotone function if, for every Hilbert space
$H$,

$\displaystyle A,B\in B(H)^{++},\;A\leq B\Rightarrow f(A)\leq f(B),$

where $f$ is defined on $B(H)^{+}$ by the functional calculus for self-adjoint
operators on $H$ [RS80, Tes14]. A deep result of Loewner [Löw34] (see also
[Don74, Sim19]) says that every positive operator monotone function $g$ has a
unique analytic continuation to a function $f\in OM_{+}$ [or, equivalently,
$f(x)=g(x)$ for all $x>0$] and conversely. Given this correspondence between
positive operator monotone functions and the elements of $OM_{+}$, we will
abuse notation throughout the rest of this paper and not distinguish them
unless necessary.

More precisely, the map $f\mapsto f|_{(0,\infty)}$ sending $f\in OM_{+}$ to
its restriction $f|_{(0,\infty)}:(0,\infty)\rightarrow[0,\infty)$ is a
bijection of $OM_{+}$ onto the class of positive operator monotone functions.
Furthermore, the map $m\mapsto f$, defined by

$\displaystyle
f(z)=a+bz+\int_{(0,\infty)}\frac{z(1+\lambda)}{z+\lambda}dm(\lambda),\;\text{for
}z\in\mathbb{C}\setminus(-\infty,0],$ (1.6)

where $a=m(\\{0\\})$ and $b=m(\\{\infty\\})$, establishes a bijection between
the class of finite (positive) Borel measures on $[0,\infty]$ onto this class
$OM_{+}$ of functions.

Now a result of [KA80] says that for any Hilbert space $H$, every positive
operator monotone function $f$ gives rise to a unique connection
$\sigma=\sigma_{f}$ on $B(H)^{+}$ by the formula

$\displaystyle A\sigma B=A^{1/2}f(A^{-1/2}BA^{-1/2})A^{1/2},$

for all $B\in B(H)^{+},A\in B(H)^{++}$ which is extended uniquely to all
$B(H)^{+}$ by property (III). In particular, $f$ can be recovered from
$\sigma$ by the formula

$\displaystyle f(x)I=I\sigma(xI),\;\text{for }x>0.$ (1.7)

The function $f$ is called the representing function of $\sigma$. Furthermore,
if $H$ is infinite dimensional then by a deep result of [KA80], each and every
connection $\sigma$ on $B(H)^{+}$ arises in this way, i.e.,
$\sigma=\sigma_{f}$ for a unique operator monotone function $f$. Moreover, by
[KA80, Theorem 3.4] (see also [KA80, Lemma 3.1]) it is known that, in terms of
the representation (1.6),

$\displaystyle A\sigma
B=aA+bB+\int_{(0,\infty)}\frac{1+\lambda}{\lambda}[(\lambda
A):B]dm(\lambda),\;\text{for }A,B\in B(H)^{+},$

where $a=m(\\{0\\})$, $b=m(\\{\infty\\})$, $:$ denotes the parallel sum as
def. by (1.5), and this establishes a bijection, $m\mapsto\sigma$, between the
class of finite (positive) Borel measures on $[0,\infty]$ onto the class of
connections.

For instance, by (1.7) and [KA80, Corollary 4.2], it is known that a
representing function $f$ corresponding to a connection $\sigma$ that is a
mean or symmetric is one which satisfies, respectively,

$\displaystyle f(1)=1;\quad f(x)=xf\left(\frac{1}{x}\right),\;\text{for }x>0.$

For example, the representing functions $f_{\nabla},f_{!},f_{\\#}$ of the
arithmetic mean $\nabla,$ harmonic mean $!$, and geometric mean $\\#$,
respectively, are given by

$\displaystyle f_{\nabla}(x)=\frac{1}{2}(1+x);$ $\displaystyle
f_{!}(x)=\frac{2x}{1+x};$ $\displaystyle f_{\\#}(x)=\sqrt{x}.$

A natural question arises that was considered by Kubo-Ando [KA80, Sec. 4 and
5] and, more recently, by L. Molnár [Mol19]: What properties of a symmetric
Kubo-Ando mean $\sigma=\sigma_{f}$ characterize it as the arithmetic mean
$\nabla$, harmonic mean $!$, or geometric mean $\\#$? Equivalently, in terms
of representing functions $f$ (or, equivalently, on operator monotone
functions $f$), what are necessary and sufficient conditions that guarantee
$f\in\\{f_{\nabla},f_{!},f_{\\#}\\}$?

To address this question, L. Molnár [Mol19] considered an algebraic
characterization of those means in terms of a weak form of an associativity
law of a binary operation on $B(H)^{++}$. His main results in this regard can
be summarized as follows (see [Mol19, Theorems 6 and 8]).

######  Theorem 2 (L. Molnár).

Let $H$ be a complex Hilbert space with $\dim H\geq 2$ and $\sigma$ be a
symmetric Kubo-Ando mean on $B\left(H\right)^{++}$ with representing operator
monotone function $f$. Assume that there exists a continuous strictly
increasing and surjective function $g:(0,\infty)\rightarrow(0,\infty)$ such
that the operation

$\displaystyle\diamond:\left(A,B\right)\mapsto g\left(A\sigma
B\right),\;\text{for }A,B\in B\left(H\right)^{++}$

is either associative, i.e.,

$\left(A\diamond C\right)\diamond B=A\diamond\left(C\diamond B\right),\text{
}\forall A,B,C\in B\left(H\right)^{++},$ (1.8)

or satisfies the weaker form of associativity

$\left(A\diamond I\right)\diamond B=A\diamond\left(I\diamond B\right),\text{
}\forall A,B\in B\left(H\right)^{++}.$ (1.9)

If (1.8) is satisfied then $\sigma$ is the arithmetic or harmonic mean. On the
other hand, if it satisfies (1.9) then either we have
$g\left(f\left(x\right)\right)=x$, $x>0$ [meaning that $A\diamond I=I\diamond
A=A$, $A\in B\left(H\right)^{++}$] or we have one of the following three
possibilities:

  * (a)

there is a positive scalar $c\not=1$ such that
$f\left(c^{2}x\right)=cf\left(x\right)$, for $x>0$;

  * (b)

$\sigma$ is the arithmetic mean;

  * (c)

$\sigma$ is the harmonic mean.

He also proves (see [Mol19, pp. 160-161]) the following partial converse of
this theorem.

######  Theorem 3 (L. Molnár).

If $\sigma$ is a symmetric Kubo-Ando mean with representing operator monotone
function $f$ such that (a), (b), or (c) is true in Theorem 2 then there is a
continuous strictly increasing and surjective function
$g:(0,\infty)\rightarrow(0,\infty)$ such that the operation
$\diamond:\left(A,B\right)\mapsto g\left(A\sigma B\right)$, $A,B\in
B\left(H\right)^{++}$ satisfies (1.9).

This motivates the following definition of a special class of positive
operator means considered by L. Molnár.

######  Definition 4 (Molnár mean).

A symmetric Kubo-Ando mean $\sigma$ is called a Molnár mean if its
representing function $f$ has property (a) in Theorem 2. The set of all Molnár
means will be called the Molnár class of means.

Notice that the geometric mean $\\#$ is a Molnár mean whose representing
function $f_{\\#}$ has the property $\lim_{x\rightarrow 0^{+}}f_{\\#}(x)=0$.
Because of this, L. Molnár [Mol19, p. 161] posed the following problems, which
we have rephrased in terms of Definition 4.

######  Problem 5 (L. Molnár).

Is the geometric mean $\\#$ the only Molnár mean? If $\sigma$ is a Molnár mean
with representing function $f$ satisfying $\lim_{x\rightarrow 0^{+}}f(x)=0$,
does this imply $\sigma=\\#$?

In [Mol19, p. 161], L. Molnár says in regard to the last part of the question
above the following which motivated our paper: “However, we still do not know
if the answer to the question is positive or negative. If it were affirmative,
then we would get an interesting common characterization of the three
fundamental operator means, the arithmetic, harmonic and geometric means.” We
are able to answer his question (in the negative) by proving the following
theorem:

######  Theorem 6 (First Main Result).

There are infinitely many Molnár means in the subclass having a representing
function $f$ that satisfies $\lim_{x\rightarrow 0^{+}}f(x)=0$.

The main goal of our paper is to prove this and, furthermore, to completely
characterize the class of Molnár means in terms of their representing
functions which we do with Theorem 21. This theorem is stated and proved at
the end of Section 2. Then, in Section 3, we construct a whole class of
examples of Molnár means using the theory developed in Sec. 2. A summary of
our approach, by reducing the problem via a series of invertible
transformations, is illustrated in Fig. 1.

Figure 1: A Molnár mean $\sigma=\sigma_{f}$ corresponds to a representing
function $f\in\mathcal{M}_{c}$ for some $c\in(1,\infty)$. As represented in
the transition from (a) to (b) in the figure, by using the invertible
transform $w=\log z$ (with inverse $z=e^{w}$) from
$\mathbb{C}\setminus(-\infty,0]$ onto the strip
$D=\\{w\in\mathbb{C}:|\operatorname{Im}w|<\pi\\}=\mathbb{R}\times(-\pi,\pi)$,
the analytic functions $f\in\mathcal{M}_{c}$ are mapped bijectively onto the
analytic functions $S\in\mathcal{W}_{p}$ via $W_{p}(f)=S$, where $p=2\log c$.
Next, in terms of the open rectangle $\Omega_{p}=(0,p/2)\times(0,\pi)$, the
restriction map $V=\operatorname{Im}S|_{\Omega_{p}}$ defines a bijection
$R_{p}:\mathcal{W}_{p}\rightarrow\mathcal{V}_{p}$ into a subclass of bounded
harmonic functions $h^{\infty}(\Omega_{p})$, $\mathcal{V}_{p}\subseteq\\{V\in
h^{\infty}(\Omega_{p}):||V||_{\infty}\leq\pi/2\\}$, which is represented in
going from (b) to (c) in the figure. Using the invertibility of the Poisson
integral operator $P_{\Omega_{p}}$ and of the zero extension map
$E_{p}:\mathcal{H}_{p}\rightarrow P_{\Omega_{p}}^{-1}(\mathcal{V}_{p})$ on
$\mathcal{H}_{p}=\\{h\in
L^{\infty}([0,p/2],dx,\mathbb{R}):||h||_{\infty}\leq\pi/2\\}$, we obtain a
bijection $P_{\Omega_{p}}\circ
E_{p}:\mathcal{H}_{p}\rightarrow\mathcal{V}_{p}$ which can be described
alternatively as follows [as illustrated in the figure by going from (c) to
(d)]: If $V\in\mathcal{V}_{p}$ then $V=P_{\Omega_{p}}(v)$ is the unique
solution to the (generalized) Dirichlet problem for the Laplacian on
$\Omega_{p}$, i.e., $\Delta V=0$ on $\Omega_{p}$ and $V=v$ on
$\partial\Omega_{p}$, in which $v=E_{p}(h)$, i.e., $v=0$ on
$\partial\Omega_{p}\setminus[0,p/2]\times\\{\pi\\}$ and $v(x,\pi)=h(x)$ for
$x\in[0,p/2]$, where $h\in\mathcal{H}_{p}$. All of these transformations
achieve our goal, namely, a bijection from $\mathcal{H}_{p}$ onto
$\mathcal{M}_{c}$ via the composition of the following bijections:
$M_{c}=W_{p}^{-1}\circ R_{p}^{-1}\circ P_{\Omega_{p}}\circ
E_{p}:\mathcal{H}_{p}\rightarrow\mathcal{M}_{c}$. In addition, if
$h\in\mathcal{H}_{p}$ then the unique Molnár mean $\sigma=\sigma_{f}$ with
representing function $f=M_{c}(h)$ has the property that $\lim_{x\rightarrow
0^{+}}f(x)=0$. Finally, as the geometric mean $\\#$ has the representing
function $f_{\\#}=M_{c}(0)$, this provides a summary of our complete
characterization of Molnár means and solution to the open problem of L.
Molnár.

##  2 Characterization of the Molnár class of means

In this section we completely characterize the Molnár class of means in terms
of their representing functions. The next lemma is an important first step in
this regard (whose proof is immediate from our discussion above) and motivates
the definition that follows. Also, as mentioned in the introduction, we will
abuse notation and not distinguish between elements of $OM_{+}$ and positive
operator monotone functions, unless necessary.

######  Lemma 7.

A connection $\sigma$ with representing function $f$ (i.e.,
$\sigma=\sigma_{f}$) is a Molnár mean if and only if all the following
statements hold:

  * (i)

$f\in OM_{+}$;

  * (ii)

$f(1)=1$;

  * (iii)

$xf(1/x)=f(x),\text{ for }x>0$;

  * (iv)

there exists a positive scalar $c\not=1$ such that
$f\left(c^{2}x\right)=cf\left(x\right)$, for $x>0$.

######  Definition 8 (Molnár class of functions).

A function $f$ having properties (i)-(iv) in Lemma 7 will be called a Molnár
function. For a positive scalar $c\not=1$, we say $f$ is a Molnár function of
type $c$ if it is a Molnár function and $f(c^{2}x)=cf(x),$ for $x>0$. The set
of all Molnár functions of type $c$ will be denoted by $\mathcal{M}_{c}$ and
the Molnár class of functions is
$\mathcal{M}\equiv\cup_{c\in(0,\infty)\setminus\\{1\\}}\mathcal{M}_{c}$.

As a consequence of the following lemma, we need only consider those scalars
$c$ in $(1,\infty)$ if we want to characterize Molnár means using the Molnár
class $\mathcal{M}$.

######  Lemma 9.

For any positive scalar $c\not=1$,

$\displaystyle\mathcal{M}_{c}=\mathcal{M}_{1/c}.$

In particular,

$\displaystyle\mathcal{M}=\cup_{c\in(1,\infty)}\mathcal{M}_{c}=\cup_{c\in(0,1)}\mathcal{M}_{c}.$

###### Proof.

Let $c$ be a positive scalar $c\not=1$. If $f(c^{2}x)=cf(x)$ for all $x>0$
then $x>0$ implies $c^{-2}x>0$ and hence $f(x)=f(c^{2}(c^{-2}x))=cf(c^{-2}x)$
so that $f((c^{-1})^{2}x)=c^{-1}f(x)$ for all $x>0$. ∎

Next, we denote the principal branch of the logarithm and square root by
$\log(\cdot)$ and $\sqrt{\cdot}$, i.e.,

$\displaystyle\log(z)=\log|z|+i\operatorname{Arg}(z),\;\operatorname{Arg}(z)\in(-\pi,\pi),\;z\not=0,$
$\displaystyle\sqrt{z}=e^{\frac{1}{2}\log(z)},\;z\in\mathbb{C}\setminus(-\infty,0].$

Now have some preliminary results.

######  Lemma 10.

Let $f\in OM_{+}\setminus\\{0\\}$. Then

$\displaystyle\overline{f(\overline{z})}=f(z)\text{ and }f(z)\not=0,\text{ if
}z\in\mathbb{C}\setminus(-\infty,0].$

In addition,

$\displaystyle 0\leq\operatorname{Arg}f(z)\leq\operatorname{Arg}z<\pi,\text{
if }\operatorname{Im}z>0.$ (2.1)

###### Proof.

Let $f\in OM_{+}\setminus\\{0\\}$. First, as $\overline{f(\overline{x})}=f(x)$
for all $x>0$, it follows by analyticity that
$\overline{f(\overline{z})}=f(z)$ for all
$z\in\mathbb{C}\setminus(-\infty,0]$. Next, it follows from the integral
representation (1.6) of $f$ that $f(z)\not=0$ for all
$z\in\mathbb{C}\setminus(-\infty,0]$ and that $f_{\perp}\in
OM_{+}\setminus\\{0\\}$, where
$f_{\perp}:\mathbb{C}\setminus(-\infty,0]\rightarrow\mathbb{C}$ is defined by
(cf. [KA80, Corollary 4.3] and [Hia10, p. 194] on the dual of a non-zero
connection):

$\displaystyle f_{\perp}(z)=\frac{z}{f(z)},\text{ for
}z\in\mathbb{C}\setminus(-\infty,0].$

As $f,f_{\perp}\in OM_{+}\setminus\\{0\\}$, then the compositions $\log\circ
f$ and $\log\circ f_{\perp}$ are analytic functions on
$\mathbb{C}\setminus(-\infty,0]$ satisfying $(\log\circ
f_{\perp})(x)=\log(x)-\log f(x)$ for all $x>0$. By analyticity this implies
$\log\circ f_{\perp}=\log-\log\circ f$ on $\mathbb{C}\setminus(-\infty,0]$ so
that

$\displaystyle 0\leq\operatorname{Im}[\log
f_{\perp}(z)]=\operatorname{Im}(\log z)-\operatorname{Im}[\log
f(z)]=\operatorname{Arg}z-\operatorname{Arg}f(z)$

for every $z\in\mathbb{C}\setminus(-\infty,0]$. This, and the fact
$\operatorname{Im}f(z)\geq 0$ if $\operatorname{Im}z>0$, imply the
inequalities (2.1) hold. ∎

######  Lemma 11.

If $f\in OM_{+}\setminus\\{0\\}$ then the function $S:D\rightarrow\mathbb{C}$
defined by

$\displaystyle S(w)=\log f(e^{w})-\frac{1}{2}w,\text{ for }w\in D,$ (2.2)
$\displaystyle D=\\{w\in\mathbb{C}:-\pi<\operatorname{Im}w<\pi\\},$ (2.3)

is analytic on $D$ and satisfies

$\displaystyle|\operatorname{Im}S(w)|\leq\frac{1}{2}\operatorname{Im}w,\;\text{if
}0\leq\operatorname{Im}w<\pi,$ (2.4) $\displaystyle f(z)=\sqrt{z}e^{S(\log
z)},\;\text{for }z\in\mathbb{C}\setminus(-\infty,0].$ (2.5)

Conversely, if $S:D\rightarrow\mathbb{C}$ is an analytic function satisfying
(2.4) then the function
$f:\mathbb{C}\setminus(-\infty,0]\rightarrow\mathbb{C}$ defined by (2.5) is in
$OM_{+}\setminus\\{0\\}$ and $S$ is given in terms of $f$ by (2.2).

###### Proof.

($\Rightarrow$): Let $f\in OM_{+}\setminus\\{0\\}$. Then by Lemma 10,
$f:\mathbb{C}\setminus(-\infty,0]\rightarrow\mathbb{C}$ is an analytic
function with range
$f(\mathbb{C}\setminus(-\infty,0])\subseteq\mathbb{C}\setminus(-\infty,0]$ so
that the function $z\mapsto\log f(z)$ is analytic on the domain
$\mathbb{C}\setminus(-\infty,0]$. Next, the function $w\mapsto e^{w}$ is
analytic on the domain $D$ with range $e^{D}=\mathbb{C}\setminus(-\infty,0]$.
It follows that the composition of functions $w\mapsto\log f(e^{w})$ is well-
defined and analytic on $D$. From this it follows immediately that the
function $S:D\rightarrow\mathbb{C}$ defined by (2.2), (2.3) is analytic on $D$
and satisfies (2.5). Also, if $\operatorname{Im}w=0$ then $w\in D$ and since
$f(x)\geq 0$ for $x>0$, then by (2.2) we have
$\operatorname{Im}S(w)=\operatorname{Arg}f(e^{\operatorname{Re}w})=0$. To
complete the proof of (2.4), we note that by the hypotheses on $f$ it follows
from Lemma 10 that $0\leq\operatorname{Arg}f(z)\leq\operatorname{Arg}z<\pi$ if
$\operatorname{Im}z>0$ and since

$\displaystyle\operatorname{Arg}e^{w}=\operatorname{Im}w,\;\operatorname{Im}e^{w}=e^{\operatorname{Re}w}\sin(\operatorname{Im}w)>0,\;\text{if
}0<\operatorname{Im}w<\pi,$

it follows that

$\displaystyle-\frac{1}{2}\operatorname{Im}w\leq\operatorname{Im}S(w)=\operatorname{Arg}f(e^{w})-\frac{1}{2}\operatorname{Im}w\leq\frac{1}{2}\operatorname{Im}w,\;\text{if
}0<\operatorname{Im}w<\pi,$

which proves (2.4).

($\Leftarrow$): Conversely, suppose $S:D\rightarrow\mathbb{C}$ is an analytic
function [where $D$ is defined by (2.3)] satisfying (2.4). Let
$f:\mathbb{C}\setminus(-\infty,0]\rightarrow\mathbb{C}$ be the function
defined by (2.5). As $\log z$ and $\sqrt{z}$ are analytic on the domain
$\mathbb{C}\setminus(-\infty,0]$ with $\log z\in D$ for
$z\in\mathbb{C}\setminus(-\infty,0]$ then it follows that
$f:\mathbb{C}\setminus(-\infty,0]\rightarrow\mathbb{C}$ is analytic and cannot
be the zero function since $f(z)=\sqrt{z}e^{S(\log z)}\not=0$ for all $z$ in
the domain of $f$. Next, (2.4) implies that $S(\log x)\in\mathbb{R}$ for $x>0$
so that $f(x)=\sqrt{x}e^{S(\log x)}>0$ for $x>0$. Also, (2.4) implies that for
$\operatorname{Im}z>0$ we have

$\displaystyle 0\leq\operatorname{Im}\left[S(\log z)+\frac{1}{2}\log
z\right]\leq\operatorname{Im}\log z=\operatorname{Arg}z<\pi$ (2.6)

and hence

$\displaystyle f(z)=\sqrt{z}e^{S(\log z)}=e^{S(\log z)+\frac{1}{2}\log z},$
$\displaystyle\operatorname{Im}f(z)=e^{\operatorname{Re}[S(\log
z)+\frac{1}{2}\log z]}\sin\left\\{\operatorname{Im}\left[S(\log
z)+\frac{1}{2}\log z\right]\right\\}\geq 0.$

Now to complete the proof, it remains only to prove that the function
$S:D\rightarrow\mathbb{C}$ is given in terms of $f$ by the formula (2.2). It
follows by (2.5) that there exists a constant $m\in\mathbb{Z}$ such that

$\displaystyle\log f(z)=S(\log z)+\frac{1}{2}\log z+i2\pi m,\text{ for
}z\in\mathbb{C}\setminus(-\infty,0].$ (2.7)

This implies that

$\displaystyle 2\pi m+\operatorname{Im}\left[S(\log z)+\frac{1}{2}\log
z\right]=\operatorname{Im}\log f(z)=\operatorname{Arg}f(z),\text{ for
}z\in\mathbb{C}\setminus(-\infty,0].$

As we know that if $\operatorname{Im}z>0$ then (2.6) holds and
$\operatorname{Arg}f(z)\in[0,\pi)$, this implies $m=0$. Hence, from (2.7) it
follows that

$\displaystyle\log f(e^{w})=S(\log e^{w})+\frac{1}{2}\log
e^{w}=S(w)+\frac{1}{2}w,\text{ for }w\in D,$

which proves the equality (2.2). This completes the proof. ∎

######  Lemma 12.

Let $f\in OM_{+}\setminus\\{0\\}$ and denote by $S:D\rightarrow\mathbb{C}$ the
corresponding function defined by (2.2). Then the following statements are
true:

  * (i)

$f(1)=1$ if and only if $S(0)=0$.

  * (ii)

$f(x)=xf(1/x),$ for $x>0$ if and only if $S(-w)=S(w),\text{ for }w\in D$.

  * (iii)

If $c\in(1,\infty)$ then $f\left(c^{2}x\right)=cf\left(x\right)$, for $x>0$ if
and only if $S(w+2\log c)=S(w),$ for $w\in D$ [i.e., $S$ is periodic with a
period $2\log c$], in which case

$\displaystyle\lim_{x\rightarrow 0^{+}}f(x)=0.$

###### Proof.

(i): If $f(1)=1$ then $S(0)=\log f(e^{0})-\frac{1}{2}(0)=\log 1=0.$
Conversely, if $S(0)=0$ then $f(1)=\sqrt{1}e^{S(\log 1)}=e^{S(0)}=1$.

(ii): Suppose $f(x)=xf(1/x)$ for all $x>0$. If $w\in\mathbb{R}$ then

$\displaystyle S(-w)=\log
f(e^{-w})-\frac{1}{2}(-w)=\log[e^{-w}f(e^{w})]+\frac{1}{2}w$
$\displaystyle=\log[f(e^{w})]+\log[e^{-w}]+\frac{1}{2}w=\log[f(e^{w})]-\frac{1}{2}w=S(w).$

By the Riemann identity theorem [Con78, Corollary 3.8, p. 79], it follows that
$S(w)=S(-w)$ for all $w\in D$. Conversely, suppose that $S(w)=S(-w)$ for all
$w\in D$. Then, for every $x>0$,

$\displaystyle xf(1/x)=x\sqrt{x^{-1}}e^{S(\log(x^{-1}))}=\sqrt{x}e^{S(-\log
x)}=\sqrt{x}e^{S(\log x)}=f(x).$

(iii): Let $c\in(1,\infty)$. Suppose $f\left(c^{2}x\right)=cf\left(x\right)$
for all $x>0$. If $w\in\mathbb{R}$ then

$\displaystyle S(w+2\log c)=\log f(e^{w+2\log c})-\frac{1}{2}(w+2\log c)$
$\displaystyle=\log f(c^{2}e^{w})-\log c-\frac{1}{2}w=\log[cf(e^{w})]-\log
c-\frac{1}{2}w$ $\displaystyle=\log c+\log f(e^{w})-\log c-\frac{1}{2}w=S(w).$

By the Riemann identity theorem [Con78, Corollary 3.8, p. 79], it follows that
$S(w+2\log c)=S(w)$ for all $w\in D$. Conversely, suppose $S(w+2\log c)=S(w)$
for all $w\in D$. Then, for every $x>0$,

$\displaystyle f(c^{2}x)=\sqrt{c^{2}x}e^{S(\log(c^{2}x))}=c\sqrt{x}e^{S(\log
x+2\log c)}=c\sqrt{x}e^{S(\log x)}=cf(x).$

Thus, in particular, if this the case then the restriction
$S|_{\mathbb{R}}:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous periodic
function on $\mathbb{R}$ implying it is bounded so that $x\mapsto e^{S(\log
x)}$ is a bounded function on $(0,\infty)$ and therefore,

$\displaystyle\lim_{x\rightarrow 0^{+}}f(x)=\lim_{x\rightarrow
0^{+}}\sqrt{x}e^{S(\log x)}=0.$ ∎

These results motivate the following definition and proposition.

######  Definition 13 ($\mathcal{W}$-functions).

An analytic function $S:D\rightarrow\mathbb{C}$ having the properties:

  * (i)

$|\operatorname{Im}S(w)|\leq\frac{1}{2}\operatorname{Im}w,\;\text{if
}0\leq\operatorname{Im}w<\pi,$

  * (ii)

$S(0)=0$,

  * (iii)

$S(-w)=S(w)$ for all $w\in D$,

  * (iv)

there exists a scalar $p>0$ such that $S(w+p)=S(w)$ for all $w\in D$,

will be called a $\mathcal{W}$-function with period $p$. The set of all
$\mathcal{W}$-functions with period $p$ will be denoted by $\mathcal{W}_{p}$
and the class of all $\mathcal{W}$-functions is
$\mathcal{W}=\cup_{p\in(0,\infty)}\mathcal{W}_{p}$.

######  Proposition 14.

For each $f\in\mathcal{M}$, denote by $S_{f}$ the function $S$ defined by
(2.2) and (2.3). Then the map

$\displaystyle W:\mathcal{M}\rightarrow\mathcal{W},$ $\displaystyle
W(f)=S_{f},\;f\in\mathcal{M}$

is a bijection from $\mathcal{M}$ onto $\mathcal{W}$ such that

$\displaystyle f\in\mathcal{M}_{c}\Leftrightarrow S_{f}\in\mathcal{W}_{p},$

where $c\in(1,\infty),p\in(0,\infty)$ are related by the formula:

$\displaystyle p=2\log c.$

In particular, the restriction map $W_{p}$ defined by

$\displaystyle
W_{p}=W|_{\mathcal{M}_{c}}:\mathcal{M}_{c}\rightarrow\mathcal{W}_{p},$
$\displaystyle W_{p}(f)=W(f)=S_{f},\;f\in\mathcal{M}_{c}$

is a bijection from $\mathcal{M}_{c}$ onto $\mathcal{W}_{p}$.

###### Proof.

The proof of this proposition follows immediately from Definitions 8 and 13
and Lemmas 7, 11, and 12. ∎

Our next goal is to characterize the class of all $\mathcal{W}$-functions in
terms of the boundary values of their imaginary parts. In order to do this we
begin by recalling some basic notations and a theorem from harmonic analysis
that we need. For this, we’ve collected together some well-known results from
multiple sources (see [Dur70, Con78, Pom92, Con95, GM05]) as no single source
had the precise statement we need. After this we proceed with our
characterization.

######  Notation 15.

Let $\Omega$ be an open set in $\mathbb{C}$. Denote the Laplacian of a real-
valued function $V:\Omega\rightarrow\mathbb{R}$ with continuous second
derivatives on $\Omega$ by

$\displaystyle\Delta
V=\frac{\partial^{2}V}{dx^{2}}+\frac{\partial^{2}V}{dy^{2}}$

and say $V$ is harmonic on $\Omega$ if satisfies the Laplace equation, i.e.,

$\displaystyle\Delta V=0\text{ on }\Omega.$

The set of all harmonic functions on $\Omega$ is denoted by

$\displaystyle h(\Omega)=\\{V\;|\;V:\Omega\rightarrow\mathbb{R},\;\Delta
V=0\text{ on }\Omega\\}$

and the subset of all bounded harmonic functions on $\Omega$ is denoted by

$\displaystyle h^{\infty}(\Omega)=\\{V\in
h(\Omega):\sup_{w\in\Omega}|V(w)|<\infty\\}$

which is a Banach space with norm

$\displaystyle||V||_{\infty}=\sup_{w\in\Omega}|V(w)|,\;\text{ for }V\in
h^{\infty}(\Omega).$

Let $\partial\Omega$ denote the boundary of $\Omega$. For any
$\zeta\in\partial\Omega,\theta\in[0,2\pi),\beta\in(0,\pi/2)$, and
$\varepsilon>0$, we define (truncated cone at $\zeta$)

$\displaystyle\Gamma_{\beta}^{\varepsilon}(\zeta,\theta)=\\{\zeta+re^{i(\theta+\phi)}:0<r<\varepsilon,|\phi|<\beta\\}.$

We say that $\partial\Omega$ has an inner tangent (with inner normal
$e^{i\theta}$) at $\zeta\in\partial\Omega$ if for every $\beta\in(0,\pi/2)$
there exists an $\varepsilon=\varepsilon(\beta)>0$ such that

$\displaystyle\Gamma_{\beta}^{\varepsilon}(\zeta)=\Gamma_{\beta}^{\varepsilon}(\zeta,\theta)\subseteq\partial\Omega.$

A function $V\in h(\Omega)$ is said to have a nontangential (n.t.) limit $L$
at $\zeta\in\partial\Omega$ if $\partial\Omega$ has an inner tangent at
$\zeta$ such that

$\displaystyle\lim_{\Gamma_{\beta}^{\varepsilon}(\zeta)\ni
w\rightarrow\zeta}V(w)=L,\;\text{for every }\beta\in(0,\pi/2)$

in which case we write

$\displaystyle\lim_{\Omega\ni
w\overset{\text{n.t.}}{\longrightarrow}\zeta}V(w)=L.$

The boundary $\partial\Omega$ of $\Omega$ is said to be a Jordan curve (i.e.,
a simple closed curve) if there exists a continuous function
$\gamma:[a,b]\rightarrow\mathbb{C}$ such that
$\gamma([a,b))=\partial\Omega,\gamma(a)=\gamma(b),$ and
$\gamma(x)\not=\gamma(y)$ whenever $x\not=y$ and $x,y\in[a,b)$. In addition,
if the curve $\gamma$ is rectifiable (i.e., has finite arclength) then
$\partial\Omega$ is said to be a rectifiable Jordan curve.

######  Theorem 16 (Solvability of the Dirichlet problem).

Let $\Omega$ be an open, bounded, and simply connected set in $\mathbb{C}$
whose boundary $\partial\Omega$ is a rectifiable Jordan curve. Then the
following statements are true:

  * (a)

For each $v\in L^{\infty}(\partial\Omega,ds,\mathbb{R})$, there is a unique
solution $V\in h(\Omega)$ to the (generalized) Dirichlet problem for $v$ on
$\Omega$, i.e.,

$\displaystyle\Delta V=0\text{ on }\Omega,$ (2.8)
$\displaystyle\lim_{\Omega\ni
w\overset{\text{n.t.}}{\longrightarrow}\zeta}V(w)=v(\zeta),\;\text{for almost
every }\zeta\in\partial\Omega.$ (2.9)

In addition,

$\displaystyle V\in h^{\infty}(\Omega),$
$\displaystyle\sup_{w\in\Omega}|V(w)|=||V||_{\infty}=||v||_{\infty}.$

Also, if $v:\partial\Omega\rightarrow\mathbb{R}$ is continuous at
$\zeta\in\partial\Omega$, then

$\displaystyle\lim_{\Omega\ni w\rightarrow\zeta}V(w)=v(\zeta).$

  * (b)

The map $P_{\Omega}:L^{\infty}(\partial\Omega,ds,\mathbb{R})\rightarrow
h^{\infty}(\Omega)$ defined by $P_{\Omega}(v)=V$, where $V$ is the solution to
the Dirichlet problem for $v$ on $\Omega$, is an invertible bounded linear
operator, from the Banach space $L^{\infty}(\partial\Omega,ds,\mathbb{R})$
onto the Banach space $h^{\infty}(\Omega)$, which is both norm and order
preserving.

######  Definition 17.

The Poisson integral operator on $\Omega$ is defined as the operator
$P_{\Omega}:L^{\infty}(\partial\Omega,ds,\mathbb{R})\rightarrow
h^{\infty}(\Omega)$ defined in Theorem 16.(b). In particular, for each
$p\in(0,\infty)$,

$\displaystyle
P_{\Omega_{p}}:L^{\infty}(\partial\Omega_{p},ds,\mathbb{R})\rightarrow
h^{\infty}(\Omega_{p})$ (2.10)

will denote the Poisson integral operator on the open rectangle

$\displaystyle\Omega_{p}=\left(0,\frac{p}{2}\right)\times(0,\pi).$ (2.11)

######  Corollary 18.

Let $p\in(0,\infty)$, $S\in\mathcal{W}_{p}$, and define the restriction

$\displaystyle
V=\operatorname{Im}S|_{\Omega_{p}}:\Omega_{p}\rightarrow\mathbb{R}$

where $\Omega_{p}$ is the open rectangle defined by (2.11). Then

$\displaystyle V\in h(\Omega_{p}),$ (2.12)
$\displaystyle\sup_{w\in\Omega_{p}}|V(w)|\leq\frac{\pi}{2},$ (2.13)
$\displaystyle\lim_{\Omega_{p}\ni w\rightarrow\zeta}V(w)=0,\;\text{for every
}\zeta\in\partial\Omega_{p}\setminus\left[0,\frac{p}{2}\right]\times\\{\pi\\}.$
(2.14)

Furthermore, there exists a function
$v:\partial\Omega_{p}\rightarrow\mathbb{R}$ with the properties

$\displaystyle v\in L^{\infty}(\partial\Omega_{p},ds,\mathbb{R}),$ (2.15)
$\displaystyle||v||_{\infty}\leq\frac{\pi}{2},$ (2.16) $\displaystyle
v(\zeta)=0,\;\text{for all
}\zeta\in\partial\Omega_{p}\setminus\left[0,\frac{p}{2}\right]\times\\{\pi\\}$
(2.17)

such that

$\displaystyle V=P_{\Omega_{p}}(v),$ (2.18)

where $P_{\Omega_{p}}$ is the Poisson integral operator (2.10) on
$\Omega_{p}$. Moreover, there exists a unique function

$\displaystyle h:\left[0,\frac{p}{2}\right]\rightarrow\mathbb{R}$

such that

$\displaystyle
v(\zeta)=\left\\{\begin{array}[]{cl}h(\operatorname{Re}\zeta),&\text{if
}\operatorname{Re}\zeta\in\left[0,\frac{p}{2}\right]\text{and
}\operatorname{Im}\zeta=\pi,\\\ 0,&\text{if
}\zeta\in\partial\Omega_{p}\setminus\left[0,\frac{p}{2}\right]\times\\{\pi\\},\end{array}\right.$
(2.21)

for every $\zeta\in\Omega_{p}$. In addition,

$\displaystyle h(x)=v(x+i\pi),\forall x\in\left[0,\frac{p}{2}\right],$ (2.22)
$\displaystyle h\in
L^{\infty}\left(\left[0,\frac{p}{2}\right],dx,\mathbb{R}\right),$ (2.23)
$\displaystyle||h||_{\infty}\leq\frac{\pi}{2}.$ (2.24)

###### Proof.

Fix $p\in(0,\infty)$ and $S\in\mathcal{W}_{p}$. Then, by Def. 13,
$S:D\rightarrow\mathbb{C}$ is an analytic function on the open region $D$
which implies that its imaginary part
$\operatorname{Im}S:D\rightarrow\mathbb{R}$ is a harmonic function on $D$.
Hence as $\Omega_{p}\subseteq D$ and $\Omega_{p}$ is also open region, then
the restriction of $\operatorname{Im}S$ to $\Omega_{p}$, i.e., the function
$V=\operatorname{Im}S|_{\Omega_{p}}:\Omega_{p}\rightarrow\mathbb{R}$, is a
harmonic function on $\Omega_{p}$, that is, according to Notation 15, $V\in
h(\Omega_{p})$. Next, since $S\in\mathcal{W}_{p}\subseteq\mathcal{W}$, it
follows from Def. 13 that
$|\operatorname{Im}S|\leq\frac{1}{2}\operatorname{Im}w$ if
$0\leq\operatorname{Im}w<\pi$ from which it follows that

$\displaystyle|V(w)|=|\operatorname{Im}S|\leq\frac{1}{2}\operatorname{Im}w<\frac{\pi}{2},\;\forall
w\in\Omega_{p}$

which implies that $\sup_{w\in\Omega_{p}}|V(w)|\leq\frac{\pi}{2}$. Next, as
$\partial\Omega_{p}\setminus\left[0,\frac{p}{2}\right]\times\\{\pi\\}\subseteq
D$ and $\operatorname{Im}S$ is continuous on $D$, it follows that to prove
(2.14) we need only prove that

$\displaystyle\operatorname{Im}S(w)=0,\;\forall
w\in\partial\Omega_{p}\setminus\left[0,\frac{p}{2}\right]\times\\{\pi\\}.$
(2.25)

First, we have

$\displaystyle\partial\Omega_{p}\setminus\left[0,\frac{p}{2}\right]\times\\{\pi\\}=L_{1}\cup
L_{2}\cup L_{3},$ $\displaystyle
L_{1}=\\{0\\}\times[0,\pi)=\left\\{w\in\mathbb{C}:\operatorname{Re}w=0,\operatorname{Im}w\in[0,\pi)\right\\},$
$\displaystyle
L_{2}=\left[0,\frac{p}{2}\right]\times\\{0\\}=\left\\{w\in\mathbb{C}:\operatorname{Re}w\in\left[0,\frac{p}{2}\right],\operatorname{Im}w=0\right\\},$
$\displaystyle
L_{3}=\left\\{\frac{p}{2}\right\\}\times[0,\pi)=\left\\{w\in\mathbb{C}:\operatorname{Re}w=\frac{p}{2},\operatorname{Im}w\in[0,\pi)\right\\}.$

Second, by property (i) of Def. 13, we know that

$\displaystyle\operatorname{Im}S(w)=0,\;\text{if }w\in D\text{ and
}\operatorname{Im}w=0$

which implies the following two facts:

$\displaystyle\operatorname{Im}S(w)=0,\;\forall w\in L_{2},$

and (because of the analyticity of $S$ and $\overline{D}=D$)

$\displaystyle\overline{S(\overline{w})}=S(w),\;\forall w\in D.$ (2.26)

Next, from this and property (iii) of Def. 13 it follows that

$\displaystyle\operatorname{Im}S(iy)=\frac{\overline{S(iy)}-S(iy)}{2i}=\frac{S(\overline{iy})-S(iy)}{2i}=\frac{S(-iy)-S(iy)}{2i}=0,$

for every $y\in(-\pi,\pi)$. From this we conclude that

$\displaystyle\operatorname{Im}S(w)=0,\;\forall w\in L_{1}.$

Next, as $S\in\mathcal{W}_{p}$ it follows from (2.26) and properties (iii) and
(iv) of Def. 13 that

$\displaystyle\operatorname{Im}S(w)=\operatorname{Im}\overline{S(-\overline{w})}=\operatorname{Im}\overline{S(-\overline{w}+p)}=-\operatorname{Im}S(-\overline{w}+p),\;\forall
w\in D$

implying

$\displaystyle\operatorname{Im}S\left(\frac{p}{2}+iy\right)=-\operatorname{Im}S\left[-\overline{\left(\frac{p}{2}+iy\right)}+p\right]=-\operatorname{Im}S\left(\frac{p}{2}+iy\right),\;\forall
y\in(-\pi,\pi)$

which proves that

$\displaystyle\operatorname{Im}S(w)=0,\;\forall w\in L_{3}.$

Thus, we have shown that (2.25) is true which proves (2.14).

The proof of the existence of a function
$v:\partial\Omega_{p}\rightarrow\mathbb{R}$ with the properties (2.15)-(2.18)
now follows immediately from Theorem 16. Next, by (2.17), its clear this
function $v$ is uniquely determined its restriction

$\displaystyle
v|_{\left[0,\frac{p}{2}\right]\times\\{\pi\\}}:\left[0,\frac{p}{2}\right]\times\\{\pi\\}\rightarrow\mathbb{R}.$

Furthermore, by (2.15) and (2.16), it follows that

$\displaystyle v|_{\left[0,\frac{p}{2}\right]\times\\{\pi\\}}\in
L^{\infty}\left(\left[0,\frac{p}{2}\right]\times\\{\pi\\},ds,\mathbb{R}\right),$
$\displaystyle\left\|v|_{\left[0,\frac{p}{2}\right]\times\\{\pi\\}}\right\|_{\infty}\leq\frac{\pi}{2}.$

Now define a function $h:\left[0,\frac{p}{2}\right]\rightarrow\mathbb{R}$ by
(2.22), i.e.,

$\displaystyle
h(x)=v|_{\left[0,\frac{p}{2}\right]\times\\{\pi\\}}(x+i\pi),\forall
x\in\left[0,\frac{p}{2}\right].$

Then it follows immediately from this and the properties of
$v|_{\left[0,\frac{p}{2}\right]\times\\{\pi\\}}$ that (2.23) and (2.24) are
true. Finally, its clear that this function
$h:\left[0,\frac{p}{2}\right]\rightarrow\mathbb{R}$ is uniquely determined by
(2.21). This completes the proof of the corollary. ∎

Notice Corollary 18 (together with Theorem 16) tells us that
$V=\operatorname{Im}S|_{\Omega_{p}}$ is the solution to the (generalized)
Dirichlet problem for $v$ on $\Omega_{p}$. Our next goal is to determine to
what extent the properties of the boundary value function $v$ (and, more
specifically, the function $h$) uniquely determine $S=S_{f}=W(f)$ and hence
$W^{-1}(S)=f\in\mathcal{M}_{c}$, where $p=2\log c$. As we shall see, Corollary
18 already has all the properties of $v$ (and $h$) that we need and, in fact,
the boundary values on $[0,\log c]\times\\{\pi\\}$ and the $L^{\infty}$-norm
bound there is the only information we need. This is partially clear by
considering extension by zero [cf. (2.21)] to the rest of the boundary
$\partial\Omega_{p}=[0,\log
c]\times\\{\pi\\}\cup\partial\Omega_{p}\setminus[0,\log c]\times\\{\pi\\}$.

######  Definition 19.

Let $p\in(0,\infty)$. Define the restriction map $R_{p}$ by

$\displaystyle\mathcal{V}_{p}=\\{V:\Omega_{p}\rightarrow\mathbb{R}\;|\;V=\operatorname{Im}S|_{\Omega_{p}}\text{
for some }S\in\mathcal{W}_{p}\\},$ $\displaystyle
R_{p}:\mathcal{W}_{p}\rightarrow\mathcal{V}_{p},\;R_{p}(S)=\operatorname{Im}S|_{\Omega_{p}},\;\text{for
}S\in\mathcal{W}_{p}.$

Define the extension map $E_{p}$ by

$\displaystyle\mathcal{H}_{p}=\left\\{h\in
L^{\infty}\left(\left[0,\frac{p}{2}\right],dx,\mathbb{R}\right):||h||_{\infty}\leq\frac{\pi}{2}\right\\},$
$\displaystyle E_{p}:\mathcal{H}_{p}\rightarrow
P_{\Omega_{p}}^{-1}(\mathcal{V}_{p})\subseteq
L^{\infty}(\partial\Omega_{p},ds,\mathbb{R}),$

where, for each $h\in\mathcal{H}_{p}$,

$\displaystyle
v(\zeta)=E_{p}(h)(\zeta)=\left\\{\begin{array}[]{cl}h(\operatorname{Re}\zeta),&\text{if
}\operatorname{Re}\zeta\in\left[0,\frac{p}{2}\right]\text{and
}\operatorname{Im}\zeta=\pi,\\\ 0,&\text{if
}\zeta\in\partial\Omega_{p}\setminus\left[0,\frac{p}{2}\right]\times\\{\pi\\},\end{array}\right.$
(2.29)

for every $\zeta\in\partial\Omega_{p}$.

######  Proposition 20.

Let $p\in(0,\infty)$. The function $E_{p}:\mathcal{H}_{p}\rightarrow
P_{\Omega_{p}}^{-1}(\mathcal{V}_{p})$ is well-defined. Moreover, the maps
$R_{p}:\mathcal{W}_{p}\rightarrow\mathcal{V}_{p}$ and
$E_{p}:\mathcal{H}_{p}\rightarrow P_{\Omega_{p}}^{-1}(\mathcal{V}_{p})$ are
invertible.

###### Proof.

Let us begin by showing that $R_{p}:\mathcal{W}_{p}\rightarrow\mathcal{V}_{p}$
is invertible. First, it follows immediately from the definition of
$\mathcal{V}_{p}$ and $R_{p}$ in Def. 19 that $R_{p}$ is surjective. Next, we
prove $R_{p}$ is injective. Suppose $S,T\in\mathcal{W}_{p}$ and
$R_{p}(S)=R_{p}(T)$. Then $S,T:D\rightarrow\mathbb{C}$ are analytic functions
on the open connected set $D$ such that on the open connected set
$\Omega_{p}\subseteq D$ we have $\operatorname{Im}S=\operatorname{Im}T$
implying that there exists a constant $C\in\mathbb{C}$ such that $T=S+C$ on
$D$. As $S,T\in\mathcal{W}_{p}\subseteq\mathcal{W}$ then $S(0)=T(0)=0$ which
implies $0=T(0)=S(0)+C=C$ and hence $S=T$. This proves $R_{p}$ is injective.
Therefore, $R_{p}$ is bijective.

Now we will prove the function $E_{p}:\mathcal{H}_{p}\rightarrow
P_{\Omega_{p}}^{-1}(\mathcal{V}_{p})$ is well-defined. Let
$h\in\mathcal{H}_{p}$ and define $v:\partial\Omega_{p}\rightarrow\mathbb{R}$
by (2.29). Then $v$ satisfies (2.15), (2.16), and (2.17). Hence, we can define
$V:\Omega_{p}\rightarrow\mathbb{R}$ by $V=P_{\Omega_{p}}(v)$ and conclude from
Theorem 16 that $V$ is the solution to the (generalized) Dirichlet problem for
$v$ on $\Omega_{p}$, i.e., satisfies (2.8) on $\Omega=\Omega_{p}$ and (2.9) on
$\partial\Omega=\partial\Omega_{p}$, and $V$ the properties (2.12), (2.13),
and (2.14). Next, as

$\displaystyle-\frac{\operatorname{Im}\zeta}{2}\leq
v(\zeta)\leq\frac{\operatorname{Im}\zeta}{2},\text{ for a.e.\
}\zeta\in\partial\Omega_{p},$

then it follows from Theorem 16.(b) [more specifically, that the map
$P_{\Omega}:L^{\infty}(\partial\Omega,ds,\mathbb{R})\rightarrow
h^{\infty}(\Omega)$ is order preserving] that

$\displaystyle-\frac{\operatorname{Im}w}{2}\leq
V(w)\leq\frac{\operatorname{Im}w}{2},\;\forall w\in\Omega_{p}.$ (2.30)

Next, since $V\in h(\Omega_{p})$ and satisfies (2.14), it follows by the
Schwarz reflection principle [Con78, Sec. IX.1 and Exercise 10, p. 256, Sec.
X.1] that $V$ has an extension to a harmonic function
$V_{1}:\left(0,\frac{p}{2}\right)\times(-\pi,\pi)\rightarrow\mathbb{R}$ which
satisfies $V_{1}(x+iy)=-V_{1}(x-iy)$ for all
$(x,y)\in\left(0,\frac{p}{2}\right)\times(-\pi,\pi)$. Similarly, by the
Schwarz reflection principle, $V_{1}$ has an extension to a harmonic function
$V_{1,2}:\left(-\frac{p}{2},\frac{p}{2}\right)\times(-\pi,\pi)\rightarrow\mathbb{R}$
which satisfies $V_{1,2}(-x+iy)=-V_{1,2}(x+iy)$ for all
$(x,y)\in\left(-\frac{p}{2},\frac{p}{2}\right)\times(-\pi,\pi)$. Now, since
$V_{1,2}$ is harmonic on
$\left(-\frac{p}{2},\frac{p}{2}\right)\times(-\pi,\pi)$ and satisfies both

$\displaystyle\lim_{\left(-\frac{p}{2},\frac{p}{2}\right)\times(-\pi,\pi)\ni
w\rightarrow\pm\frac{p}{2}+iy}V_{1,2}(w)=0,\;\forall y\in(-\pi,\pi),$
$\displaystyle
V_{1,2}(-x+iy)=-V_{1,2}(x+iy),\;\forall(x,y)\in\left(-\frac{p}{2},\frac{p}{2}\right)\times(-\pi,\pi)$

it follows from repeated use of the Schwarz reflection principle (first over
the line $x=p/2$, then over $x=-p/2$, followed by over the line $x=3p/2$, then
over $x=-3p/2$, and so on) that $V_{1,2}$ has an extension to a harmonic
function $V_{1,2,3}:D\rightarrow\mathbb{R}$ which satisfies

$\displaystyle V_{1,2,3}(w+p)=V_{1,2,3}(w),\;\forall w\in D.$

As the open region $D$ is simply connected, it follows (by [Con78, Theorem
2.2.(j), p. 202, Sec. VIII.2] or, more directly, one can use a similar method
in the proof of [Con78, Theorem 2.30, p. 43, Sec. III.2]) that $V_{1,2,3}$ has
a unique harmonic conjugate $U:D\rightarrow\mathbb{R}$ satisfying $U(0)=0$ and
hence

$\displaystyle S(w)=U(w)+iV_{1,2,3}(w),\;\text{for }w\in D,$

defines an analytic function $S:D\rightarrow\mathbb{C}$. We claim that
$S\in\mathcal{W}_{p}$. First, the functions $S(\cdot\pm
p):D\rightarrow\mathbb{C}$ are analytic on $D$ such that
$\operatorname{Im}S(w\pm p)=V_{1,2,3}(w\pm
p)=V_{1,2,3}(w)=\operatorname{Im}S(w)$ for all $w\in D$ which implies that
both $U(\cdot\pm p)$ and $U(\cdot)$ are harmonic conjugates of
$V_{1,2,3}(\cdot)$ so that there exists real constant $C_{\pm}$ such that
$U(\cdot\pm p)=U(\cdot)+C_{\pm}$ so that $U(\pm p)=U(0)+C_{\pm}=C_{\pm}$ and
$0=U(0)=U(-p+p)=U(-p)+C_{+}=C_{-}+C_{+}$ implying $C_{-}=-C_{+}$. Next, as
$\operatorname{Im}S(x)=V_{1,2,3}(x)=V(x)=v(x)=0$ for all
$x\in(0,\frac{p}{2})$, then it follows from this and the analyticity of $S$ on
$D=\overline{D}$ that $\overline{S(\overline{w})}=S(w)$ for all $w\in D$.
Similarly, as $\operatorname{Im}S(iy)=V_{1,2,3}(iy)=V(iy)=v(iy)=0$ for all
$y\in(0,\pi)$, i.e., $S(w)=\overline{S(-\overline{w})}$ for all
$w=iy,y\in(0,\pi)$, then it follows from this and the analyticity of $S$ on
$D=-\overline{D}$ that $S(w)=\overline{S(-\overline{w})}$ for all $w\in D$.
These facts together with $D=-D$, imply that
$S(-w)=\overline{S(-\overline{(-w)})}=\overline{S(\overline{w})}=S(w)$ for all
$w\in D$. This proves that $S$ has property (iii) in Def. 13. It also follows
from this that $-C_{+}=C_{-}=U(-p)=S(-p)=S(p)=U(p)=C_{+}$ so that $C_{+}=0$
and thus
$S(\cdot+p)=U(\cdot+p)+iV_{1,2,3}(\cdot+p)=U(\cdot)+C_{+}+iV_{1,2,3}(\cdot)=U(\cdot)+iV_{1,2,3}(\cdot)=S(\cdot)$.
This proves that $S$ has property (iv) in Def. 13. Also,
$S(0)=U(0)+iV_{1,2,3}(0)=iV(0)=iv(0)=0$ which proves that $S$ has property
(ii) in Def. 13. Next, it follows from the property (2.30) that

$\displaystyle|\operatorname{Im}S(w)|=|V_{1,2,3}(w)|=|V(w)|\leq\frac{1}{2}\operatorname{Im}w,\;\text{if
}0\leq\operatorname{Im}w<\pi,\;0\leq\operatorname{Re}w\leq\frac{p}{2}.$ (2.31)

Next, it follows from this and since $S(w)=\overline{S(-\overline{w})}$ for
all $w\in D$ that

$\displaystyle|\operatorname{Im}S(w)|\leq\frac{1}{2}\operatorname{Im}w,\;\text{if
}0\leq\operatorname{Im}w<\pi,\;-\frac{p}{2}\leq\operatorname{Re}w\leq 0.$
(2.32)

Now, it follow from (2.31), (2.32), and since
$\operatorname{Im}S(w+p)=\operatorname{Im}S(w)$ for all $w\in D$, that

$\displaystyle|\operatorname{Im}S(w)|\leq\frac{1}{2}\operatorname{Im}w,\;\text{if
}0\leq\operatorname{Im}w<\pi,$

which proves that $S$ has property (i) in Def. 13. This completes the proof of
the claim that $S\in\mathcal{W}_{p}$.

Now we have shown that
$P_{\Omega_{p}}(v)=V=\operatorname{Im}S|_{\Omega_{p}}\in\mathcal{V}_{p}$ from
which it follows that $E_{p}(h)=v=P_{\Omega_{p}}^{-1}(V)\in
P_{\Omega_{p}}^{-1}(\mathcal{V}_{p})$. This proves the function
$E_{p}:\mathcal{H}_{p}\rightarrow P_{\Omega_{p}}^{-1}(\mathcal{V}_{p})$ is
well-defined.

Finally, we prove $E_{p}:\mathcal{H}_{p}\rightarrow
P_{\Omega_{p}}^{-1}(\mathcal{V}_{p})$ is invertible. First, if
$h,g\in\mathcal{H}_{p}$ and $E_{p}(h)=E_{p}(g)$ then it follows that
$h(x)=E_{p}(h)(x+i\pi)=E_{p}(g)(x+i\pi)=g(x)$ for a.e.
$x\in\left[0,\frac{p}{2}\right]$ and so $h=g$ in
$L^{\infty}\left(\left[0,\frac{p}{2}\right],dx,\mathbb{R}\right)$. This proves
$E_{p}$ is injective. Next, let $v\in P_{\Omega_{p}}^{-1}(\mathcal{V}_{p})$.
Then there exists a $V\in\mathcal{V}_{p}$ such that $V=P_{\Omega_{p}}(v)$.
Hence, by definition of the set $\mathcal{V}_{p}$ in Def. 19, there exists
$S\in\mathcal{W}_{p}$ such that
$V=\operatorname{Im}S|_{\Omega_{p}}:\Omega_{p}\rightarrow\mathbb{R}$. By
Corollary 18 and invertibility (by Theorem 16) of the Poisson integral
$P_{\Omega_{p}}:L^{\infty}(\partial\Omega_{p},ds,\mathbb{R})\rightarrow
h^{\infty}(\Omega_{p})$, it follows that there exists an $h\in\mathcal{H}_{p}$
such that $E_{p}(h)=v$ in
$L^{\infty}\left(\left[0,\frac{p}{2}\right],dx,\mathbb{R}\right)$. This proves
$E_{p}$ is surjective. Therefore, $E_{p}$ is bijective. This completes the
proof. ∎

Now we come to Theorem 21, which provides (together with Lemma 9) the explicit
characterization of all Molnár means in Theorem 6:

######  Theorem 21 (Second Main Result).

Let $c\in(1,\infty)$ and define

$\displaystyle p=2\log c.$

Then the function

$\displaystyle M_{c}:\mathcal{H}_{p}\rightarrow\mathcal{M}_{c}$

defined by

$\displaystyle M_{c}=W_{p}^{-1}\circ R_{p}^{-1}\circ P_{\Omega_{p}}\circ
E_{p}$

is a bijection from $\mathcal{H}_{p}$ onto $M_{c}$. In particular, if
$f\in\mathcal{M}$ then $f\in\mathcal{M}_{c}$ for some $c\in(1,\infty)$, there
exists a unique $h\in\mathcal{H}_{2\log c}$ such that $f=M_{c}(h)$, and

$\displaystyle\lim_{x\rightarrow 0^{+}}f(x)=0.$

###### Proof.

The proof of these statements follows immediately from Proposition 20, Lemma
9, and Lemma 12.(iii). ∎

##  3 A Class of Examples

In this section we will construct, for each $c\in(1,\infty)$, an explicit
class of examples of Molnár means $\sigma$ with representing function
$f\in\mathcal{M}_{c}$ such that $\sigma\not=\\#$.

Let $c\in(1,\infty)$ and define $p=2\log c$. Let $h\in\mathcal{H}_{p}$, i.e.,
$h\in L^{\infty}\left(\left[0,\frac{p}{2}\right],dx,\mathbb{R}\right)$ with
$||h||_{\infty}\leq\frac{\pi}{2}$. Let $V$ be the unique solution to Dirichlet
boundary-value problem for the Laplacian on the rectangle
$\Omega_{p}=(0,a)\times(0,b)$, where $a=\frac{p}{2},b=\pi$:

$\displaystyle\Delta V=0,\;0<x<a,\;0<y<b,$ $\displaystyle
V(x,0)=V(0,y)=V\left(a,y\right)=0,\;0<x<a,\;0<y<b,$ $\displaystyle
V(x,b)=h(x),\;0<x<a.$

The formal solution $V$, using the standard method of separation of variables
and Fourier series, can be represented by the infinite series

$\displaystyle V(x,y)=\sum_{n=1}^{\infty}c_{n}\sin(\mu_{n}x)\sinh(\mu_{n}y),$
(3.1)
$\displaystyle\mu_{n}=\frac{n\pi}{a},\;c_{n}=\frac{2}{a\sinh(\mu_{n}b)}\int_{0}^{a}h(x)\sin(\mu_{n}x)dx,\;\forall
n\in\mathbb{N}.$

Next, we extend this $V$ as discussed in Proposition 20: first, from
$\Omega_{p}$ to $(0,a)\times(-b,b)$ and using the fact
$\sinh(\mu_{n}y)=-\sinh(\mu_{n}(-y))$ gives us $V_{1}$ with the same series
representation above, then from $(0,a)\times(-b,b)$ to $(-a,a)\times(-b,b)$
and using the fact $\sin(\mu_{n}(-x))=-\sin(\mu_{n}x)$ gives us $V_{1,2}$ with
the same series representation above, and finally from $(-a,a)\times(-b,b)$ to
all $D=\mathbb{R}\times(-b,b)$ and using the fact that
$\sin(\mu_{n}(x+2a))=\sin(\mu_{n}x)$ gives us $V_{1,2,3}$ with the same series
representation above. For simplicity, we will continue using the notation $V$
for $V_{1,2,3}$.

As discussed in Proposition 20, we next find the harmonic conjugate $U$ of
$V=V_{1,2,3}$ on $D$ satisfying $U(0)=0$: using the Cauchy-Riemann equations
we must have

$\displaystyle U_{x}=V_{y},\;U_{y}=-V_{x}.$

To solve for $U$, we integrate the first equation with respect to $x$ to get

$\displaystyle U(x,y)=\int_{[0,x]}V_{y}(t,y)dt+g(y)$

for some differentiable (real) function $g(y)$. Next, we differentiate this
with respect to $y$ and use the second equation to get (using the fact that
$V$ satisfies Laplace’s equation on $D$)

$\displaystyle-V_{x}(x,y)=U_{y}(x,y)=\int_{[0,x]}V_{yy}(t,y)dt+g^{\prime}(y)$
$\displaystyle=-\int_{[0,x]}V_{xx}(t,y)dt+g^{\prime}(y)$
$\displaystyle=-V_{x}(x,y)+V_{x}(0,y)+g^{\prime}(y)$

implying

$\displaystyle g^{\prime}(y)=-V_{x}(0,y).$

Integrating this with respect to $y$ yields

$\displaystyle g(y)=-\int_{[0,y]}V_{x}(0,s)ds+C,$

for a real constant $C$ so that

$\displaystyle U(x,y)=\int_{[0,x]}V_{y}(t,y)dt-\int_{[0,y]}V_{x}(0,s)ds+C,$

and so $0=U(0)=C$. Hence, using the series representation (3.1) of $V$ we get
the formal representation of its harmonic conjugate $U$ in the following form:

$\displaystyle U(x,y)=\int_{[0,x]}V_{y}(t,y)dt-\int_{[0,y]}V_{x}(0,s)ds$
$\displaystyle=\sum_{n=1}^{\infty}c_{n}[1-\cos(\mu_{n}x)]\cosh(\mu_{n}y)-\sum_{n=1}^{\infty}c_{n}[\cosh(\mu_{n}y)-1]$
$\displaystyle=\sum_{n=1}^{\infty}c_{n}[1-\cos(\mu_{n}x)\cosh(\mu_{n}y)].$

Now we want to find a converse to the above approach. First, we want to define
a convergent series by

$\displaystyle V(x,y)=\sum_{n=1}^{\infty}c_{n}\sin(\mu_{n}x)\sinh(\mu_{n}y).$

Then as

$\displaystyle|\sinh(\mu_{n}y)|=\sinh(\mu_{n}y)\leq\sinh(\mu_{n}b),\forall
y\in[0,b]$

it suffices to choose any real coefficients $c_{n}$ such that:

$\displaystyle\sum_{n=1}^{\infty}|c_{n}|\sinh(\mu_{n}b)\leq\frac{\pi}{2}.$
(3.2)

This proves absolute convergence of the series and the series converges
uniformly. A similar reasoning, and using the fact that

$\displaystyle\cosh(t)=\sinh(t)+e^{-t},\forall t;$ $\displaystyle e^{-t}\leq
C\sinh(t),\forall t\gg 1,$

for any fix constant $C>2$, proves we can define a convergent series by

$\displaystyle
U(x,y)=\sum_{n=1}^{\infty}c_{n}[1-\cos(\mu_{n}x)\cosh(\mu_{n}y)],$

which converges absolutely and uniformly. As such the above analysis is
rigorous starting from

$\displaystyle h(x)=\lim_{y\rightarrow
b^{-}}V(x,y)=\sum_{n=1}^{\infty}c_{n}\sin(\mu_{n}x)\sinh(\mu_{n}b)\in\mathcal{H}_{p}.$

This gives us a family of Molnár means $\sigma=\sigma_{f}$ with representing
function $f\in\mathcal{M}_{c}$ such that $\sigma\not=\\#$ by choosing any
sequence of real numbers $(c_{n})_{n\in\mathbb{R}}$, not all zero, satisfying
the inequality (3.2) and then defining

$\displaystyle f(z)=\sqrt{z}e^{S(\operatorname{Log}(z))},\text{ for
}z\in\mathbb{C}\setminus(-\infty,0],\text{ where }$ $\displaystyle
S(w)=U(x,y)+iV(x,y)$
$\displaystyle=\sum_{n=1}^{\infty}c_{n}[1-\cos(\mu_{n}x)\cosh(\mu_{n}y)]+ic_{n}\sin(\mu_{n}x)\sinh(\mu_{n}y),$
$\displaystyle\text{for }w=x+iy\in D.$

######  Example 22.

Here we construct an simple explicit counterexample to the Molnár open problem
based on our class of examples above. First, we can choose $c_{n}=0,n\geq 2$.
We now choose any $c_{1}$ satisfying

$\displaystyle|c_{1}|\leq\frac{1}{\sinh(\mu_{1}b)}\frac{\pi}{2},$

and as such we can use, for example,

$\displaystyle c_{1}=\frac{1}{\sinh(\mu_{1}b)}\frac{\pi}{2}.$

Then we have a Molnár mean $\sigma=\sigma_{f}$ with representing function
$f\in\mathcal{M}_{c}$ such that $\sigma\not=\\#$ with $f$ defined by

$\displaystyle f(z)=\sqrt{z}e^{S(\operatorname{Log}(z))},\text{ for
}z\in\mathbb{C}\setminus(-\infty,0],\text{ where }$ $\displaystyle
S(w)=c_{1}[1-\cos(\mu_{1}x)\cosh(\mu_{1}y)]+ic_{1}\sin(\mu_{1}x)\sinh(\mu_{1}y),$
$\displaystyle\text{for }w=x+iy\in D,$

and

$\displaystyle\mu_{1}=\frac{\pi}{a},a=\frac{p}{2},b=\pi,p=2\log c,$

for any $c\in(1,\infty)$.

## Acknowledgement

GWM is grateful to the National Science Foundation for support through grant
DMS-2107926. The work of the AW was supported by the Simons Foundation grant
MPS-TSM-00002799. This paper was initiated at CIRM during the conference
entitled “Herglotz-Nevanlinna Functions and their Applications to Dispersive
Systems and Composite Materials,” which was held on May 23-27, 2022. Both
authors wish to thank the co-organizers of that conference as well as CIRM for
hosting us and for their hospitality.

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