#  On an integral representation of the normalized trace of the $k$-th
symmetric tensor power of matrices and some applications

Hassan Issa∗ ,  Hassan Abbas∗ and  Bassam Mourad∗ *Department of Mathematics,
Faculty of Science I, Lebanese University, Beirut, Lebanon  hissa@uni-
math.gwdg.de

###### Abstract.

Let $A$ be an $n\times n$ matrix and let $\vee^{k}A$ be its $k$-th symmetric
tensor product. We express the normalized trace of $\vee^{k}A$ as an integral
of the $k$-th powers of the numerical values of $A$ over the unit sphere
$\mathbb{S}^{n}$ of $\mathbb{C}^{n}$ with respect to the normalized Euclidean
surface measure. Equivalently, this expression in turn can be interpreted as
an integral representation for the (normalized) complete symmetric polynomials
over $\mathbb{C}^{n}$. As applications, we present a new proof for the
MacMahon Master Theorem in enumerative combinatorics. Then, our next
application deals with a generalization of the work of Cuttler et al. in [10]
concerning the monotonicity of products of complete symmetric polynomials. In
the process, we give a solution to an open problem that was raised by I.
Rovenţa and L. E. Temereanca in [31].

###### Key words and phrases:

Integral formula, trace of symmetric powers, homogeneous polynomials.

###### 2010 Mathematics Subject Classification:

Primary 15A15, 15A60; Secondary 32A26

##  1\. Introduction

Given $A$ in the space of all $n\times n$ complex matrices
$M_{n}(\mathbb{C})$, we denote its tace by $tr(A)$ and we shall write
$Tr(A):=\frac{tr(A)}{n}$ for the normalized trace. It is well known ([7, 20])
that $Tr(A)$ can be obtained via the following functional representation

$\int_{\mathbb{S}^{n}}<A\xi,\xi>d\sigma(\xi)=Tr(A),$ (1.1)

where $\mathbb{S}^{n}$ is the unit sphere of $\mathbb{C}^{n}$, $\sigma$ is the
probability Lebesgue measure on $\mathbb{S}^{n}$ and $<\cdot,\cdot>$ is the
standard inner product on $\mathbb{C}^{n}$. The above formula states that the
normalized trace of a matrix is the expectation of the numerical values of the
matrix computed with respect to $d\sigma$. Noting the linearity on both sides
of (1.1), one obtains the above formula by checking it on matrix bases. When
linearity is lost due to some perturbation of the numerical values, new
techniques are needed to obtain a reasonable formula. Our main objective
inthis paper is to find the expectation of natural powers of the numerical
values computed with respect to $d\sigma$. More precisely, given two natural
numbers $n,k\in\mathbb{N}$ we aim to find an expression for

$\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{k}d\sigma(\xi)$ (1.2)

The preceding integral is of special interest notably in the analysis of some
reproducing kernel Hilbert spaces of analytic functions. More precisely, when
addressing the analysis of the Hardy space $H^{2}(\mathbb{S}^{n})$ over the
unit ball $B(0,1)$ of $\mathbb{C}^{n}$, we recall that the Szegö projection
$P_{S}$ is the orthogonal projection of $L^{2}(\mathbb{S}^{n})$ onto
$H^{2}(\mathbb{S}^{n})$ and is given by

$P_{S}f(z)=\int_{\mathbb{S}^{n}}\frac{f(\xi)}{(1-<v,\xi>)^{n}}d\sigma(\xi),\quad\mbox{for
}v\in B(0,1).$ (1.3)

This integral representation for $P_{S}$ is due to a classical series
representation for the reproducing kernel in terms of an orthonormal basis
(cf. [33]). Note that for a fixed $v\in B(0,1)$, if $\delta_{v}$ is the
pointwise evaluation at $v$ then the linear functional $\delta_{v}\circ P_{S}$
is expressed as an infinite series of integral (scalar valued) operators of
the form

$c_{n,k}\int_{\mathbb{S}^{n}}(<v,\xi>)^{k}f(\xi)d\sigma(\xi),$

where $c_{n,k}=\binom{n+k-1}{k}$. Thus, it is natural to refer to such
operators as the ”building blocks” for the Szegö projection. In order to
clarify the connection of such analysis to our work, we consider a collection
of integral transforms $\\{T_{A,k}\\}_{{}_{A\in
M_{n}(\mathbb{C}),k\in\mathbb{N}}}$ on $L^{2}(\mathbb{S}^{n})$ defined by

$T_{A,k}f(\mu)=\int_{\mathbb{S}^{n}}(<A\mu,\xi>)^{k}f(\xi)d\sigma(\xi),\quad\mbox{for
}f\in L^{2}(\mathbb{S}^{n}),\quad\mu\in\mathbb{S}^{n}.$ (1.4)

Let $e_{1}=(1,0,\cdots,0)\in\mathbb{C}^{n}$, then
$\\{T_{A,k}f(e_{1})\\}_{{}_{A\in M_{n}(\mathbb{C})}}$ provides a
generalization for the ”building blocks” of $P_{S}$. Indeed, given a non-zero
$v\in B(0,1)$ consider $A=\lVert v\rVert_{2}U$ where $\lVert\cdot\rVert_{2}$
is the Euclidean norm and $U$ is a unitary matrix satisfying
$Ue_{1}=\frac{v}{\lVert v\rVert_{2}}$, then

$T_{A,k}f(e_{1})=\int_{\mathbb{S}^{n}}(<v,\xi>)^{k}f(\xi)d\sigma(\xi).$

It is easy to see that $T_{A,k}$ is a finite rank operator and apparently the
permutation in the $\mu$-coordinate might not allow us to get a canonical
decomposition of $T_{A,k}$ for general $A$. This in turn leads to difficulties
in computing the trace of $T_{A,k}$. Alternatively, as the kernel
$(<A\mu,\xi>)^{k}$ is continuous in the $\mu$ and $\xi$ variables then the
trace of $T_{A,k}$ is given by (1.2). A similar argument holds for the
Hilbert-Schmidt norm $\lVert\cdot\rVert_{(2)}$ of $T_{A,k}$ which in support
of our line of investigation, turns out to satisfy the following identity;

$c^{\frac{1}{2k}}_{n,k}\lVert
T_{A,k}\rVert^{\frac{1}{k}}_{(2)}=\left(\int_{\mathbb{S}^{n}}\lVert
A\xi\rVert_{2}^{2k}d\sigma(\xi)\right)^{\frac{1}{2k}}.$ (1.5)

In this paper, we shall use matrix analysis techniques to find the integral
(1.2) and then we apply our results to enumerative combinatorics. More
explicitly, our main objective is to prove the following main result.

######  Theorem 1.1.

Let $A\in M_{n}(\mathbb{C})$ and let $k\in\mathbb{N}$. Denote by
$Tr(\vee^{k}A)=\frac{tr(\vee^{k}A)}{c_{n,k}}$ the normalized trace of the k-th
symmetric power of A then

$\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{k}d\sigma(\xi)=Tr(\vee^{k}A).$
(1.6)

Since $Tr(\vee^{k}A)$ is simply the evaluation of the normalized complete
$k$-homogenous polynomials $H_{k}$ at the eigenvalues of $A$, the preceding
integral can be then interpreted as a representation for such polynomials. A
direct consequence of the preceding equation is the positivity of $H_{2k}$ on
$\mathbb{R}^{n}$ which has been proved by D. B. Hunter in [15]. A simpler
proof to this fact that is based on Schur convexity of the complete symmetric
polynomials, was obtained by I. Rovena and L. E. Temereanca in [31]. However,
we believe that in the context of our work, we offer the simplest proof among
the previously mentioned work.

In connection with the analysis of symmetric polynomials, we recall that the
Harish-Chandra-Itzykson-Zuber integral which was used by S. Sra in [35] to
express the normalized Schur polynomials. Via such a representation together
with a technique of Schur’s convexity, the author in [35] presented the
sufficiency for a conjecture that was proposed in [10] on the monotonicity
(via a lexicographic order) for (a product of) normalized Schur polynomials.
More explicitly, in [10] the authors presented many inequalities for symmetric
functions and in particular, they proved the monotonicity for (a product of)
normalized complete symmetric polynomials on $\mathbb{R}^{n}_{+}$. Another
consequence of our results lies in exploiting our integral representation
together with a result due Y. L. Tong [38] on Schur convexity to prove that
this monotonicity holds also for (a product of) normalized complete symmetric
polynomials of even degrees on $\mathbb{R}^{n}$ not only on
$\mathbb{R}^{n}_{+}$.

The paper is organized as follows. Section 2 is devoted to proving our main
theorem. In Section 3 we provide a new proof for Macmahon Master Theorem and
we obtain an integral representation for a particular determinant. In the last
section, we present a generalization to the monotonicity result in [10] and in
the process, we give a solution to an open problem that was raised in [31].

##  2\. Integral formula for the trace of symmetric powers

There is a natural way to provide an exact formula for (1.2) by integrating
polynomials over the unit sphere of arbitrary dimensional complex spaces. For
this reason, it is more convenient to use the standard notation from the
theory of complex analysis of several variables. For a multi-index
$\alpha=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})\in\mathbb{N}^{n}_{0}$ of
non-negative integers and for $z\in\mathbb{C}^{n}$, we shall write

$\overline{z}=(\overline{z_{1}},\overline{z_{2}},\cdots,\overline{z_{n}}),\quad
z^{\alpha}=z_{1}^{\alpha_{1}}.z_{2}^{\alpha_{2}}\cdots
z_{n}^{\alpha_{n}},\quad\lvert\alpha\rvert=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n},\quad\mbox{and}\quad\alpha!=\alpha_{1}!\alpha_{2}!\cdots\alpha_{n}!.$

In connection to matrix analysis, given $A=(a_{ij})\in M_{n}(\mathbb{C})$ we
denote by $a:=vec(A^{T})$ the vectorization of its transpose i.e. $a$ is the
vector obtained by stacking the rows of A on top of each other (cf. [16, 17])
so that

$a=(a_{11},a_{12},\cdots,a_{1n},a_{21},a_{22}\cdots,a_{2n},\cdots,a_{n1},\cdots,a_{nn})^{T}\in\mathbb{C}^{n^{2}}.$

Whenever $\alpha\in M_{n^{2}}(\mathbb{N}_{0})$, we will not distinguish
between its matrix form and its representation
$vec(\alpha^{T})\in\mathbb{N}_{0}^{n^{2}}$. For example, given $A=(a_{ij})\in
M_{n}(\mathbb{C})$ and $\alpha=(\alpha_{ij})\in M_{n^{2}}(\mathbb{N}_{0})$ we
write $|\alpha|=\sum_{ij}\alpha_{ij}$ and

$a^{\alpha}=\prod_{i,j}^{n^{2}}a^{\alpha_{ij}}_{ij}.$

Preliminary to our consideration of the integral formula, we shall express the
integrand in (1.2) using the above notation.

######  Proposition 2.1.

Let $A\in M_{n}(\mathbb{C})$ and $k\in\mathbb{N}$. Given $z\in\mathbb{C}^{n}$,
the following holds:

$\Big{(}<Az,z>\Big{)}^{k}=\sum_{\begin{subarray}{c}|\alpha|=k\\\ \alpha\in
M_{n^{2}}(\mathbb{N}_{0})\end{subarray}}\binom{k}{\alpha}a^{\alpha}z^{\big{(}\sum_{i}\alpha_{i1},\sum_{i}\alpha_{i2},\cdots,\sum_{i}\alpha_{in}\big{)}}\
\overline{z}^{\big{(}\sum_{j}\alpha_{1j},\sum_{j}\alpha_{2j},\cdots,\sum_{j}\alpha_{nj}\big{)}},$
(2.1)

where $\binom{k}{\alpha}=\frac{k!}{\alpha!}.$

###### Proof.

Applying the multinomial theorem to
$\Big{(}<Az,z>\Big{)}^{k}=\Bigg{(}\sum_{i,j}a_{ij}\overline{z_{i}}z_{j}\Bigg{)}^{k}$
leads to:

$\displaystyle\Big{(}<Az,z>\Big{)}^{k}$
$\displaystyle=\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}_{0}^{n^{2}}\end{subarray}}\binom{k}{\alpha}\prod_{i,j}^{n^{2}}a^{\alpha_{ij}}_{ij}\
\overline{z_{i}}^{\alpha_{ij}}z_{j}^{\alpha_{ij}}=\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}_{0}^{n^{2}}\end{subarray}}\binom{k}{\alpha}\Bigg{[}\prod_{i,j}^{n^{2}}a^{\alpha_{ij}}_{ij}\Bigg{]}\prod_{i,j}^{n^{2}}\overline{z_{i}}^{\alpha_{ij}}z_{j}^{\alpha_{ij}}$
$\displaystyle=\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}_{0}^{n^{2}}\end{subarray}}\binom{k}{\alpha}a^{\alpha}\prod_{j=1}^{n}\Bigg{[}\prod_{i=1}^{n}\overline{z_{i}}^{\alpha_{ij}}z_{j}^{\alpha_{ij}}\Bigg{]}=\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}_{0}^{n^{2}}\end{subarray}}\binom{k}{\alpha}a^{\alpha}\prod_{j=1}^{n}z_{j}^{\sum_{i}\alpha_{ij}}\
\overline{z}^{\big{(}\alpha_{1j},\alpha_{2j},\cdots,\alpha_{nj}\big{)}}$
$\displaystyle=\sum_{\begin{subarray}{c}|\alpha|=k\\\ \alpha\in
M_{n^{2}}(\mathbb{N}_{0})\end{subarray}}\binom{k}{\alpha}a^{\alpha}z^{\big{(}\sum_{i}\alpha_{i1},\sum_{i}\alpha_{i2},\cdots,\sum_{i}\alpha_{in}\big{)}}\
\overline{z}^{\big{(}\sum_{j}\alpha_{1j},\sum_{j}\alpha_{2j},\cdots,\sum_{j}\alpha_{nj}\big{)}}.$

∎

######  Remark 2.1.

Suppose that $A$ is an upper triangular matrix then the sum on the right hand
side of (2.1) reduces to those $\alpha\in M_{n^{2}}(\mathbb{N}_{0})$
satisfying $|\alpha|=k$ and $\alpha$ is only upper triangular. Indeed, if
$\alpha_{i_{0}j_{0}}\neq 0$ for certain $i_{0}>j_{0}$ then as $A$ is upper
triangular we have $a^{\alpha}=0$.

Due to the orthogonality of the monomials on $L^{2}(\mathbb{S}^{n})$, a
natural condition is then employed on the row sums as well as the column sums
of $\alpha\in M_{n^{2}}(\mathbb{N}_{0})$ when integrating (2.1). The next
lemma examines the type of $\alpha$ under such condition.

######  Lemma 2.1.

Let $\alpha\in M_{n^{2}}(\mathbb{N}_{0})$ be an upper triangular matrix
satisfying the condition:

$\sum_{i=1}^{n}\alpha_{il}=\sum_{j=1}^{n}\alpha_{lj}\ \ \mbox{ for every
}l=1,2\cdots,n,$ (2.2)

then $\alpha$ is a diagonal matrix.

###### Proof.

We shall proceed by induction. For $l=1$, we have
$\alpha_{11}=\alpha_{11}+\sum_{j=2}^{n}\alpha_{1j}$ so that $\alpha_{1j}=0$
for all $j=2,\cdots n$. Assume that for all $k\leq l_{0}$ we have
$\alpha_{kj}=0$ whenever $j\neq k$. By (2.2) the $(l_{0}+1)$ column sum of
$\alpha$ satisfies

$\sum_{i=1}^{n}\alpha_{i,l_{0}+1}=\sum_{j=1}^{n}\alpha_{l_{0}+1,j}$

Notice that,

$\sum_{i=1}^{n}\alpha_{i,l_{0}+1}=\sum_{i\leq
l_{0}}^{n}\underbrace{\alpha_{i,l_{0}+1}}_{=0}+\alpha_{l_{0}+1,l_{0}+1}+\sum_{i>l_{0}+1}^{n}\underbrace{\alpha_{i,l_{0}+1}}_{=0}$

The first sum on the right hand side of the above equation vanishes by making
se of the induction hypothesis and the second sum vanishes as $\alpha$ is
upper triangular. Similarly, by using the fact that $\alpha$ is upper
triangular we get

$\sum_{j=1}^{n}\alpha_{l_{0}+1,j}=\sum_{j\leq
l_{0}}^{n}\underbrace{\alpha_{l_{0}+1,j}}_{=0}+\alpha_{l_{0}+1,l_{0}+1}+\sum_{j>l_{0}+1}^{n}\alpha_{l_{0}+1,j}.$

We conclude that

$\alpha_{l_{0}+1,l_{0}+1}=\alpha_{l_{0}+1,l_{0}+1}+\sum_{j>l_{0}+1}^{n}\alpha_{l_{0}+1,j},$

which in turn yields that $\alpha_{l_{0}+1,j}=0$ for all $j>l_{0}+1$. Thus,
$\alpha$ is diagonal and the proof is complete. ∎

Now, for our purposes we need a well-known result for the exact value for
integration of monomials on spheres. Such explicit value plays an important
role in real, complex and harmonic analysis (cf. for example [2, 12, 21]). In
particular, for the study of (commuting) Toeplitz operators on the Segal-
Bargmann space [5, 18, 19]. A simple proof using complex analysis for the next
lemma could be found in [11] and is due to V. Bargmann and E. Nelson as
indicated by G.B. Folland in [11] (see also Proposition 1.4.9 in [32]).

######  Lemma 2.2.

Let $\sigma$ be the normalized probability measure on the unit sphere
$\mathbb{S}^{n}$ of $\mathbb{C}^{n}$. For any
$\alpha,\beta\in\mathbb{N}^{n}_{0}$ we have

$\int_{\mathbb{S}^{n}}\xi^{\alpha}\overline{\xi}^{\beta}d\sigma(\xi)=\begin{cases}\dfrac{(n-1)!\alpha!}{(n-1+\lvert\alpha\rvert)!}\quad&\alpha=\beta;\\\
0\quad&\alpha\neq\beta.\end{cases}$ (2.3)

In order to prove our main theorem, we shall follow the notation used in [6]
for the symmetric tensor product (or power) of matrices. Recall that, for
$k\in\mathbb{N}$ the symmetric tensor product of $\mathbb{C}^{n}$, denoted by
$\vee^{k}\mathbb{C}^{n}$ is the subspace of $\otimes^{k}\mathbb{C}^{n}$
spanned by all elementary symmetric tensor products $x_{1}\vee
x_{2}\vee\cdots\vee x_{k}$ of vectors $x_{i}\in\mathbb{C}^{n}$. Given $A\in
M_{n}(\mathbb{C})$, we denote by $\vee^{k}A$ the $k$th-symmetric tensor power
of $A$ i.e. $\vee^{k}A$ is the matrix in $M_{c_{n,k}}(\mathbb{C})$
representing the linear operator
$\vee^{k}A:\vee^{k}\mathbb{C}^{n}\longrightarrow\vee^{k}\mathbb{C}^{n}$ whose
first definition on the elementary symmetric tensor product of vectors is
given by

$\vee^{k}A(x_{1}\vee x_{2}\vee\cdots\vee x_{k}):=Ax_{1}\vee
Ax_{2}\vee\cdots\vee Ax_{k}.$ (2.4)

Recall that the k-th complete elementary symmetric polynomial with n-complex
variables is a homogeneous polynomial of order k given by

$h_{k}(z)=\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}^{n}_{0}\end{subarray}}z^{\alpha},\quad
z\in\mathbb{C}^{n}.$

Here, and in the sequel, if the spectrum $sp(A)=\\{\lambda_{i}\mid
i=1,\cdots,n\\}$ we shall denote the (un-ordered) n-tuple by
$\lambda=\lambda(A):=(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in\mathbb{C}^{n}$
counted with their multiplicities. Using (2.4) it is easy to check that
$sp(\vee^{k}A)=\\{\lambda_{i_{1}}\lambda_{i_{2}}\cdots\lambda_{i_{k}}\mid
1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq n\\}$. This leads to the well-
known formula for the trace of $\vee^{k}A$ in terms of
$\lambda_{1},\lambda_{2},\cdots,\lambda_{n}$:

$tr(\vee^{k}A)=\sum_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq
n}\lambda_{i_{1}}\lambda_{i_{2}}\cdots\lambda_{i_{k}}=h_{k}(\lambda),$ (2.5)

Now we are in a position to prove our main result which is Theorem 1.1. For
the sake of the reader, we shall restate it again here as follows.

######  Theorem 2.1.

Let $A\in M_{n}(\mathbb{C})$ and let $k\in\mathbb{N}$. Denote by
$Tr(\vee^{k}A)=\frac{tr(\vee^{k}A)}{c_{n,k}}$ the normalized trace of the k-th
symmetric power of A, then

$\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{k}d\sigma(\xi)=Tr(\vee^{k}A).$
(2.6)

###### Proof.

Using Schur’s decomposition, one can write $A=U^{\star}TU$ with $T$ upper
triangular and $U$ is unitary. So that the change of variable $u=U\xi$
provides

$\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{k}d\sigma(\xi)=\int_{\mathbb{S}^{n}}\big{(}<TU\xi,U\xi>\big{)}^{k}d\sigma(U^{\star}u)=\int_{\mathbb{S}^{n}}\big{(}<Tu,u>\big{)}^{k}d\sigma(u),$

where the last equality follows from the invariance of $\sigma$ under unitary
transformation. Thus, we may assume that $A$ is upper triangular and
$\lambda=(a_{11},a_{22},\cdots,a_{nn})$. By Proposition 2.1 and Remark 2.1, we
obtain that

$\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{k}d\sigma(\xi)$

is given by

$\sum_{\begin{subarray}{c}|\alpha|=k\\\ \alpha\in M_{n^{2}}(\mathbb{N}_{0})\\\
\alpha\text{ upper
triangular}\end{subarray}}\binom{k}{\alpha}a^{\alpha}\int_{\mathbb{S}^{n}}\xi^{\big{(}\sum_{i}\alpha_{i1},\sum_{i}\alpha_{i2},\cdots,\sum_{i}\alpha_{in}\big{)}}\
\overline{\xi}^{\big{(}\sum_{j}\alpha_{1j},\sum_{j}\alpha_{2j},\cdots,\sum_{j}\alpha_{nj}\big{)}}d\sigma(\xi).$

Following Lemma 2.2, the preceding integral terms vanishes for all $\alpha$
except when

$\sum_{i=1}^{n}\alpha_{il}=\sum_{j=1}^{n}\alpha_{lj},\ \ \ \mbox{ for every
}l=1,2\cdots,n.$

By Lemma 2.1, $\alpha$ is a diagonal matrix so that

$a^{\alpha}=\lambda_{1}^{\alpha_{11}}\lambda_{2}^{\alpha_{22}}\cdots\lambda_{2}^{\alpha_{nn}}=\lambda^{\alpha},$

where in the last equality we considered
$\alpha=(\alpha_{1},\alpha_{2},\cdots,\alpha_{n})=(\alpha_{11},\alpha_{22},\cdots,\alpha_{nn})$
as an $n$-tuple in $\mathbb{N}^{n}_{0}$. This reduces the preceding integral
to the following form

$\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{k}d\sigma(u)=\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}^{n}\end{subarray}}\binom{k}{\alpha}\lambda^{\alpha}\int_{\mathbb{S}^{n}}\xi^{\big{(}\alpha_{11},\alpha_{22},\cdots,\alpha_{nn}\big{)}}\
\overline{\xi}^{(\alpha_{11},\alpha_{22},\cdots,\alpha_{nn}\big{)}}d\sigma(\xi).$

Another application of Lemma 2.2 for the norms of monomials, yields

$\displaystyle\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{k}d\sigma(\xi)$
$\displaystyle=\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}^{n}_{0}\end{subarray}}\binom{k}{\alpha}\lambda^{\alpha}\dfrac{(n-1)!\alpha!}{(n-1+\lvert\alpha\rvert)!}$
$\displaystyle=\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}^{n}_{0}\end{subarray}}\frac{k!}{\alpha!}\lambda^{\alpha}\dfrac{(n-1)!\alpha!}{(n-1+k)!}$
$\displaystyle=\frac{k!(n-1)!}{(n-1+k)!}\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}^{n}_{0}\end{subarray}}\lambda^{\alpha}$ (2.7)
$\displaystyle=\frac{tr(\vee^{k}A)}{c_{n,k}}.$

∎

Formula (2.6) merits (at least a quick) discussion on its important role in
the analysis of function spaces. It is trivial that (2.6) generalizes (1.1)
from $k=1$ to arbitrary $k\in\mathbb{N}$. With this in mind, a direct
application of ((1.1)) to $\vee^{k}A\in M_{c_{n,k}}(\mathbb{C})$ together with
(2.6) provides

$\int_{\mathbb{S}^{c_{n,k}}}<\vee^{k}A\mu,\mu>d\sigma(\mu)=\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{k}d\sigma(\xi).$
(2.8)

The above formula can be viewed as a reduction from integrals over
$\mathbb{S}^{c_{n,k}}$ to integrals over $\mathbb{S}^{n}$. This leads to
various applications in the study of the analysis of reproducing kernel
Hilbert spaces of analytic function over specific domains. For example, the
left hand side of (2.8) is the Szegö projection onto
$H^{2}(\mathbb{S}^{c_{n,k}})$ of

$\mathbb{S}^{c_{n,k}}\ni\mu\longrightarrow<\vee^{k}A\mu,\mu>$

at $z=0\in\mathbb{C}^{c_{n,k}}$, while the right hand side is the Szegö
projection onto $H^{2}(\mathbb{S}^{n})$ of

$\mathbb{S}^{n}\ni\xi\longrightarrow\big{(}<A\xi,\xi>\big{)}^{k}$

at $z=0\in\mathbb{C}^{n}.$ Another application is connected to the study of
the composition of Toeplitz operators on the Segal-Bargmann space. We refer
the reader to [1, 4, 9, 19] where integral reduction was frequently used to
obtain a composition formula for some classes of Toeplitz operators.

Notice that Formula (2.6) shows that the integral depends on the spectrum of
the matrix. However, by expressing the complete symmetric polynomials as power
sums we obtain that the integral depends only on moments of the eigenvalues.
For this reason, we present the following well-known result on representing
the complete symmetric polynomials in terms of power sums.

######  Lemma 2.3 ([24] pp. 24-25).

Let $k\in\mathbb{N}$ and let $h_{k}(x)=\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}^{n}\end{subarray}}x^{\alpha}$. For each $r\in\mathbb{N}$
put $p_{r}(x)=\sum_{i=1}^{n}x_{i}^{r}.$ Given a non-zero compactly supported
sequence arranged in decreasing order
$\beta=(\beta_{1},\beta_{2},\cdots,\beta_{l},0,0,0\cdots)\in\mathbb{N}^{\infty}$,
define

$p_{\beta}(x)=p_{\beta_{1}}(x)p_{\beta_{2}}(x)\cdots p_{\beta_{l}}(x),$

with $\beta_{l}$ is the leading entry of $\beta$. Then,

$h_{k}(x)=\sum_{|\beta|=k}z^{-1}_{\beta}p_{\beta}(x),$ (2.9)

where

$z_{\beta}=\prod_{i\geq 1}i^{m_{i}}m_{i}!$ (2.10)

and $m_{i}$ is the number for which $i$ is occurring in $\beta$.

We shall apply the preceding lemma to express the trace of the symmetric
tensor product of a matrix $A$ in-terms of product of traces of powers of $A$.
Following the notation in the above lemma, Formula (2.6) can then be
reformulated in-terms of the $rth$-moments of the eigenvalues
$p_{r}(\lambda)=\sum_{i=1}^{n}\lambda_{i}^{r}$ as follows:

$c_{n,k}\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{k}d\sigma(\xi)=\sum_{\begin{subarray}{c}|\beta|=k\\\
\beta_{1}\geq\beta_{2}\geq\cdots\geq\beta_{k}\geq
0\end{subarray}}z^{-1}_{\beta}p_{\beta}(\lambda)=\sum_{\begin{subarray}{c}|\beta|=k\\\
\beta_{1}\geq\beta_{2}\geq\cdots\geq\beta_{k}\geq
0\end{subarray}}z^{-1}_{\beta}\prod_{t=1}^{l}tr(A^{\beta_{t}}).$ (2.11)

Notice that the condition for which $\beta\in\mathbb{N}^{k}_{0}$ is trivial
since $\beta$ is decreasing with $|\beta|=k$. Using the basic fact that the
product of traces is the trace of tensor products, Eq. (2.11) can be rewritten
as follows

$c_{n,k}\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{k}d\sigma(\xi)=\sum_{\begin{subarray}{c}|\beta|=k\\\
\beta_{1}\geq\beta_{2}\geq\cdots\geq\beta_{k}\geq
0\end{subarray}}z^{-1}_{\beta}tr\Big{[}\bigotimes_{t=1}^{l}A^{\beta_{t}}\Big{]},$
(2.12)

where $\bigotimes$ denotes the tensor (Kronecker) product of matrices.

With an inspection of equations (2.11) and (2.12) one obtains an ”exhaustive
behavior” of the $l$-index of $\beta$. Indeed, in both equations there is no-
need to worry about $\beta_{i}$ whenever $i>l$ so fixing $l$ will be helpful
in computation. Moreover, regarding (2.12) one might be interested in summing
up tensor powers of matrices (of same dimension). This also amounts to
collecting the multi-indices having the same index in the leading term. For
this reason, we consider

$S^{k}:=\\{\beta=(\beta_{1},\beta_{2},\cdots,\beta_{k})\in\mathbb{N}_{0}^{k}\mid\beta_{1}\geq\beta_{2}\geq\cdots\geq\beta_{k}\mbox{
and }|\beta|=k\\}$

and for each $l=1,2,\cdots,k$ we set

$S^{k}_{l}=\\{\beta\in S^{k}\mid\beta_{l}\neq 0\mbox{ and }\beta_{i}=0\mbox{
for all }i>l\\}.$

It is worthy to mention here that $\\{S^{k}_{l}\\}_{l=1,2,\cdots,k}$ forms a
partition for $S^{k}$ and thus we conclude from the previous discussion the
following new formulation of Theorem 2.1.

######  Corollary 2.1.

Let $A\in M_{n}(\mathbb{C})$ and let $k\in\mathbb{N}$ then the following holds

$\displaystyle
c_{n,k}\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{k}d\sigma(\xi)$
$\displaystyle=\sum_{l=1}^{k}\sum_{\beta\in
S^{k}_{l}}\frac{1}{z_{\beta}}\prod_{t=1}^{l}tr(A^{\beta_{t}})$ (2.13)
$\displaystyle=\sum_{l=1}^{k}tr\left[\Big{(}\sum_{\beta\in
S^{k}_{l}}\frac{1}{z_{\beta}}\bigotimes_{t=1}^{l}A^{\beta_{t}}\Big{)}\right].$
(2.14)

The next example is devoted to finding the explicit value of the integral
formula for the cases $k=2,\cdots 6$. Since going from (2.13) to (2.14) is
direct, we shall only apply (2.14) for the cases $k=2$ and $k=3$.

######  Example 2.1.

For a given $k,n\in\mathbb{N}$ put $c_{n,k}=\binom{n+k-1}{k}$ then for any
$A\in M_{n}(\mathbb{C})$ the following formulas hold []

$\begin{split}c_{n,2}\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{2}d\sigma(\xi)=\frac{1}{2}tr(A^{2})+\frac{1}{2}tr(A\otimes
A)=\frac{1}{2}tr(A^{2})+\frac{1}{2}(trA)^{2}\end{split}$ (2.15)
$\begin{split}c_{n,3}\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{3}d\sigma(\xi)&=\frac{1}{3}tr(A^{3})+\frac{1}{2}tr(A^{2}\otimes
A)+\frac{1}{6}tr(A\otimes A\otimes A)\\\
&=\frac{1}{3}tr(A^{3})+\frac{1}{2}[tr(A^{2})]tr(A)+\frac{1}{6}[trA]^{3}.\end{split}$
(2.16)
$c_{n,4}\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{4}d\sigma(\xi)=\frac{1}{4}tr(A^{4})+\frac{1}{3}tr(A^{3})tr(A)+\frac{1}{8}[tr(A^{2})]^{2}+\frac{1}{4}tr(A^{2})[tr(A)]^{2}+\frac{1}{24}[tr(A)]^{4}.$
(2.17)
$\begin{split}c_{n,5}\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{5}d\sigma(\xi)=&\frac{1}{5}tr(A^{5})+\frac{1}{4}[tr(A^{4})]tr(A)+\frac{1}{6}[tr(A^{3})][tr(A^{2})]+\frac{1}{6}[tr(A^{3})][tr(A)]^{2}\\\
&+\frac{1}{8}[tr(A^{2})]^{2}(trA)+\frac{1}{12}tr(A^{2})[tr(A)]^{3}+\frac{1}{120}[tr(A)]^{5}.\end{split}$
(2.18)
$\begin{split}c_{n,6}\int_{\mathbb{S}^{n}}\big{(}<A\xi,\xi>\big{)}^{6}d\sigma(\xi)=&\frac{1}{6}tr(A^{6})+\frac{1}{5}[tr(A^{5})]tr(A)+\frac{1}{8}[tr(A^{4})][tr(A^{2})]+\frac{1}{18}[tr(A^{3})]^{2}\\\
&+\frac{1}{8}[tr(A^{4})][tr(A)]^{2}+\frac{1}{6}[tr(A^{3})][tr(A^{2})][tr(A)]+\frac{1}{48}[tr(A^{2})]^{3}\\\
&+\frac{1}{18}[tr(A^{3})][tr(A)]^{3}+\frac{1}{16}[tr(A^{2})]^{2}[tr(A)]^{2}\\\
&+\frac{1}{48}[tr(A^{2})][tr(A)]^{4}+\frac{1}{720}[tr(A)]^{6}.\end{split}$
(2.19)

We conclude this section by noting that Eq. (2.15) can be obtained from
Corollary 2.3 and Theorem 2.4 in [7], while to the best of our knowledge
Equations (2.16)-(2.19) are new.

##  3\. A new proof of MacMahon Master Theorem

This section is devoted to provide a simple proof for MacMahon Master theory.
This theory has been central in combinatorics and in the theory of angular
momentum of systems of particles as well. The original proof is due to P. A.
MacMahon [25] while another proof depending on complex analysis was provided
by I. J. Good in [14]. Our proof seems to be the simplest. Making use of the
results in the previous section, let us first present an integral
representation of a particular determinant as follows.

######  Corollary 3.1.

Let $B\in M_{n}(\mathbb{C})$ with $\lVert B\rVert_{2}=\sup_{\lVert
x\rVert=1}\lVert Bx\rVert_{2}<1$ then

$\int_{\mathbb{S}^{n}}\dfrac{d\sigma(\xi)}{\big{(}<(I_{n}-B)\xi,\xi>\big{)}^{n}}=\det(I_{n}-B)^{-1}.$
(3.1)

###### Proof.

By Theorem 2.1, we know that for each $k\in\mathbb{N}$ the following identity
holds

$\int_{\mathbb{S}^{n}}\binom{n+k-1}{k}\big{(}<B\xi,\xi>\big{)}^{k}d\sigma(\xi)=\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}^{n}\end{subarray}}\lambda^{\alpha}.$

Since $\lVert B\rVert_{2}<1$ the series
$\sum_{k=0}^{\infty}\binom{n+k-1}{k}\big{(}<B\xi,\xi>\big{)}^{k}$ converges
absolutely and is dominated by $\dfrac{1}{(1-\lVert B\rVert_{2})^{n}}$. Hence
by Lebesgue dominated convergence theorem we get

$\int_{\mathbb{S}^{n}}\sum_{k=0}^{\infty}\binom{n+k-1}{k}\big{(}<B\xi,\xi>\big{)}^{k}=\sum_{k=0}^{\infty}\sum_{\begin{subarray}{c}|\alpha|=k\\\
\alpha\in\mathbb{N}^{n}\end{subarray}}\lambda^{\alpha}$

i.e.

$\int_{\mathbb{S}^{n}}\big{(}1-<B\xi,\xi>\big{)}^{-n}d\sigma(\xi)=\sum_{\alpha\in\mathbb{N}^{n}}\lambda^{\alpha}=\prod_{j=1}^{n}\dfrac{1}{1-\lambda_{j}}=\dfrac{1}{\det(I_{n}-B)}.$
(3.2)

∎

######  Remark 3.1.

Note that the monotone convergence theorem ensures that Eq. (3.1) remains true
for any positive semidefinite matrix B. Also, it is easy to see that for any
$c\in\mathbb{C}$ with $|c|>1$ and for any non-zero matrix $A\in
M_{n}(\mathbb{C})$, we have

$\int_{\mathbb{S}^{n}}\dfrac{d\sigma(\xi)}{\big{(}<(c\lVert
A\rVert_{2}I_{n}-A)\xi,\xi>\big{)}^{n}}=\det(c\lVert
A\rVert_{2}I_{n}-A)^{-1}.$

Now, we are ready to present our proof for MacMahon Master theorem.

######  Theorem 3.1 (MacMahon).

Let $A=(a_{ij})\in M_{n}(\mathbb{C})$, $\alpha\in\mathbb{N}^{n}$ and
$x=(x_{1},x_{2},\cdots,x_{n})^{T}$ be a formal variable. Then the coefficient
of $x^{\alpha}intheexpressionof\big{(}Ax\big{)}^{\alpha}$ is equal to the
coefficient of $x^{\alpha}$ in the expansion of $\det(I_{n}-\Delta(x)B)^{-1}$,
where $\Delta(x)$ is the diagonal matrix given by
$\Delta(x)=\texttt{diag}(x_{1},_{2},\cdots,x_{n})$.

###### Proof.

For$k\in\mathbb{N}$ and for each $\beta\in\mathbb{N}^{n}$ with $|\beta|=k$,
denote by $c_{\beta}$ the coefficient of $x^{\beta}$ in
$\big{(}Ax\big{)}^{\beta}$. Since
$\lambda\big{(}\Delta(x)A\big{)}=\lambda\big{(}A\Delta(x)\big{)}$ then by
(2.7) we have

$\displaystyle h_{k}\Big{(}\lambda\big{(}\Delta(x)A\big{)}\Big{)}$
$\displaystyle=c_{n,k}\int_{\mathbb{S}^{n}}\big{(}<A\Delta(x)\xi,\xi>\big{)}^{k}d\sigma(\xi)$
$\displaystyle=c_{n,k}\int_{\mathbb{S}^{n}}\bigg{[}\Big{<}\big{(}\sum_{j=1}^{n}a_{1j}x_{j}\xi_{j},\sum_{j=1}^{n}a_{2j}x_{j}\xi_{j},\cdots,\sum_{j=1}^{n}a_{nj}x_{j}\xi_{j}\big{)}^{T},\xi\Big{>}\bigg{]}^{k}d\sigma(\xi)$

Applying again the multinomial theorem, we get

$\displaystyle h_{k}\Big{(}\lambda\big{(}\Delta(x)A\big{)}\Big{)}$
$\displaystyle=c_{n,k}\sum_{\begin{subarray}{c}|\beta|=k\\\
\beta\in\mathbb{N}^{n}\end{subarray}}\binom{k}{\beta}\int_{\mathbb{S}^{n}}c_{\beta}x^{\beta}\xi^{\beta}\overline{\xi}^{\beta}d\sigma(\xi)=c_{n,k}\sum_{\begin{subarray}{c}|\beta|=k\\\
\beta\in\mathbb{N}^{n}\end{subarray}}\binom{k}{\beta}\dfrac{(n-1)!\beta!}{(n-1+\lvert\beta\rvert)!}$
(3.3)
$\displaystyle=\dfrac{(n+k-1)!}{(n-1)!k!}\sum_{\begin{subarray}{c}|\beta|=k\\\
\beta\in\mathbb{N}^{n}\end{subarray}}\frac{k!}{\beta!}c_{\beta}x^{\beta}\dfrac{(n-1)!\beta!}{(n-1+k)!}=\sum_{\begin{subarray}{c}|\beta|=k\\\
\beta\in\mathbb{N}^{n}\end{subarray}}c_{\beta}x^{\beta},$ (3.4)

where the second equality in (3.3) follows from (2.3). Finally, taking the sum
over $k\in\mathbb{N}$ and using (3.2) we obtain

$\det(I_{n}-\Delta(x)B)^{-1}=\sum_{\alpha\in\mathbb{N}^{n}}\lambda^{\alpha}\Big{(}(\Delta(x)A\Big{)}=\sum_{k=0}^{\infty}h_{k}\Big{(}\lambda\big{(}\Delta(x)A\big{)}\Big{)}=\sum_{\beta\in\mathbb{N}^{n}}c_{\beta}x^{\beta}.$

∎

##  4\. On the monotonicity of complete symmetric polynomials

In this section, we extend the definition of k-homogeneous complete symmetric
polynomials where $k\in\mathbb{N}$ to arbitrary $p\in\mathbb{R}^{+}$. We then
exploit the definition to introduce a natural extension for the term-
normalized homogeneous polynomial. Using the theory of majorization, we obtain
a Schur convexity of this extension. As a consequence , we obtain a
generalization of Theorem 7.3 in [10] and we answer an open problem raised in
[31] as well.

Throughout this section, we shall follow the standard notation used in the
study of symmetric polynomials and the theory of majorization. Thus, for
$m$-tuples $\lambda,\mu\in\mathbb{R}^{m}_{+}$ we say that $\lambda$ is
majorized by $\mu$ and we will write $\lambda\preceq\mu$ if

$\sum^{k}_{i=1}\lambda_{[i]}\leq\sum^{j}_{i=1}\mu_{[i]},\ j=1\cdots
n-1\quad\mbox{ and }\quad|\lambda|=|\mu|,$

where $x_{[i]}$ is the $i$he component obtained from $x=(x_{1},\cdots,x_{n})$
after arranging its components in decreasing order. In the case where
$\mu\in\mathbb{R}^{n}_{+}$ and $n<m$, then we shall complete the components of
$\mu$ to an $m$-tuple by adding zeros so that the preceding notion is still
well defined. If $\lambda$ and $\mu$ are partitions of (possibly different)
natural integers then we shall write

$\lambda\sqsubseteq\mu\quad\mbox{if}\quad\frac{\lambda}{|\lambda|}\preceq\frac{\mu}{|\mu|},$

For each $k\in\mathbb{N}$, we shall denote by $H_{k}$ to be the normalized
complete symmetric polynomial on $\mathbb{C}^{n}$ i.e.

$H_{k}(z)=\frac{h_{k}(z)}{c_{n,k}}.$ (4.1)

Given $\lambda=(\lambda_{1},\cdots,\lambda_{m})\in\mathbb{N}^{m}$, the term-
normalized homogeneous complete symmetric function is defined on
$\mathbb{C}^{n}$ by

$H_{\lambda}(z)=\prod_{i-1}^{m}H_{\lambda_{i}}(z).$ (4.2)

In addition, for $x\in\mathbb{R}^{n}_{+}$ and for the map defined by

$\mathfrak{H}_{\lambda}(x):=\sqrt[|\lambda|]{H_{\lambda}(x)},$ (4.3)

the authors in [10] proved the monotonicity of $\mathfrak{H}_{\lambda}$ on
$\mathbb{R}^{n}_{+}$ as follows.

######  Theorem 4.1 (Theorem 7.3, [10]).

Given integer partitions $\lambda$ and $\mu$ with $\lambda\sqsubseteq\mu$,
then for any $x\in\mathbb{R}^{n}_{+}$ we have

$\mathfrak{H}_{\lambda}(x)\leq\mathfrak{H}_{\mu}(x).$ (4.4)

In [35], the author used the Harish-Chandra-Itzykson-Zuber (HCIZ) integral to
represent the term-normalized Schur polynomials in order to obtain a
monotonicity result concerning a conjecture formulated in [10]. As mentioned
earlier, we shall next use the integral representation obtained in Section 2
to provide a generalization of the preceding theorem. Our work is based on
majorization methods for moments and thus differs from the algebraic approach
used in [10]. Since the integrand in HCIZ involves two matrices, the author in
[35] was able to express the term-normalized Schur polynomial as a single
integral (over the set of unitary matrices). The existence of the exponential
term in HCIZ was needed to obtain a Schur convexity result. In our case, the
integral is over a finite product of spheres and the integrand is given by
powers of numerical values and so many difficulties naturally arise.

For an arbitrary $z\in\mathbb{C}^{n}$, we shall write
$Z:=\texttt{diag}(z_{1},\cdots,z_{n})\in M_{n}(\mathbb{C})$. Thus, by (2.5)
and (2.6) we have the following representation for the normalized complete
symmetric polynomials

$H_{k}(z)=\int_{\mathbb{S}^{n}}\big{(}<Z\xi,\xi>\big{)}^{k}d\sigma(\xi)$ (4.5)

In [15], the author proved that $H_{2k}$ is positive definite on
$\mathbb{R}^{n}$ using a differential operator approach and some identities
between symmetric polynomials. Following a post by T. Tao concerning this
issue, two proofs were presented by I. Roventa and L. E. Temereanca for the
positivity but still based on polynomial identities and some computations. In
the next corollary, we present our own proof not only for the positivity but
also for the definitness of this property for $H_{2k}$.

######  Corollary 4.1.

The even degree complete symmetric polynomials are positive definite on
$\mathbb{R}^{n}$.

###### Proof.

Using the representaion (4.5), we know that for any $x\in\mathbb{R}^{n}$ we
have

$H_{2k}(x)=\int_{\mathbb{S}^{n}}\big{(}<X\xi,\xi>\big{)}^{2k}d\sigma(\xi),$
(4.6)

where $X=\texttt{diag}(x_{1},\cdots,x_{n})$ so that positivity is clear. If
$H_{2k}(x)=0$ then the map

$\mathbb{S}^{n}\ni\xi\longrightarrow\big{(}<X\xi,\xi>\big{)}^{2}\in\mathbb{R}_{+}$

is equal to zero almost everywhere on $\mathbb{S}^{n}$. By continuity of the
preceding map, we obtain $<X\xi,\xi>=0$ for all $\xi\in\mathbb{S}^{n}$ or
equivalently $x=0$. ∎

Motivated by the latter property, we introduce the following definition and we
shall use the same notation as in (4.3)

######  Definition 4.1.

Given $\nu=(\nu_{1},\nu_{2},\cdots,\nu_{m})\in\mathbb{N}^{m}$, let the map
$\mathfrak{H}_{2\nu}$ be defined on $\mathbb{R}^{n}$ by

$\mathfrak{H}_{2\nu}(x):=\sqrt[2|\nu|]{H_{2\nu}(x)}.$

In the following proposition, we examine the possibility of an extension
Formula (4.6) from the case of integer powers to the case of arbitrary real
powers.

######  Proposition 4.1.

Let $X\in\ M_{n}(\mathbb{C})$ be a non-zero positive semidefinite matrix. Then
for any $p\in\mathbb{R}$, the integral

$\int_{\mathbb{S}^{n}}\big{(}<X\xi,\xi>\big{)}^{p}d\sigma(\xi)$ (4.7)

is well defined with values in $]0,\infty[$.

###### Proof.

Suppose first that $p\geq 0$, then clearly the proposition follows from the
inequality

$\int_{\mathbb{S}^{n}}\big{(}<X\xi,\xi>\big{)}^{p}d\sigma(\xi)\leq\int_{\mathbb{S}^{n}}\big{(}w(X)\big{)}^{p}d\sigma(\xi)=\big{(}w(X)\big{)}^{p},$

where $w(\cdot)$ denotes the numerical radius norm. On the other hand, if
$p<0$ and if $v$ denotes the Lebesgue measure on $\mathbb{C}^{n}$. then we
consider the following set

$F:=\Big{\\{}z\in\mathbb{C}^{n}\mid<Xz,z>=0\Big{\\}}.$

As $X\geq 0$ and $X\neq 0$ then $F=\ker X$ which is an intersection of
hyperplanes in $\mathbb{C}^{n}$. So that $v(F)=0$ or equivalently
$F\backslash\\{0\\}$ is measurable and $v(F\backslash\\{0\\})=0.$ Next, we
consider the set defined by

$E:=\Big{\\{}\xi\in\mathbb{S}^{n}\mid<X\xi,\xi>=0\Big{\\}}.$

By the definition of the sigma-algebra on $\mathbb{S}^{n}$ (cf. for example
Chapter 6, Section 3 in [36]), the set $E$ is measurable if and only if the
set

$\widetilde{E}:=\Big{\\{}z\in B(0,1)\mid z\neq 0\mbox{ and }\frac{z}{|z|}\in
E\Big{\\}}$

is Lebesgue measurable in $\mathbb{C}^{n}$. Notice that by representing $z$ in
polar coordinates $z=r\xi$ with $r\in]0,\infty[$ we obtain

$\widetilde{E}=B(0,1)\cap F\backslash\\{0\\}$

and hence $E$ is measurable. Let $\mu_{1}$ be the corresponding radial measure
on $(0,\infty)$ i.e. $\mu_{1}(\theta)=\int_{\theta}r^{2n-1}dr$ for every
Lebesgue measurable set $\theta$ in $(0,\infty)$. Since
$F\backslash\\{0\\}=(0,\infty)\times E=\bigcup_{l\in\mathbb{N}}(0,l)\times E$
then

$0=v(F\backslash\\{0\\})=\lim_{l\rightarrow\infty}\mu_{1}((0,l))\sigma(E)=\lim_{l\rightarrow\infty}\frac{l}{2n}\sigma(E)$

which holds only in the case when $\sigma(E)=0$. Let $f$ be the map on
$\mathbb{S}^{n}$ defined by $f(\xi)=\big{(}<X\xi,\xi>\big{)}^{p}$ with values
in $[0,\infty]$. As $f$ is real valued continuous function on
$E^{c}:=\mathbb{S}^{n}\backslash\\{E\\}$ then $f$ is measurable on $E^{c}$ and
therefore on $\mathbb{S}^{n}$. Moreover, as $E$ is a null-$\sigma$-set then

$\int_{\mathbb{S}^{n}}\big{(}<X\xi,\xi>\big{)}^{p}d\sigma(\xi)=\int_{\mathbb{S}^{n}}\big{(}<X\xi,\xi>\big{)}^{p}\chi_{E^{c}}(\xi)d\sigma(\xi).$
(4.8)

By the unitary invariance of $\sigma$, one can assume that $X$ is diagonal and
the preceding formula remains true. Indeed, compared to the work in Section 1
the only factor that needs to be examined is $\chi_{E^{c}}$. However, writing
$X=U^{\star}\Delta U$ with $\Delta$ being diagonal matrix and $U\in U_{n}$
then by the invariance of the Euclidean norm under unitary transformation we
have

$E=\Big{\\{}\xi\in\mathbb{S}^{n}\mid<X\xi,\xi>=0\Big{\\}}=\Big{\\{}\xi\in\mathbb{S}^{n}\mid<\Delta
U\xi,U\xi>=0\Big{\\}}=\Big{\\{}\eta\in\mathbb{S}^{n}\mid<\Delta\eta,\eta>=0\Big{\\}},$

where the last equality folllows from th e fact that $U$ is an isometry. The
above equation shows that $\chi_{E}(U\xi)=1$ if and only if
$\xi\in\mathbb{S}^{n}$ and $<\Delta U\xi,U\xi>=0$ which is equivalent to say
that $\xi\in E$. Therefore, $\chi_{E^{c}}(U\xi)=\chi_{E^{c}}(\xi)$ and thus we
can assume $X=\texttt{diag}(x_{1},\cdots,x_{n})$ with $x=\lambda(X)$ being a
non-zero vector in $\mathbb{R}^{n}_{+}$. It remains to prove that the value of
the integral is strictly positive. For this, we let
$\check{x}=\min\\{x_{i}\mid x_{i}\neq 0\\}$ and $\hat{x}=\max\\{x_{i}\mid
x_{i}\neq 0\\}$ then clearly $\check{x}>0$, $\hat{x}>0$ and

$\hat{x}^{p}\leq\big{(}<X\xi,\xi>\big{)}^{p}\leq\check{x}^{p},$

where the above inequality holds for all $\xi\in E^{c}$. Therefore, by (4.8)
we get

$\hat{x}^{p}\leq\int_{\mathbb{S}^{n}}\big{(}<X\xi,\xi>\big{)}^{p}d\sigma(\xi)\leq\check{x}^{p}.$

∎

Motivated by the above result, we introduce the following generalization for
normalized (and term normalized) complete homogeneous polynomials on
$\mathbb{R}^{n}_{+}$.

######  Definition 4.2.

Let $p\in\mathbb{R}$. The $p$-homogeneous function $H_{p}$ is the map defined
on $\mathbb{R}^{n}_{+}$ by

$H_{p}(x)=\begin{cases}\int_{\mathbb{S}^{n}}\big{(}<X\xi,\xi>\big{)}^{p}d\sigma(\xi)&\mbox{if}\quad
x\neq 0\\\ 0,&\mbox{if}\quad x=0,\end{cases}$ (4.9)

where $X=\texttt{diag}(x_{1},\cdots,x_{n})$. In addition, given
$\lambda=(\lambda_{1},\cdots,\lambda_{m})\in\mathbb{R}^{m}$ we define the
term-normalized $\lambda$-function by

$H_{\lambda}(x)=\prod_{i=1}^{m}H_{\lambda_{i}}(x).$ (4.10)

In [38], and based on Muirhead Theorem, Y. L. Tong provided an inequality for
the expectation of product of independent random variables with a majorization
condition. The case where the probability measure is considered on
$[0,\infty[$ can be found in [27] (see p. 107). We formulate this result in
the context of our work and we provide a proof motivated by the work of S. Sra
in [35] which is based on a Schur’s result for convex symmetric functions.

######  Theorem 4.2.

Let $f:\mathbb{S}^{n}\longrightarrow\mathbb{R}_{+}$ be a measurable function.
Let $I\subseteq\mathbb{R}$ be an interval and assume that

$\int_{\mathbb{S}^{n}}f^{p}(\xi)d\sigma(\xi)<\infty$

for all $p\in I$. Then for any $m\in\mathbb{N}$ the map

$F(\lambda):=\prod_{i=1}^{m}\int_{\mathbb{S}^{n}}f^{\lambda_{i}}(\xi)d\sigma(\xi)$
(4.11)

is Schur convex on $I^{m}$. Moreover, if $\lambda,\mu\in I^{m}$ are integer
partitions with $\lambda\sqsubseteq\mu$ then

$\sqrt[|\lambda|]{F(\lambda)}\leq\sqrt[|\mu|]{F(\mu)}$ (4.12)

###### Proof.

As $F$ is continuous and symmetric it is sufficient to prove $F$ is convex
(cf. [27] p. 97). Now, by Fubini’s theorem we write

$F(\lambda)=\int_{\mathbb{S}^{n}\times\mathbb{S}^{n}\times\cdots\mathbb{S}^{n}}\prod_{i=1}^{m}f^{\lambda^{i}}d(\xi^{i}),$

where $\xi^{i}$ is the coordinates in the $i$-th copy of $\mathbb{S}^{n}$.
Thus, for $\lambda,\mu\in I^{m}$, the power mean inequality yields

$F(\frac{\lambda+\mu}{2})=\int_{\mathbb{S}^{n}\times\mathbb{S}^{n}\times\cdots\mathbb{S}^{n}}\sqrt{\prod_{i=1}^{m}f^{\lambda^{i}}}\sqrt{\prod_{i=1}^{m}f^{\mu^{i}}}\\\
\leq\int_{\mathbb{S}^{n}\times\mathbb{S}^{n}\times\cdots\mathbb{S}^{n}}\frac{\prod_{i=1}^{m}f^{\lambda^{i}}+\prod_{i=1}^{m}f^{\mu^{i}}}{2}=\frac{F(\lambda)}{2}+\frac{F(\mu)}{2}.$

In case where $\lambda,\mu\in I^{m}$ are integer partitions the condition
$\lambda\sqsubseteq\mu$ is equivalent to
$\lambda^{|\mu|}\preceq\mu^{|\lambda|}$ (cf. [10]). Hence applying the Schur
convexity of $F$ on $I^{m|\mu}$ we obtain the following.

$\displaystyle\prod_{i=1}^{m}\int_{\mathbb{S}^{n}}f^{\lambda_{i}}$
$\displaystyle=\Big{[}\prod_{i=1}^{m}\int_{\mathbb{S}^{n}}f^{\lambda_{i}}\Big{]}^{\frac{|\mu|}{|\mu|}}$
$\displaystyle=\Big{[}\underbrace{\int_{\mathbb{S}^{n}}f^{\lambda_{1}}\int_{\mathbb{S}^{n}}f^{\lambda_{1}}\cdots\int_{\mathbb{S}^{n}}f^{\lambda_{1}}}_{|\mu|}\underbrace{\int_{\mathbb{S}^{n}}f^{\lambda_{2}}\int_{\mathbb{S}^{n}}f^{\lambda_{2}}\cdots\int_{\mathbb{S}^{n}}f^{\lambda_{2}}}_{|\mu|}\cdots\underbrace{\int_{\mathbb{S}^{n}}f^{\lambda_{m}}\int_{\mathbb{S}^{n}}f^{\lambda_{m}}\cdots\int_{\mathbb{S}^{n}}f^{\lambda_{m}}}_{|\mu|}\Big{]}^{\frac{1}{|\mu|}}$
$\displaystyle\leq\Big{[}\underbrace{\int_{\mathbb{S}^{n}}f^{\mu_{1}}\int_{\mathbb{S}^{n}}f^{\mu_{1}}\cdots\int_{\mathbb{S}^{n}}f^{\mu_{1}}}_{|\lambda|}\underbrace{\int_{\mathbb{S}^{n}}f^{\mu_{2}}\int_{\mathbb{S}^{n}}f^{\mu_{2}}\cdots\int_{\mathbb{S}^{n}}f^{\mu_{2}}}_{|\lambda|}\cdots\underbrace{\int_{\mathbb{S}^{n}}f^{\mu_{m}}\int_{\mathbb{S}^{n}}f^{\mu_{m}}\cdots\int_{\mathbb{S}^{n}}f^{\mu_{m}}}_{|\lambda|}\Big{]}^{\frac{1}{|\mu|}}$
$\displaystyle=\Big{[}\prod_{i=1}^{m}\int_{\mathbb{S}^{n}}f^{\mu_{i}}\Big{]}^{\frac{|\lambda|}{|\mu|}}.$

∎

The following corollaries are the main results of this section. Applying the
apreceding theorem to the case $f(\xi)=<X\xi,\xi>$, whenever
$x\in\mathbb{R}^{n}_{+}$ we obtain the following generalization of Theorem 4.1
for the case of term-normalized complete homogeneous polynomials to the term-
normalized $\lambda$-function as follows.

######  Corollary 4.2.

Let $\lambda\in\mathbb{R}^{m}$ and $\mu\in\mathbb{R}^{l}$. If
$\lambda\preceq\mu$ then

$H_{\lambda}(x)\leq H_{\mu}(x),\quad x\in\mathbb{R}^{n}_{+}.$

Moreover, if $\lambda,\mu$ are integer partitions with $\lambda\sqsubseteq\mu$
then

$\mathfrak{H}_{\lambda}(x)\leq\mathfrak{H}_{\mu}(x),\quad
x\in\mathbb{R}^{n}_{+}.$

Now let $x\in\mathbb{R}^{n}$ and consider the map
$f(\xi)=\big{(}<X\xi,\xi>\big{)}^{2}$ then for any $p\geq 0$ we have

$\int_{\mathbb{S}^{n}}f^{p}(\xi)d\sigma(\xi)<\infty.$

Notice that if $\lambda=(2\nu_{1},\cdots,2\nu_{m})$ and
$\mu=(2\eta_{1},2\eta_{2},\cdots,2\eta_{l})$ are two integer partitions then
$\lambda\sqsubseteq\mu$ if and only if $\nu\sqsubseteq\eta$. Applying Theorem
4.2 to $f(\xi)=\big{(}<X\xi,\xi>\big{)}^{2}$ we obtain

######  Corollary 4.3.

Let $\lambda\in\mathbb{R}^{m}_{+}$ and $\mu\in\mathbb{R}^{l}_{+}$. If
$\lambda\preceq\mu$ then

$\prod_{i=1}^{m}\int_{\mathbb{S}^{n}}\bigg{[}\big{(}<X\xi,\xi>\big{)}^{2}\bigg{]}^{\lambda_{i}}d\sigma(\xi)\leq\prod_{i=1}^{l}\int_{\mathbb{S}^{n}}\bigg{[}\big{(}<X\xi,\xi>\big{)}^{2}\bigg{]}^{\mu_{i}}d\sigma(\xi),\quad
x\in\mathbb{R}^{n}.$

If in addition $\nu,\eta$ are integer partitions with $\nu\sqsubseteq\eta$
then

$\mathfrak{H}_{2\nu}(x)\leq\mathfrak{H}_{2\eta}(x),\quad x\in\mathbb{R}^{n}.$
(4.13)

Note that applying the above equation to the cases

$\begin{cases}\nu=k-1\quad\mbox{and}\quad\eta=k\\\
\nu=(k-1,k+1)\quad\mbox{and}\quad\eta=(k,k)\end{cases}$

provides us with an entire sequence of inequalities similar to Newton’s
identities which is valid for all $x\in\mathbb{R}^{n}$. More precisely, we
have

$\sqrt[2k-2]{H_{2k-2}(x)}\leq\sqrt[2k]{H_{2k}(x)},\quad x\in\mathbb{R}^{n}$
(4.14)

and

$H_{2k-2}(x)H_{2k+2}(x)\leq H^{2}_{2k}(x),\quad x\in\mathbb{R}^{n}.$ (4.15)

Finally,we conclude by noting that (4.14) provides a proof for the open
problem concerning the sequence of inequalities that was raised in [31].

## References

  * [1] D. Agbor and W. Bauer, Heat flow and an algebra of Toeplitz operators, _Integral Equations and Operator theory_ 70 (2015), pp. 271-299 .
  * [2] S. Axler, P. Bourdon and R. Wade, Harmonic Function Theory, _Springer Science and Business Media_ , Vol. 137 Graduate texts in Mathematics, 2013.
  * [3] G. Aubrun and I. Nechita, The multiplicative property characterizes $l_{p}$ and $L_{p}$ norms, _Conflu. Math._ , Vol. 03 No. 04 (2011), pp. 637-647.
  * [4] W. Bauer, Berezin-Toeplitz quantization and composition formulas, _Journal of Functional Analysis_ 256 (2009), pp. 3107-3142.
  * [5] W. Bauer and H. Issa, Commuting Toeplitz operators with quasi-homogeneous symbols on the Segal-Bargmann space, _J. Math. Anal. Appl._ 386 (2012), pp. 213-235.
  * [6] R. Bhatia, Matrix analysis, _Springer Science and Business Media,_ , 1996.
  * [7] R. Bhatia and J .A.R. Holbrook, Unitary invariance and spectral variation, _Linear Algebra Appl.,_ 95 (1987), pp. 43-68.
  * [8] Y.-M.Chu, G.-D.Wang and X.-H. Zhang, The Schur multiplicative and harmonic convexities of the complete symmetric function. _Math. Nachr_. 284 (5–6) (2011), pp. 653–663.
  * [9] L. A. Coburn, On the Berezin-Toeplitz calculus, _Proc. Am. Math. Soc._ 129 (11) (2007), 3331-3338.
  * [10] A. Cuttler, C. Greene and M. Skandera, Inequalities for symmetric means, _European Journal of Combinatorics_ Vol.32, I 6 (2011), pp. 745-761.
  * [11] G.B. Folland, How to integrate a polynomial over a sphere, _American Mathematical Society_ , Vol. 108 No. 5 (2001), pp. 446-448.
  * [12] G.B. Folland, Real analysis, _John Wiley, New York_ , 1999.
  * [13] C.-K. Fong and J. A.R. Holbrook, Unitarily invariant operator norms, _Canad. J. Math._ , 35 ( 1983), pp. 274-299.
  * [14] I. J. Good, A short proof of MacMahon’s ’Master Theorem,_Proc. Cambridge Philos. Soc._ 58 pp. 160.
  * [15] D.B. Hunter, The positive-definiteness of the complete symmetric functions of even order. Math. _Proc. Camb. Philos. Soc._ 82 (2) (1977), pp. 255–258.
  * [16] D.A. Harver, Matrix Algebra From a Statistician’s Perspective, _Springer Science and Business Media_ , 2008.
  * [17] H.V. Henderson and S.R. Searle, S.R., The vec-permutation, the vec-operator and Kronecker products: a review. _Linear Multilinear Algebra_ 9 (1981), pp. 271-288.
  * [18] H. Issa, Construction of some classes of commutative Banach and $C^{\star}$-algebras of Toeplitz operators,_C. R. Acad. Sci. Paris_ , Ser 1 357 (2019), pp. 389-394.
  * [19] H. Issa, The analysis of Toeplitz operators, commutative Toeplitz algebras and applications to heat kernel constructions, _S.U.B Göttingen_ , 2012.
  * [20] T. Kania, A short proof of the fact that the matrix trace is the expectation of the numerical values, _American Mathematical Monthly_ , 122 Vol. 8 (2015), pp. 782-783.
  * [21] S. G. Krantz, Function theory of several complex variables, _AMS Chelsea Publishing Series, Amer. Math. Soc._ , 2001.
  * [22] Chi-K. Li and N. K. Tsing, Norms that are invariant under unitary similarities and the C-numerical radii, _Linear and Multilinear Algebra,_ 24 (1989), pp. 209-222.
  * [23] R. Loewy and D. London, A note on the inverse problem for nonnegative matrices,_Linear Multilinear Algebra_ 6 (1978), pp. 83-90.
  * [24] I. Macdonald, Symmetric Functions and Hall Polynomials, _Oxford science publications: Oxford mathematical monographs_ , second edition, 1995.
  * [25] P. A. MacMahon, Combinatory analysis,_Cambridge University Press,_ Vol I and II 1915–16.
  * [26] M. Marcus and L. Lopes, Inequalities for symmetric functions and Hermitian matrices,_Canad. J. Math._ 9 (1957), pp. 305-312.
  * [27] A. W. Marshall, I. Olkin and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications, _Springer-Verlag New York,_ second edition, 2011.
  * [28] J. B. McLeod, On four inequalities in symmetric functions,_Proceedings of the Edinburgh Mathematical Society,_ 11-4 (1959), pp. 211-219.
  * [29] L. Mirsky, Symmetric gauge functions and unitarily invariant norms, _Quart . J. Math. , Oxford,_ Ser. 2, 11 (1960), pp. 50-59.
  * [30] C. P. Niculescu, Convexity according to the geometric means, _Math. Inequal. Appl._ 3(2) (2000), pp. 155-167.
  * [31] I. Rovenţa, and L. E. Temereanca, A Note on the Positivity of the Even Degree Complete Homogeneous Symmetric Polynomials, _Mediterr. J. Math._ (2019), pp. 1-16.
  * [32] W. Rudin, Function theory in the unit ball of $\mathbb{C}^{n}$, _A series of comprehensive studies in Mathematics, Springer_ , 2008.
  * [33] S. Saitoh, Theory of reproducing kernels and its applications, _Pitman research notes in mathematics series, Longman Scientific & Technical_, 1988.
  * [34] S. Sra, New concavity and convexity results for symmetric polynomials and their ratios, _Linear and Multilinear Algebra,_ 68 Vol 5 (2020), pp. 1031-1038.
  * [35] S. Sra., On inequalities for normalized Schur functions,_European J. Combin._ 51 (2016), pp. 492–494.
  * [36] E.M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, _Princeton, NJ, USA: Princeton Univ. Press_ , 2005.
  * [37] R.C. Thompson, Convex and concave functions of singular values of matrix sums, _Pacific J. Math._ , 66 (1976), pp. 285-290.
  * [38] Y. L. Tong, An ordering theorem for conditionally independent and identically distributed random variables, _Ann. Statist._ , 5 (1977), pp. 274-277.
  * [39] K. Zhu, Operator theory in function spaces, _Mathematical surveys and monographs, Amer. Math. Soc._ , 2007.