The minimum principle on the sequential complete local convex spaces

Abstract

The minimum principle establishes cases when the infimum of a family of functions is plurisubharmonic. In [18], the author has otained this principle on the space that is the special inductive limits of Banach spaces. In this paper, we will establish the minimum principle on some classes of Hausdorff sequential complete local convex spaces.

1 Introduction

In this section, we will review some definitions and the motivation to obtain the main result of the paper. In this paper by U we always mean the unit disk in ${\mathbb {C}}$, ${\overline{U}}$ is the closure of U and S is the unit circle. And with X is any set, we denote H(X) the set of all holomorphic mappings of a neighborhood of ${\overline{U}}$ into X, ${\overline{H}}(X)$ the set of all mappings of unit disk continuous on ${\overline{U}}$ and holomorphic on U into X

Let W be an open set in a topological vector space Z. We denote PSH(W) is the set of the plurisubharmonic (psh) functions on W, that is $u\in \text {PSH}(W)$ if anh only if it is an upper semicontinuous (usc) functions and satisfy the subaveraging inequality follow

$$\begin{aligned} u(f(0))\le \frac{1}{2\pi }\int _0^{2\pi }u(f(e^{i\theta }))d\theta , \forall f\in {\overline{H}}(W). \end{aligned}$$

(1.1)

Let $P: Z\rightarrow Y$ be a projection from Z onto a subspace $Y\subset Z$ and $\phi \in \text {PSH}(W)$. Then the subenvelope of $\phi$ is defined as follow

$$\begin{aligned} I_P\phi (z)=\inf \{\phi (w): w\in W, P(w)=z\}, \forall z\in W_0=P(W). \end{aligned}$$

(1.2)

From (1.1) we see that the supremum of any family of psh functions on W is also psh, provided it is usc. But the minimum of two psh functions on W need not to be psh. So the subenvelope $I_P\phi$ of $\phi$ is not psh in general.

In [10], C. Kiselman has been proved that under some conditions, the subenvelope is psh that he called “the minimum principle”, as follow

Theorem 1.1

Suppose that W is a pseudoconvex open set in ${\mathbb {C}}^2$ and $\phi$ is a psh function on W such that both W and $\phi$ are S-invariant, i.e., if $(z,w)\in W$, then $(z,e^{i\alpha }w)\in W$ and $\phi (z,e^{i\alpha }w)=\phi (z,w)$ for every real number $\alpha$. Also suppose that for every $(z,w)\in W$ the fiber $\{x| (z,x)\in W\}$ are connected. If $P(z,w)=z$, then the function $I_P\phi (z)=\inf _{(z,w)\in W}\phi (z,w)$ is subharmonic.

We can see in the Theorem 1.1 that all the sufficient conditions such as “pseudoconvex”, “S-invariant”, “psh” and “connected fibers” are very close to the necessary. Indeed, first clearly $\phi$ must be psh. If W consists of two disjoint components, then the function $I_P\phi$ is the minimum of two psh functions and it is easy to make the minimum not psh. In [18], the author has given two example to show that the remain conditions are important. We cite here for convenient.

Example 1.2

Let W be the unit bidisk in ${\mathbb {C}}^2$ and $\phi (z,w)=\log |2zw-1|$. Then $I_P\phi (z)=-\infty$ when $|z|\ge \frac{1}{2}$ and $I_P\phi (0)=0$ when $z=0$. This means that $I_P\phi$ is not subharmonic.

Example 1.3

Let $W=\{(z,w)\in {\mathbb {C}}^2: |z|<2, -|z|^2+4<|w|^2\}$ and $\phi (z,w)=|w|^2$. Then both W and $\phi$ are S-invariant but W is not pseudoconvex. Now $I_P\phi (0)=4$ and $I_P\phi (z)=3$ when $|z|=1$. Again we see that $I_P\phi$ is not subharmonic.

In [16] the author has been investigated the following situation: Let $Y={\mathbb {C}}^n$, let $W_0$ be an open set in Y, and let $Z=H({\mathbb {C}}^n)$. We define $W=\{f\in Z: f({{\overline{U}}})\subset W_0\}$. Let $Pf=f(0)$ be the canonical projection of Z onto Y. It was proved in [16] that, under mild conditions on a function $\phi$ on W, its subenvelope is psh. Here, we can see that the main results in [10] and [16] are very different each other because their conditions such as, by [16], Z was infinitely dimensional, $\phi$ was not psh or even usc, and W was not psuedoconvex.

In [18], the author has been given the definition P-psh function. Let us call an absolutely measurable function (see section 3) $\phi$ on W be P-psh if for every holomorphic mapping $f\in H(Z)$ such that $f(S)\subset W$ and Pf maps ${{\overline{U}}}$ into $W_0=P(W)$ we have

$$\begin{aligned} I_P\phi (Pf(0))\le \frac{1}{2\pi }\int _0^{2\pi }\phi (f(e^{i\theta }))d\theta . \end{aligned}$$

(1.3)

Comparing the inequality (1.3) with the subaveraging inequality (1.1) we see that a bigger choice of values of $\phi$ in the left side is compensated by also a bigger choice of mappings f in the right side. So the inequality (1.3) does not follow immediately from the plurisubharmonicity of $\phi$. It will if we need to verify (1.3) only for $f\in H(W)$. In this case, the function $\phi$ is called a weakly P-psh function.

By considering the P-psh function $\phi$ on a open subset in Z that is the special inductive limits of Banach spaces $Z_i$, in [18], the author has been improved the Kiselman’s framework in [10] to be a new result that include the results from [16]. In this paper, we use the same technics similar to [18] to prove the minimum principle on the Hausdorff sequential complete local convex spaces.

2 Holomorphic ${\overline{U}}-$ actions on the local convex space

First we review some basic properties in functional analysis. Let X be a compact metric space and Z be Hausdorff sequential complete local convex space. Let $F: X\rightarrow Z$ be continuous mapping. We denote the set A is the closure of the convex hull of F(X) and $p_A$ is Minkowski functional of the set A. If we set $Z_A=span(A)$ then $(Z_A, p_A)$ is a Banach space. Moreover, if $Z'$ is the topological dual space of Z and $z^*\in Z'$ then $z^*\in (Z_A, p_A)'$.

Lemma 2.1

Let Z be a Hausdorff sequential complete local convex space. If f is a continuous mapping of ${{\overline{U}}}^n$ into Z holomorphic in $U^n$ then it is holomorphic in $U^n$ as a mapping into $(Z_A, p_A)$ with $A=\overline{conv(f({\overline{U}}^n))}$.

Proof

Since f is continuous as a mapping from $U^n$ into $(Z_A, p_A)$ so if g is a function is defined as follow

$$\begin{aligned} g(\zeta )=\frac{1}{(2\pi i)^n}\int _S...\int _S \frac{f(\xi _1,\ldots ,\xi _n)}{\prod _{j=1}^n (\xi _j-\zeta _j)}d\xi _1...d\xi _n, \; \forall \zeta =(\zeta _1,\ldots ,\zeta _n)\in U^n, \end{aligned}$$

then g maps $U^n$ holomorphically into $(Z_A, p_A)$. Let $z^*\in Z'$. If $h(\zeta )=z^*(f(\zeta )), \forall \zeta \in U^n$ then h is holomorphic on $U^n$. By Cauchy integral formula we have

$$\begin{aligned} h(\zeta )=\frac{1}{(2\pi i)^n}\int _S...\int _S \frac{h(\xi _1,\ldots ,\xi _n)}{\prod _{j=1}^n(\xi _j-\zeta _j)}d\xi _1...dxi_n. \end{aligned}$$

From the definition of function g above we have $z^*(g(\zeta ))=h(\zeta )$. So $z^*(f(\zeta ))=z^*(g(\zeta )), \forall \zeta \in U^n, z^*\in Z'$. This imply that $f=g$ on $U^n$. $\square$

Suppose that $E=\{\zeta _1,\ldots , \zeta _n\}\subset U$ and f is a continuous mapping of ${\overline{U}}\setminus E$ into Z, holomorphic in $U\setminus E$. We say that f has a pole of order at most $m_j$ at $\zeta _j$ if

$$\begin{aligned} \lim _{\zeta \rightarrow \zeta _j}(\zeta -\zeta _j)^{m_j+1}f(\zeta )=0. \end{aligned}$$

Corollary 2.2

Let Z be a Hausdorff sequential complete local convex space. If f is a continuous mapping of ${\overline{U}}\setminus E$ into Z, holomorphic in $U\setminus E$ with poles of order at most $m_j$ at $\zeta _j$, then there exist $g\in {\overline{H}}(Z_A)$ such that

$$\begin{aligned} f(\zeta )=\frac{g(\zeta )}{\prod _{j=1}^n(\zeta -\zeta _j)^{m_j}}. \end{aligned}$$

Proof

We set $h(\zeta )=\prod _{j=1}^n(\zeta -\zeta _j)^{m_j+2}f(\zeta )$. Then for each $\zeta _j\in E, j=1,\ldots ,n$ we have

$$\begin{aligned} h(\zeta _j)=\lim _{\zeta \rightarrow \zeta _j}h(\zeta )=0 \end{aligned}$$

So h is a continuous mapping of ${{\overline{U}}}$ into Z, holomorphic in U. By Lemma 2.1 then $h\in {\overline{H}}(Z_A)$.

On the other hand, it is easy to see that $h'(\zeta _j)=0, \forall 1\le j\le n$. So by Taylor expansion formula of h at $\zeta _j$ we have

$$\begin{aligned} g(\zeta ):=\frac{h(\zeta )}{\prod _{j=1}^n(\zeta -\zeta _j)^2}\in {\overline{H}}(Z_A) \end{aligned}$$

Since $g(\zeta )=\prod _{j=1}^n(\zeta -\zeta _j)^{m_j}f(\zeta )\in {\overline{H}}(Z_A)$ and the corollary follows. $\square$

Let Z be a sequential complete local convex space. A continuous mapping $A(\xi ,z)$ of ${\overline{U}}\times Z$ into Z is called a holomophic ${{\overline{U}}}$-action if:

i.:

A is a holomorphhic mapping of $U\times Z$ into Z

ii.:

For every $\xi \in {{\overline{U}}}$ the mapping $A(\xi , z)$ is linear in z

iii.:

$A(1,z)=z$

iv.:

$A(0,A(\xi ,z))\equiv A(0,z)$.

It followsn that $Pz=A(0,z)$ is a projection of Z onto $Y=PZ$ and the orbit $A(\xi ,z)$ of a point $z\in Z$ lies in the fiber of P over Pz.

Given a holomorphic ${{\overline{U}}}$-action A on Z. Let $W\subset Z$ be open set and $\phi$ be function on W. The set W is called S-invariant if $A(\xi ,z)\in W$ for every $z\in W$ and $\xi \in S$. The function $\phi$ is called S-invariant if W is S-invariant and $\phi (A(\xi ,z))=\phi (z)$ for every $z\in W$ and $\xi \in S$.

Lemma 2.3

Let Z be a Hausdorff sequential complete local convex space and A be a holomorphic ${{\overline{U}}}$-action on Z. Let $Pz=A(0,z)$ be a projection in Z. Let $E=\{\zeta _1,\ldots ,\zeta _n\}\subset U$ and let f be a continuous mapping of ${{\overline{U}}}\setminus E$ into Z, holomorphic in $U{\setminus } E$, with poles of order at most $m_j$ at $\zeta _j, 1\le j\le n$. We denote by B the Blaschke product as follow

$$\begin{aligned} B(\zeta )=\prod _{j=1}^n\left( \frac{\zeta -\zeta _j}{1-\overline{\zeta _j}\zeta }\right) ^{n_j}, \end{aligned}$$

where $n_j\ge m_j, 1\le j\le n$. If the mapping $h=Pf$ is in ${\overline{H}}(Z)$, then for every $\eta \in S$ the mapping $A(\eta B(\zeta ),f(\zeta ))$ is also in ${\overline{H}}(Z)$.

Proof

By Corollary 2.2 the mapping $g(\zeta )=Q(\zeta )f(\zeta )$, where $Q(\zeta )=\prod _{j=1}^n(\zeta -\zeta _j)^{m_j}$, is in ${\overline{H}}(Z_A)$. If we set

$$\begin{aligned} F(\xi ,\zeta )=A(\xi ,f(\zeta ))-h(\zeta )=\frac{A(\xi ,g(\zeta ))-Q(\zeta )h(\zeta )}{Q(\zeta )}, \end{aligned}$$

then $F(0,\zeta )\equiv 0$.

Set $G(\xi ,\zeta )=A(\xi ,g(\zeta ))-Q(\zeta )h(\zeta )$, we have $G(0,\zeta )\equiv 0$. But G is a continuous mapping of ${{\overline{U}}}^2$ into Z, holomorphic in $U^2$. So by Lemma 2.1 we have $G\in {\overline{H}}(Z_A)$.

By the Taylor expansion formula of G at (0, 0) we can see that $G(\xi ,\zeta )=\xi G_1(\xi ,\zeta )$, where $G_1$ is holomorphic in $U^2$ and continuous on the boundary where $\xi \not = 0$.

Since $G(\eta B(\zeta ),\zeta )=\eta B(\zeta )G_1(\eta B(\zeta ),\zeta )$, the mapping

$$\begin{aligned} A(\eta B(\zeta ),f(\zeta ))=F(\eta B(\zeta ),\zeta )+h(\zeta ) \end{aligned}$$

is holomorphihc in U. It is also continuous up to the boundary because $|B(\zeta )|\equiv 1$ on S and $|\zeta _j|<1, 1\le j\le n$.

$\square$

A function $\phi$ on an open set W in Z is called sequential usc (susc) if for every point $z\in W$ and every sequence of points $z_j\in W$ convergence to z we have

$$\begin{aligned} \limsup _{j\rightarrow \infty }\phi (z_j)\le \phi (z). \end{aligned}$$

Here we note that, if a function is usc then it is also sequential usc and if the space Z is metrizable then the both notions coincide.

A function $\phi$ on an open set W in Z is called sequential psh (spsh) if it is sequential usc and satisfy the subaveraging inequality.

3 Almost upper semicontinuous functions

Let ${\overline{U}}^N=\{z=(z_1, z_2,...): z_j\in {\mathbb {C}}, |z_j|\le 1\}$ be the product of countably many closed unit disk equipped with the product topology and, consequently, a compact topological space. This topology has a countable basis: for every choice of naturals $j_1,\ldots , j_n$, positive rationals $r_1,\ldots ,r_n$, and points $\zeta _1,\ldots , \zeta _n$ in ${{\overline{U}}}$ with rational coordinates we take the set

$$\begin{aligned} V=\{z\in {\overline{U}}^N: |z_{j_k}-\zeta _k|<r_k, k=1,2,\ldots ,n\}. \end{aligned}$$

The set ${\overline{U}}^N$ carries a unique Radon measure $\tau$ with the following property: if $\{j_1,\ldots ,j_n\}$ is any set of n indices and $\phi$ is any continuous function on ${\overline{U}}^N$ such that $\phi (z)=\phi (z_{j_1},\ldots ,z_{j_n})$, then

$$\begin{aligned} \int _{{\overline{U}}^N}\phi d\tau = \frac{1}{(2\pi i)^n}\int _U...\int _U\phi (z_{j_1},\ldots ,z_{j_n})dz_{j_1}\wedge d{\bar{z}}_{j_1}...dz_{j_n}\wedge d{\bar{z}}_{j_n}. \end{aligned}$$

This measure is translational invariant, i.e., if $F\subset {\overline{U}}^N$ is a Borel set and $x\in {\overline{U}}^N$ such that $F+x\subset {\overline{U}}^N$, then $\tau (F)=\tau (F+x)$.

Let Z be a sequential compete local convex space. The sequence $E=(x_n)$ in Z is called summable if every continuous semi-norm p on Z, then $\sum _{n=1}^\infty p(x_n)<\infty$. The set

$$\begin{aligned} P_E=\{x\in Z: x=\sum _{j=1}^\infty \zeta _jx_j, |\zeta _j|\le 1, \forall j=1,2,...\} \end{aligned}$$

is called a generalized polydisk. The mapping $F: {\overline{U}}^N\rightarrow P_E$ defined as $F(z)=\sum z_jx_j$ is continuous and therefore defines the measure $\tau _E$ on $P_E$ as the pushforward of the measure $\tau$.

If $P_E$ is a generalized polydisk and

$$\begin{aligned} C=\{x\in Z: x=\sum _{j=1}^\infty \zeta _jx_j, |\zeta _j|\le c_j\le 1, \forall j=1,2,...\}, \end{aligned}$$

then C is the intersection of the decreasing sequence of closed sets

$$\begin{aligned} C_k=\{x\in Z: x=\sum _{j=1}^\infty \zeta _jx_j, |\zeta _j|\le c_j, \forall 1\le j\le k\}. \end{aligned}$$

Therefore $\tau _E(C)=\prod _{j=1}^\infty c_j^2.$

A function $\phi$ on a set $W\subset Z$ is called absolutely measureable if it is measurable with respect to any regular Borel measure with compact support on W. Given a summable sequence $E=\{x_j\}$, an open set $W\subset Z$, and a local bounded above absolutely measurable function $\phi$ on W, we define the function

$$\begin{aligned} \phi _E(z)=\int _{P_E}\phi (z+y)d\tau _E(y) \end{aligned}$$

on the set $W_E=\{z\in W: z+P_E\subset W\}$.

The linear operator $L_E: \ell ^1\rightarrow Z$ defined as

$$\begin{aligned} L_E(\zeta _1,\zeta _2,...)=\sum _{j=1}^\infty \zeta _jx_j \end{aligned}$$

is continuous. We say that a function u on W is usc along E if for every $z\in W$ the function $u(z+L_E\zeta )$ is usc at the origin of the space $\ell ^1$.

Lemma 3.1

Let Z be a sequential complete local convex space and $\phi$ be a local bounded above absolutely measurable on an open set W in Z. Let $E=\{x_j\}$ be a summable sequence in Z. Then the function $\phi _E$ is usc along E on $W_E$.

Proof

Let us take a point $z_0\in W_E$. We will assume that $\phi _E(z_0)>-\infty$. If $\phi _E(z_0)=-\infty$ then the proof is similar.

Since the set $K=z_0+P_E$ is compact, we can find a continuous seminorm p and a constant $c>0$ such that $\phi (x)<c$ when $p(x-y)<1$ for some $y\in K$. We also may assume that $z\in W_E$ when $p(z-z_0)<1$.

Since the mapping $L_E$ is continuous, there is $\delta _1>0$ such that $p(L_E\zeta )<1$ when $\sum |\zeta _j|<\delta _1$. For such $\zeta$ we let

$$\begin{aligned} F=\{y\in P_E: y=\sum _{j=1}^\infty \xi _jx_j, |\xi _j|\le 1-|\zeta _j|\}. \end{aligned}$$

By absolute continuity of the integral, for every $\epsilon >0$ there exist $\epsilon '$ such that

$$\begin{aligned} \int _A\phi (z_0+y)d\tau _E(y)>-\frac{\epsilon }{2} \end{aligned}$$

when $\tau _E(A)<\epsilon '$.

We have $\tau _E(F)=\prod _{j=1}^\infty (1-|\zeta _j|)^2$. So there is $0<\delta <\delta _1$ such that $\tau _E(P_E{\setminus } F)<\frac{\epsilon }{2c}$ when $\sum |\zeta _j|<\delta$.

Let $z_1=z_0+\sum \zeta _jx_j, \sum |\zeta _j|<\delta$, and let $z=z_1-z_0$. Then $F_1=F+z\subset P_E$. Hence

$$\begin{aligned} \begin{aligned} \phi _E(z_1)&=\int _{P_E}\phi (z_1+y)d\tau _E(y)\\&=\int _F \phi (z_1+y)d\tau _E(y)+\int _{P_E\setminus F}\phi (z_1+y)d\tau _E(y)\\&\le \int _{F+z}\phi (z_0+y)d\tau _E(y)+\frac{\epsilon }{2}\\&\le \int _{P_E}\phi (z_0+y)d\tau _E(y)+\epsilon =\phi _E(z_0)+\epsilon , \end{aligned} \end{aligned}$$

and this means that $\phi _E$ is usc along E. $\square$

Let Z be a sequential complete local convex space and Y be subspace in Z. A system ${\mathcal {E}}$ of summable sequences $E\subset Y$ will be called basic in Y if for every summable sequence $F=\{x_k\}\subset Y$ there is a family $E\in {\mathcal {E}}$ and a sequence of vectors $y_k\in \ell ^1$ convergence to 0 such that $L_Ey_k=x_k$. Here we note that, in a finite-dimensional space a basic system may consist of only one sequence E that is the basis of the space. In an infinite-dimensional space the system of all summable sequence is basic.

Lemma 3.2

Let Z be a Hausdorff sequential complete local convex space and Y be a subspace of Z and let ${\mathcal {E}}$ be a basic system in Y. For every $h\in H(Y)$ there is $E\in {\mathcal {E}}$ and $g\in H(\ell ^1)$ such that $h=L_E\circ g$.

Proof

Since $h\in H(Y)$ there exist $r>1$ such that h is a continuous mapping of ${\overline{U}}_r$ into Y, holomorphic in $U_r$. Where $U_r=\{z\in {\mathbb {C}}: |z|<r\}$. If $h_r(\zeta )=h(r^{-1}\zeta )$ then $h_r\in {\overline{H}}(Y)$. By Lemma 2.1 then $h_r\in {\overline{H}}(Y_A,p_A)$, where $A=\overline{conv(h_r({{\overline{U}}}))}=\overline{conv(h({{\overline{U}}}_r))}$ and $p_A$ is the Minkowski functional of A. And so h can be represented by its Taylor series

$$\begin{aligned} h(\zeta )=\sum _{k=0}^\infty a_k\zeta ^k,\;\text { where } \sum _{k=0}^\infty p_A(a_k)t^k<\infty , \end{aligned}$$

for some $1<t<r$. This infer that the sequence $F=\{t^ka_k\}$ is summable. Thus we can find a sequence $E\in {\mathcal {E}}$ and a sequence of vectors $b_k\in \ell ^1$ converging to 0 such that $L_Eb_k=t^ka_k$. We set

$$\begin{aligned} g(\zeta )=\sum _{k=0}^\infty c_k\zeta ^k. \end{aligned}$$

The series for g converges absolutely in $U_t$ so $g\in H(\ell ^1)$ and $L_Eg(\zeta )=h(\zeta )$ when $|\zeta |<t$. $\square$

Let ${\mathcal {E}}$ be a system of summable sequence in Z. A function $\phi$ defined on an open set W in Z is called almost upper semicontinuous with respect to ${\mathcal {E}}$ (${\mathcal {E}}$-ausc) if it is locally bounded above, absolutely measurable and for every sequence $E'\{x_j\}$ of vectors in Z which is the union of a sequence $E\in {\mathcal {E}}$ and finitely many vectors $z_1,\ldots , z_n$ in Z we have

$$\begin{aligned} \limsup _{t\rightarrow 0^+}\phi _{tE'}(x)\le \phi (x), \end{aligned}$$

where $tE'=\{tx_j\}$. By Fatou lemma, an usc function is also ${\mathcal {E}}$-ausc.

Lemma 3.3

Let Z be sequential complete local convex space and P be a projection of Z onto Y. Let ${\mathcal {E}}$ be a basic system in Y. Let W be an open set in Z and let $\phi$ be an ${\mathcal {E}}$-ausc function on W. If the function $u=I_P\phi$ is absolutely measurable on $W_0=P(W)$, then it is ${\mathcal {E}}$-ausc.

Proof

We fix a point $w_0\in W_0, E\in {\mathcal {E}}$, a finite set $y_1,\ldots , y_n$ in Y and $\epsilon >0$. We find $z_0\in W$ such that $Pz_0=w_0$ and $\phi (z_0)<u(z_0)+\epsilon$. If $E'=E\cup F$, then

$$\begin{aligned} \begin{aligned} \limsup _{t\rightarrow 0^+}\int _{P_tE'}u(w_0+y)d\tau _{tE}(y)&\le \limsup _{t\rightarrow 0^+}\int _{P_tE'}\phi (z_0+y)d\tau _{tE}(y) \\&\le \phi (z_0)<u(z_0)+\epsilon . \end{aligned} \end{aligned}$$

$\square$

4 The main results

Before starting and proving the main results we will give some notions.

Let A be a proper subset of Z. We say that a mapping F of $Z\setminus A$ into another topological vector space $Z'$ is rational with poles in A if for every $g\in H(Z)$ either $g(U)\subset A$ or the set $A_g=g^{-1}(A)\cap U$ is finite and $F\circ g$ has poles of finite order at points of $A_g$. Also for a holomorphic mapping of $h: U\setminus \{0\}\rightarrow Z$ with a pole of finite order at 0, the mapping $F\circ h$ has a pole of finite order at 0 too. In the case of finite dimensional topological vector space this definition produces standard rational mappings. Also linear mappings are rational.

Now we recall a basis result in functional analysis. If $(Z,\xi )$ is a sequential complete local convex space then there is the strongest local convex topology $\eta$ on Z such that $\xi$ and $\eta$ have the same bounded sets in Z and $(Z,\eta )$ is the inductive limit of family Banach spaces $(Z_i, i\in I)$ with some the directed set I. From this we give the notion of weakly ${\mathcal {E}}$-ausc function as follow.

Let Z be a sequential complete local convex space that is also the inductive limit of family Banach spaces $(Z_i, i\in I)$ with I is the directed set. Let P be a projection of Z onto Y. Let ${\mathcal {E}}$ be a basic system in Y. A function $\phi$ on an open set W in Z is called weakly ${\mathcal {E}}$-ausc if there is rational mappings $F_i: W_i\rightarrow Z$ with poles outside of $W_i\subset Z_i$, projections $P_i$ on $Z_i$ and ${\mathcal {E}}$-ausc functions $\phi _i$ on $W_i$ such that:

i.:

$P(Z)=P_i(Z_i)=Y, P_i(W_i)=P(W)=W_0$;

ii.:

$F_i(W_i)\subset W, P\circ F_i(z)=P_i(z)$;

iii.:

$\phi (F_i(z))\le \phi _i(z)$;

iv.:

The function $u_i=I_{P_i}\phi _i$ are absolutely measurable and from a decreasing net such that $\lim _iu_i=I_P\phi$.

The net $\{Z_i, W_i, F_i, P_i, \phi _i\}$ will be called an approximating net. We can see that a ${\mathcal {E}}$-ausc function is also weakly ${\mathcal {E}}$-ausc.

The following theorem is the main of the paper.

Theorem 4.1

Let Z be a Hausdorff sequential complete local convex space with a holomorphic ${{\overline{U}}}$-action A. Let $Pz=A(0,z)$ and let $Y=P(Z)$. Let ${\mathcal {E}}$ be a basic system in Y. Let W be an S-invariant open set in Z and let $\phi$ be an S-invariant absolutely measurable function on W. If $\phi$ is weakly ${\mathcal {E}}$-ausc with an approximating net $\{Z_i, W_i, F_i, P_i, \phi _i\}$ and the fibers $W_{iz}=\{w\in W_i: P_i(w)=z\}$ are path connected, then the function $u=I_P\phi$ is sequential psh if and only if $\phi$ is P-psh.

Proof

First, we will prove the necessity of the theorem. If $h\in H(W_0), W_0=P(W)$, and $f: S\rightarrow W$ is a continuous mapping such that $P\circ f=h$ on S, then for a psh function u we have

$$\begin{aligned} u(h(0))\le \frac{1}{2\pi }\int _0^{2\pi }u(h(e^{i\theta }))d\theta \le \frac{1}{2\pi }\int _0^{2\pi }\phi (f(e^{i\theta }))d\theta . \end{aligned}$$

Thus $\phi$ is P-psh.

The proof of sufficiency is more complicated and will follow from a sequence of the following lemmas.

Lemma 4.2

In conditions of Theorem 4.1 the function $u=I_P\phi$ is sequential usc.

Proof

Let $z_0$ be a point in $W_0$ and let $\{z_k\}$ be a sequence of points in $W_0$ converging to 0. We have $H=\{z_0, z_0+z_k\}$ is a compact set in Y. Set $G=\overline{conv(H)}$ and $Y_G=span(G)$ and $p_G$ is a Minkovsky functional of G. Then $(Y_G, p_G)$ is the Banach space.

Switching if necessary to a subsequence, we can assume that the family $F=\{kz_k\}$ is summable in $Y_G$. So we can find a sequence $E\in {\mathcal {E}}$ and a sequence of vectors $x_k\in \ell ^1$ converging to 0 such that $L_Ex_k=z_k$.

Fix some $\epsilon >0$ and find $i\in I$ and a point $w_0\in W_0$ such that $P_iw_0=z_0$ and $\phi _i(w_0)<u(z_0)+\epsilon$. For every $k\ge 1$ we set $w_k=w_0+z_k$. Let us take $s>0$ so small that $w_k+P_{sE}\subset W_i$ for all k. Let

$$\begin{aligned} \psi _t(y)=\int _{P_{tE}}\phi _i(y+x)d\tau _{tE}(x). \end{aligned}$$

Since $\phi _i$ is ${\mathcal {E}}$-ausc, for any $\epsilon >0$ we can find $0<t<s$ such that $\psi _t(w_0)<\phi _i(w_0)+\epsilon$. Since the function $\psi _t$ is usc along tE and $L_{tE}(t^{-1}x_k)=z_k$, there is $k_0$ such that $\psi _t(w_k)<\phi _i(w_0)+2\epsilon$ when $k>k_0$.

For every $e^{i\theta }\in S$ we have

$$\begin{aligned} \int _{P_{tE}}\phi _i(w_k+e^{i\theta }x)d\tau _{tE}(x)=\psi _t(w_k). \end{aligned}$$

Hence by Fubini’s and the mean theorems there exist $x_0\in P_{tE}$ such that

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }\phi _i(w_k+e^{i\theta }x_0)d\theta \le \phi _i(y_0)+2\epsilon ,\;\text { when } k>k_0. \end{aligned}$$

Let $f_i(\xi )=w_k+\xi x_0$ be a mapping of the unit disk into $W_i$ and let $f=F_i\circ f_i$. Since $Pf(0)=P_if_i(0)=z_0+z_k$ and $\phi$ is weakly P-psh,

$$\begin{aligned} u(z_0+z_k)\le \frac{1}{2\pi }\int _0^{2\pi }\phi (f(e^{i\theta }))d\theta \le \frac{1}{2\pi }\int _0^{2\pi }\phi _i(f_i(e^{i\theta }))d\theta \le u(z_0)+2\epsilon . \end{aligned}$$

Thus u is sequential usc. $\square$

Now to conclude the proof of the theorem for a mapping $h\in H(W_0)$ such that $h(0)=z_0$, we will establish for every $\epsilon >0$ we have

$$\begin{aligned} u(z_0)\le \frac{1}{2\pi }\int _0^{2\pi }u(h(e^{i\theta }))d\theta +\epsilon . \end{aligned}$$

(4.1)

For this we will replace the integrand u in inequality above by a topologically better function $u_i$. Note that by Lemma 3.3 the function $u_i$ are ${\mathcal {E}}$-ausc.

Lemma 4.3

Let Z be a Hausdorff sequential complete local convex space and let P be a projection of Z onto Y. Let ${\mathcal {E}}$ be a basic system in Y. If a decreasing net of ${\mathcal {E}}$-ausc functions $u_i, i\in I$, on an open set $W_0\subset Y$ converges to an usc function u, then for every $h\in H(W_0)$ and $\epsilon >0$ there are a sequence of vectors $y_n\in Y$ converging to 0 and a sequence of $i_n\in I$ such that

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }u_{i_n}(h_n(e^{i\theta }))d\theta \le \frac{1}{2\pi }\int _0^{2\pi }u(h(e^{i\theta }))d\theta +\epsilon , \end{aligned}$$

where $h_n(\zeta )=h(\zeta )+\zeta y_n$.

Proof

By Lemma 3.2 we take $E\in {\mathcal {E}}$ and a mapping $g\in H(\ell ^1)$ such that $L_Eg=h$. For $n=1, 2,...$ let $W_0^n=\{x\in W_0: x+n^{-1}E\subset W_0\}$ and

$$\begin{aligned} u_{in}(x)=\int _{P_{n^{-1}E}}u_i(x+y)d\tau _{n^{-1}E}(y), \; \forall x\in W_0^n. \end{aligned}$$

By Lemma 3.1 the functions $u_{in}$ are usc on h(S).

Since the functions $u_i$ are decreasing and locally bounded above, $u_{in}\le u_{jn}$ when $i\ge j$ and for any $i_0\in I$ the functions $u_{in}, i\ge i_0$, are uniformly bounded above on h(S). Moreover, since the functions $u_i$ are ${\mathcal {E}}$-ausc,

$$\begin{aligned} \limsup _{n\rightarrow \infty }u_{in}\le u_i \end{aligned}$$

on h(S). By Lebesgue’s monotone convergence theorem for every $n\ge n_0$ there is $i_n\in I$ such that

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }u_{i_nn}(h(e^{i\theta }))d\theta \le \frac{1}{2\pi }\int _0^{2\pi }u(h(e^{i\theta }))d\theta +\epsilon . \end{aligned}$$

By the definition of $u_{i_nn}$, Fubini’s theorem, and the mean velue theorem we can find a vector $y_n\in P_{n^{-1}E}$ such that

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }u_{i_n}(h(e^{i\theta })+e^{i\theta }y_n)d\theta \le \frac{1}{2\pi }\int _0^{2\pi }u(h(e^{i\theta }))d\theta +2\epsilon . \end{aligned}$$

$\square$

We use Lemma 4.3 to find a mapping $h'\in {\overline{H}}(W_0)$ with $h'(0)=z_0$ and $i\in I$ such that

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }u_i(h'(e^{i\theta }))d\theta \le \frac{1}{2\pi }\int _0^{2\pi }u(h(e^{i\theta }))d\theta +\epsilon . \end{aligned}$$

Now we see that the subaveraging inequality (4.1) will be proved if we establish that

$$\begin{aligned} u(z_0)\le \frac{1}{2\pi }\int _0^{2\pi }u_i(h'(e^{i\theta }))d\theta +\epsilon . \end{aligned}$$

(4.2)

Let us denote $h'$ by h.

Now we claim the existence of a continuous selection minimizing some functional.

Lemma 4.4

Let P be a projection on a Hausdorff sequential complete local convex space and let $W\subset Z$ be an open with path connected fibers. Let ${\mathcal {E}}$ be a basic system in $Y=P(Z)$ and $W_0=P(W)$. If $\phi$ is an ${\mathcal {E}}$-ausc function on W, then for every $h\in H(W_0)$ and every $\epsilon >0$ there are a family $E\in {\mathcal {E}}$, a finite set $F\subset Z$, a positive number $s<\epsilon$, a vector $z_s\in P_{sE}$, a continuous mapping $q: S\rightarrow \ell ^1$, and $g\in H(\ell ^1)$ such that for $E'=F\cup E$ and $f=L_{E'}q$ we have: $h=L_{E'}g, P\circ f(\xi )=h(\xi )+\xi z$ on S, and

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }\phi (f(e^{i\theta }))d\theta \le \frac{1}{2\pi }\int _0^{2\pi }u(h(e^{i\theta }))d\theta +\epsilon , \; \text { where } u=I_P\phi . \end{aligned}$$

Proof

By Lemma 3.2 we choose for h a system $E\in {\mathcal {E}}$ and a mapping $g\in H(\ell ^1)$ such that $L_E\circ g=h$. For $t>0$ let $W^t=\{x\in W: x+tP_E\subset W\}$ and let

$$\begin{aligned} \psi _t(z)=\int _{P_{tE}}\phi (x+y)d\tau _{tE}(y). \end{aligned}$$

We may assume that $h({{\overline{U}}})\subset P(W^t)$ and we let

$$\begin{aligned} v_t(z)=\inf \{\psi _t(w): Pw=z, w\in W^t\}. \end{aligned}$$

Since $\phi$ is ${\mathcal {E}}$-ausc, $\limsup _{t\rightarrow 0^+}\psi _t(z)\le \phi (z), \forall z\in W$. Thus $\limsup _{t\rightarrow 0^+}v_t(z)\le u(z), \forall z\in W_0$.

Moreover, by Lemma 3.1 the functions $\psi _t(x)$ are usc along E. Therefore the functions $v_t$ are usc along E, and consequently the functions $v_t(h(\xi ))$ are usc on S.

Since the function $\phi$ is locally bounded above, we can find a constant C and open sets $W_1,W_2,\ldots , W_n$ in W such that $\phi < C$ on each of $W_j$ and the union of $P(W_j)$ covers h(S). Hence $v_t(h(\xi ))<C$ on S when t is sufficiently small, and by Fatou’s lemma for $\epsilon >0$ we can find s so small that

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }v_s(h(e^{i\theta }))d\theta \le \frac{1}{2\pi }\int _0^{2\pi }u(h(e^{i\theta }))d\theta +\epsilon . \end{aligned}$$

(4.3)

Let us take a continuous function v on S such that $v\ge v_s\circ h$ and

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }v(e^{i\theta })d\theta \le \frac{1}{2\pi }\int _0^{2\pi }v_s(h(e^{i\theta }))d\theta +\epsilon . \end{aligned}$$

(4.4)

For every $\xi \in S$ we take $z_\xi \in W^s$ such that $Pz_\xi =h(\xi )$ and $\psi _s(z_\xi )<v(\xi )+\epsilon$. Since $\psi _s$ is usc along E, there is an open arc $V_\xi \subset S$ containing $\xi$ such that $p_\xi (\zeta )=z_\xi + h(\zeta )-h(\xi )\in W^s$ and $\psi _s(p_\xi (\zeta ))<v(\zeta )+\epsilon$ on $V_\xi$.

We can find finitely many points $\xi _1, \xi _2,\ldots , \xi _n$ such that the arcs $V_j=V_{\xi _j}$ cover S. Let us take closed arcs $V_j^{'} \subset V_j$ that still cover S. We may assume that none of these arcs contains another. We let $\gamma _1 = V_1$ and delete the interior of $\gamma _1$ from away all $V_j^{'}, j\ge 2$. We continue to denote the obtained arcs by $V_j^{'}$. Then we throw away all $V_j^{'}$ that are empty or consist of one point. We take one of remaining arcs, denote it by $\gamma _2$, and repeat the process deleting from all arcs except $\gamma _1$ and $\gamma _2$ the interior of $\gamma _2$. In at most n steps we will get closed arcs $\gamma _j, 1\le j\le m$, covering S and with disjoint interiors.

Each of the arcs $\gamma _j$ was obtained from some arc $V_k$. Hence the mappings $p_{\xi _k}$ are defined on $\gamma _j$. We will denote them by $p_j$. After a renumbering and rotation we may assume that $\gamma _j=\{e^{i\theta }: \alpha _{j-1}\le \theta \le \alpha _j\}$.

Let us denote by p the mapping of the interiors of the arcs $\gamma _j$ into $W^s$ equal to $p_j$ on $\gamma _j$. By the definition of $\psi _s$, Fubini’s, theorem and the mean value theorem there is a vector $z_s\in P_{sE}$ such that

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }\phi (p(e^{i\theta })+e^{i\theta }z_s)d\theta \le \frac{1}{2\pi }\int _0^{2\pi }\phi _s(p(e^{i\theta })d\theta +\epsilon . \end{aligned}$$

Since $\psi _s(p(\zeta ))<v(\zeta )+\epsilon$, it follows from (4.3) and (4.4) that

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }\phi (p(e^{i\theta })+e^{i\theta }z_s)d\theta \le \frac{1}{2\pi }\int _0^{2\pi }u(h(e^{i\theta }))d\theta +3\epsilon . \end{aligned}$$

(4.5)

Let $p'_j(\zeta )=p_j(\zeta )+\zeta z_s$ and let $p'\equiv p$ on the interior of the arcs $\gamma _j$. Since fibers of W are path connected, for every $1\le j\le m$ there is a piecewise linear continuous mapping $\rho _j$ of [0, 1] into the fiber over $p'_j(e^{i\alpha _j})$ such that $\rho _j(0)=p'_j(e^{i\alpha _j})$ and $\rho _j(1)=p'_{j+1}(e^{i\alpha _j})$. The union of all sets $p'_j(\gamma _j)$ and $\rho _j([0, 1])$ is compact, and therefore has a neighborhood $V\subset W$ where the function $\phi$ is bounded above by some constant, say, $A>0$.

We take as F the set of all points $z_j=z_{\xi _j}$ and all vertices of the paths $\rho _j, 1\le j\le m$. Let $E'=F\cup E$. Then points $z_j$ lie in the image of $L_{E'}$ and we choose $w_j, 1\le j\le m$, and $w_s$ such that $L_{E'}w_s=z_s$. For every $1\le j\le m$ we define the mapping $q_j$ of $\gamma _j$ into $\ell ^1$ as $q_j(\zeta )=g(\zeta )-g(\xi _j)+w_j+\zeta w_s$ (g was introduced in the beginning of the proof). Note that $L_{E'}q_j=p'_j$. Since all vertices of the paths $\rho _j$ are in F, we can find a piecewise linear continuous mapping $\sigma _j$ of [0, 1] into $\ell ^1$ such that $\sigma _j(0)=q_j(e^{i\alpha _j}), \sigma _j(1)=q_{j+1}(e^{i\alpha _j})$ and $L_{E'}\sigma _j=\rho _j$.

Let us choose points $\alpha _{j-1}<\beta '_j<\alpha _j<\beta ''_j<\alpha _{j+1}$ so close to each other that the length of the union $G'$ of the arcs $[\beta '_j, \beta ''_j]$ is less than $\epsilon /A$ and

$$\begin{aligned} \frac{1}{2\pi }\int _{G'}\phi (p'(e^{i\theta }))d\theta \le \epsilon . \end{aligned}$$

We set $h_1(\zeta )=h(\zeta )+\zeta z_s$ and $g_1(\zeta )=g(\zeta )+\zeta w_s$. Let $m_j(t)$ be a linear function on $[\beta '_j, \beta ''_j]$ equal to 0 at the left end and to 1 at the right one. We define a mapping $q: S\rightarrow \ell ^1$ as $q_j(e^{i\alpha })$ when $\beta ''_{j-1}\le \alpha \le \beta '_j$, and for $\alpha \in [\beta '_j, \beta ''_j]$ we let

$$\begin{aligned} q(e^{i\alpha })&=(1-m_j(\alpha ))(q_j(e^{i\beta '_j})-q_j(e^{i\alpha _j}))+\sigma _j(m_j(\alpha ))\\&\quad +m_j(\alpha )(q_{j+1}(e^{i\beta ''_j})-q_{j+1}(e^{i\alpha _j}))+g_1(e^{i\alpha })\\&\quad -(1-m_j(\alpha ))g_1(e^{\beta '_j})-m_j(\alpha )g_1(e^{\beta ''_j}). \end{aligned}$$

It easy to verify that q is continuous and $PL_{E'}q=h_1$. We may also choose points $\beta '_j$ and $\beta ''_j$ so close that q maps S into V.

Let $f=L_{E'}q$. We denote by G the complement of $G'$ in S. Recalling the choice of the sets $G'$ and of the mapping q and also (4.5), we see that

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi } \phi (f(e^{i\theta }))d\theta&=\frac{1}{2\pi }\int _G \phi (p'(e^{i\theta }))d\theta +\frac{1}{2\pi }\int _{G'} \phi (f(e^{i\theta }))d\theta \\&\le \frac{1}{2\pi }\int _0^{2\pi } \phi (p'(e^{i\theta }))d\theta +2\epsilon \\&\le \frac{1}{2\pi }\int _0^{2\pi } u(p'(e^{i\theta }))d\theta +5\epsilon . \end{aligned}$$

$\square$

Let us return to prove Theorem 4.1 and prove (4.2). We apply Lemma 4.4 to $\phi _i$, take any s satisfying the conclusion of the lemma, and denote $h(\zeta )+\zeta z_s$ by $h_1(\zeta )$. Let

$$\begin{aligned} \psi _t(z)=\int _{P_{tE'}}\phi _i(z+y)d\tau _{tE'}(y) \end{aligned}$$

and let $\psi '_t=\psi _t\circ L_{E'}$. Since $\phi _i$ is ${\mathcal {E}}$-ausc, we can find $t>0$ such that

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }\psi _t(f(e^{i\theta }))d\theta \le \frac{1}{2\pi }\int _0^{2\pi }\phi _i(f(e^{i\theta }))d\theta +\epsilon . \end{aligned}$$

(4.6)

We can choose t so small that f(S) lies in the set $W^t=\{x\in W: x+tP_{E'}\subset W\}$. The function $\psi '_t$ is usc on $W_t=L^{-1}_{E'}(W^t)$. By Lemma 2.4 in [18], there exist a mapping

$$\begin{aligned} q_1(\zeta )=g(\zeta )+\zeta w_s+\sum _{k=-N}^N c_k\zeta ^k \end{aligned}$$

of ${\mathbb {C}}\setminus \{0\}$ into $\ell ^1$ such that $L_{E'}(g(\zeta )+\zeta w_s)=h_1(\zeta ), P_iL_{E'}q_1=h_1$ on $S, q_1(S)\subset W_t$, and

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }\psi '_t(q_1(e^{i\theta }))d\theta \le \frac{1}{2\pi }\int _0^{2\pi }\psi _t(f(e^{i\theta }))d\theta +\epsilon . \end{aligned}$$

(4.7)

Let

$$\begin{aligned} f_1(\zeta )=L_{E'}q_1(\zeta )=h_1(\zeta )+\sum _{k=-N}^N d_k\zeta ^k. \end{aligned}$$

Fubini’s theorem and the mean value theorem applied to (4.6) and (4.7) show that there is $y_0\in P_{tE'}$ such that

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }\phi _i(f_1(e^{i\theta })+e^{i\theta }y_0)d\theta \le \frac{1}{2\pi }\int _0^{2\pi }\phi _i(f(e^{i\theta }))d\theta + 3\epsilon . \end{aligned}$$

The mapping $f_2(\zeta )=f_1(\zeta )+\zeta y_0$ maps S into $W_i$, and has a pole of order at most N at the origin. Hence the mapping $f_3=F_i\circ f_2$ has no poles on S and only finitely many in U. By Lemma 2.3 there is a Blaschke product B such that the mapping $f_4(\zeta )=A(B(\zeta ), f_3(\zeta ))$ is in H(Z). Since the function $\phi$ is S-invariant, $\phi (f_4(\zeta ))=\phi (f_3(\zeta ))\le \phi _i(f_2(\zeta ))$ when $|\zeta |=1$. Therefore

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }\phi (f_4(e^{i\theta }))d\theta \le \frac{1}{2\pi }\int _0^{2\pi }\phi _i(f_2(e^{i\theta }))d\theta \le \frac{1}{2\pi }\int _0^{2\pi }u_i(h(e^{i\theta }))d\theta +4\epsilon . \end{aligned}$$

Note that on S we have:

$$\begin{aligned} Pf_4(\zeta )=Pf_3(\zeta )=P_if_2(\zeta )=h_1(\zeta )+\zeta P_iy_0. \end{aligned}$$

Thus $Pf_4(0)=z_0$. The function $\phi$ is P-psh, and therefore

$$\begin{aligned} u(h(0))\le \frac{1}{2\pi }\int _0^{2\pi }u_i(h(e^{i\theta }))d\theta +4\epsilon . \end{aligned}$$

This proves (4.2) and the theorem. $\square$

According to this theorem, we must verify that a function $\phi$ is P-psh to establish the minimum principle. Clearly it could be done easier if it will suffice to verify the weak P-plurisubharmonicity. The following, we are going to prove in the case of ${{\overline{U}}}$-invariant domains, i. e., $A(\zeta ,z)\in W$ whenever $z\in W$ and $|\zeta |\le 1$, the weak P-plurisubharmonicity implies P-plurisubharmonicity.

Theorem 4.5

Let Z be a Hausdorff sequential complete local convex space with a holomorphic ${{\overline{U}}}$-action A and let $Pz=A(0,z)$. Let W be an ${{\overline{U}}}$-invariant open set in Z and let $\phi$ be a weakly P-psh S-invariant function on W. Then $\phi$ is P-psh.

Proof

Note that now $W_0=P(W)\subset W$. For $f\in H(Z)$ such that $f(0)=z, P\circ f=h\in H(W_0)$, and $f(S)\subset W$ we consider $F(\zeta ,\xi )=A(\zeta , f(\xi ))$. The preimage D of W under the mapping F is an open set containing neighborhoods of sets $\{(\zeta , \xi ): |\xi |\le 1, \zeta =0\}$. Therefore there is an integer $N>0$ such that the mapping $g(\xi )=(\xi ^N,\xi )$ is in H(D). Let $G=F\circ g$. Then $G\in H(W), P\circ G=h$, and $\phi (G(\xi ))=\phi (f(\xi ))$ when $|\xi |=1$. Since $\phi$ is weakly P-psh,

$$\begin{aligned} I_P\phi (z)\le \frac{1}{2\pi }\int _0^{2\pi }\phi (G(e^{i\theta }))d\theta =\frac{1}{2\pi }\int _0^{2\pi }\phi (f(e^{i\theta }))d\theta , \end{aligned}$$

and therefore $\phi$ is P-psh. $\square$