A Sharp Inequality of Hardy-Littlewood Type Via Derivatives

TL;DR

This paper investigates a generalized Hardy-Littlewood inequality for holomorphic functions, analyzing the behavior of associated norms via derivatives, and establishes the inequality for specific function classes with numerical support.

Contribution

It introduces a derivative-based approach to verify a generalized Carleman's inequality for holomorphic functions, including cases involving linear combinations of kernels.

Findings

Norms $\

Derivative of norms $\

Abstract

In this paper we consider a generalized version of Carleman's inequality. An equivalent version of it states that $\|f\|_{A_\alpha^{2\alpha}}\leq\|f\|_{H^2}$, where $f$ is a holomorphic function and $\alpha>1$. If the norms $\|f\|_{A_\alpha^{2\alpha}}$ are decreasing in $\alpha$, then the inequality holds for $f$. For a dense set of functions, we calculate the derivative of the norms $\|f\|_{A_\alpha^{2\alpha}}$ in $\alpha$ and give sufficient conditions for this derivative to be non-positive. As an application, we prove the inequality for linear combinations of two reproducing kernels. Some numerical evidences are also provided.

Equations

$$\|f\|_{H^{p}}:=\sup_{0<r<1}\bigg{(}\int_{0}^{2\pi}|f(re^{i\theta})|^{p}\frac{\mathrm{d}\theta}{2\pi}\bigg{)}^{\frac{1}{p}}<\infty.$$

$$\|f\|_{A_{\alpha}^{p}}:=\bigg{(}\int_{\mathbb{D}}|f(z)|^{p}(\alpha-1)(1-|z|^{2})^{\alpha}\mathrm{d}\mu(z)\bigg{)}^{\frac{1}{p}}<\infty,$$

$$\Vert f{\Vert }_{A_{2/p}^2}\le \Vert f{\Vert }_{H^p}.$$

$$\Vert f{\Vert }_{A_{2/p}^2}\le C\Vert f{\Vert }_{H^p}$$

$$\Vert f{\Vert }_{A_2^2}\le \Vert f{\Vert }_{H^1}.$$

$$\Vert f{\Vert }_{A_{\alpha }^{2\alpha }}\le \Vert f{\Vert }_{H^2}.$$

$$\Vert f{\Vert }_{H^2}^{2\alpha }-\Vert f{\Vert }_{A_{\alpha }^{2\alpha }}^{2\alpha }$$

$$=$$

$$N_{f}(\alpha)=\|f\|_{A_{\alpha}^{2\alpha}}=\bigg{(}\int_{\mathbb{D}}|f(z)|^{2\alpha}(\alpha-1)(1-|z|^{2})^{\alpha}\mathrm{d}\mu(z)\bigg{)}^{\frac{1}{2\alpha}}.$$

$$\frac{\partial }{\partial \alpha }{N}_f(\alpha )\le 0$$

$$F^{\alpha }=c_1{K}_{w_1,\alpha }+c_2{K}_{w_2,\alpha }.$$

$$\Vert F{\Vert }_{A_{\alpha }^{2\alpha }}\le \Vert F{\Vert }_{A_{\beta }^{2\beta }}.$$

$$\Vert f{\Vert }_{A_{\alpha }^{2\alpha }}\le \Vert f{\Vert }_{H^2}.$$

$$\underset{\alpha \to 1+}{\lim }\Vert f{\Vert }_{A_{\alpha }^p}=\Vert f{\Vert }_{H^p}$$

$$\underset{\alpha \to 1+}{\lim }\Vert f{\Vert }_{A_{\alpha }^{\alpha p}}=\Vert f{\Vert }_{H^p}.$$

$$\underset{\alpha \to 1+}{\lim }{\int }_{\mathbb{D}}|f(z){|}^p(\alpha -1)(1-|z{|}^2{)}^{\alpha }\mathrm{d}\mu (z)=1.$$

$$\displaystyle\bigg{|}\int_{\mathbb{D}}|f(z)|^{p}(\alpha-1)(1-|z|^{2})^{\alpha}\mathrm{d}\mu(z)-\int_{\mathbb{D}}|f(z)|^{\alpha p}(\alpha-1)(1-|z|^{2})^{\alpha}\mathrm{d}\mu(z)\bigg{|}$$

$$\Vert f{\Vert }_{A_{\alpha }^{p\alpha }}^{p\alpha }\to 1,\alpha \to 1+.$$

$$\frac{\partial}{\partial\alpha}N_{f}(\alpha)=\frac{\alpha-1}{2\alpha}N_{f}^{1-2\alpha}(\alpha)\bigg{(}-\frac{2}{\alpha-1}N_{f}^{2\alpha}(\alpha)\mathrm{Log}N_{f}(\alpha)+\frac{1}{(\alpha-1)^{2}}N_{f}^{2\alpha}(\alpha)+I_{f}(\alpha)\bigg{)},$$

$$I_{f}(\alpha)=\int_{\mathbb{D}}|f(z)|^{2\alpha}(1-|z|^{2})^{\alpha}\mathrm{Log}\big{(}|f(z)|^{2}(1-|z|^{2})\big{)}\mathrm{d}\mu(z).$$

$$\Vert f{\Vert }_{A_{\alpha }^{2\alpha }}^{\alpha }=\Vert f^{\alpha }{\Vert }_{A_{\alpha }^2}.$$

$$f^{\alpha }(z)=\sum _{i=1}^k{c}_i\frac{1}{(1-\overline{w_i}z{)}^{\alpha }},z\in \mathbb{D}.$$

$$\frac{\partial }{\partial \alpha }{N}_f(\alpha )=\frac{1}{2{\alpha }^2}{N}_f^{1-2\alpha }(\alpha )D_f(\alpha ),$$

$$D_{f}(\alpha)=\sum_{i,j=1}^{k}c_{i}\overline{c_{j}}\frac{1}{(1-\overline{w_{i}}w_{j})^{\alpha}}\bigg{(}\overline{\log f^{\alpha}(w_{i})}+\log f^{\alpha}(w_{j})+\alpha\mathrm{Log}(1-\overline{w_{i}}w_{j})-2\alpha\mathrm{Log}N_{f}(\alpha)\bigg{)}.$$

$$I_f(\alpha )$$

$${\int }_{\mathbb{D}}(1-|\lambda {|}^2{)}^{\alpha -2}\mathrm{Log}(1-|\lambda {|}^2)\frac{\mathrm{d}m(\lambda )}{\pi }=-(\alpha -1{)}^{-2}.$$

$${\int }_{\mathbb{D}}(1-|\lambda {|}^2{)}^{\alpha -2}\frac{\mathrm{d}m(\lambda )}{\pi }=\frac{1}{\alpha -1}$$

$$N_f^{2\alpha }(\alpha )=(1-|w{|}^2{)}^{-\alpha }.$$

$$I_f(\alpha )=-(\alpha -1{)}^{-2}(1-|w{|}^2{)}^{-\alpha }-(\alpha -1{)}^{-1}(1-|w{|}^2{)}^{-\alpha }\mathrm{Log}(1-|w{|}^2).$$

$$f^{\alpha }(z)=\sum _{i=1}^k{c}_i\frac{1}{(1-\overline{w_i}z{)}^{\alpha }}.$$

$$II=\frac{1}{\alpha }{\int }_{\mathbb{D}}|f^{\alpha }(z){|}^2(1-|z{|}^2{)}^{\alpha }\mathrm{Log}|f^{\alpha }(z){|}^2\mathrm{d}\mu (z)$$

Full text

A Sharp Inequality of Hardy-Littlewood Type Via Derivatives

Hui Dan, Kunyu Guo and Yi Wang

Abstract:

In this paper we consider a generalized version of Carleman’s inequality. An equivalent version of it states that $\|f\|_{A_{\alpha}^{2\alpha}}\leq\|f\|_{H^{2}}$, where $f$ is a holomorphic function and $\alpha>1$. If the norms $\|f\|_{A_{\alpha}^{2\alpha}}$ are decreasing in $\alpha$, then the inequality holds for $f$. For a dense set of functions, we calculate the derivative of the norms $\|f\|_{A_{\alpha}^{2\alpha}}$ in $\alpha$ and give sufficient conditions for this derivative to be non-positive. As an application, we prove the inequality for linear combinations of two reproducing kernels. Some numerical evidences are also provided.

Keywords: Carleman’s inequality, inequality of Hardy and Littlewood type

2010 AMS Subject Classification: 30H20, 30H10

1 Introduction

In this paper we consider a sharp inequality concerning the weighted Bergman norms and the Hardy norm on the unit disc $\mathbb{D}$. Recall that for $0<p<\infty$, the Hardy space $H^{p}$ consists of all holomorphic functions $f$ on $\mathbb{D}$ such that

[TABLE]

For $\alpha>1$, the weighted Bergman space $A_{\alpha}^{p}$ consists of all holomorphic functions $f$ on $\mathbb{D}$ such that

[TABLE]

where $\mathrm{d}\mu(z)=(1-|z|^{2})^{-2}\frac{\mathrm{d}x\mathrm{d}y}{\pi}$ is the Möbius invariant measure of the unit disc. The inequality we are considering in this paper is the following.

Conjecture 1. For any $0<p\leq 2$ and any $f\in H^{p}$,

[TABLE]

In the case when $p=1$, (1.1) is the Carleman’s inequality (cf. [14]). For $p=1/k$ where $k$ is any positive integer, Burbea [9] showed that (1.1) holds true. If one releases the restriction on the controlling constants, that is, if one asks whether

[TABLE]

for some constant $C$, then using interpolation techniques, Brevig, Ortega-Cerdà, Seip and Zhao has proved that (1.2) holds for $0<p<1$ and $C$ as close to $1$ as $C=(2/(e\log 2))^{1/2}=1.030279\ldots$. In [6], the authors also gave and discussed about several interesting related conjectures and questions.

In the case when $p=1$, inequality (1.1) becomes

[TABLE]

This is known as the Carleman’s inequality. In 1921, Carleman [10] proved this inequality and used it to give the first complex-analytic proof of the famous isoperimetric theorem. For a different purpose, in 1932, Hardy and Littlewood showed that $H^{p}\subset A_{2}^{2p}$ (in particular, $H^{1}\subset A_{2}^{2}$) in [11]. See [14] for an excellent exposition of the relation between the two problems. Various generalizations were proved, for example, in [1][7][8][9][13][15].

In recently years, Inequality (1.1) has regained attention because of its application in number theory. Via an iterating process [2] [12], contractive inequalities like (1.1) may “lift” multiplicatively to interesting inequalities for Hardy spaces on the infinite-dimensional torus, which in turn, by the Bohr transform, translates into inequalities of Dirichlet polynomials[3][4][5].

Next, let us go to the technical side. An immediate observation is that for Conjecture 1, it suffices to consider any outer function $f$, because multiplying an inner function on $f$ does not make a difference on the right hand side of (1.1), but makes the left hand side smaller. For an outer function, one can consider its powers. By replacing $f$ with $f^{\alpha}$ where $\alpha=2/p$, it is easy to show that Conjecture 1 is equivalent to the following (cf. [6]).

Conjecture 2. For any $\alpha>1$ and any $f\in H^{2}$,

[TABLE]

Then Burbea’s result [9] is equivalent to that (1.3) holds when $\alpha$ is any integer that is greater than $1$. A straight-forward proof was given in [6, Corollary 3]. In the case when $\alpha$ is not an integer, the problem becomes very hard. The following computation may give us a clue. In the case when $\alpha>1$ is an integer, and suppose that $f$ is an outer function, $f=\sum_{n=0}^{\infty}a_{n}z^{n}$ and $a_{0}=1$. Then one can compute that

[TABLE]

This gives an alternative proof of Burbea’s result. For non-integer valued $\alpha$, we have the same equation (under some convergence assumption). However, the coefficients $\alpha\choose k$ and $\alpha\choose l$ may be negative. Similar obstructions occur when one tries to extend other proofs of Burbea’s result to a non-integer valued $\alpha$.

In [6], the authors gave several related conjectures (including the Conjectures 1 and 2 above) and questions. In particular, in [6, Question 1], they asked whether $\|f\|_{A_{\alpha}^{2\alpha}}^{2\alpha}$ is non-increasing in the parameter $\alpha$, for an outer function $f$ with $\|f\|_{H^{2}}=1$. A positive answer to the question above will lead to a positive answer to Conjecture 2. In this paper, we will mainly consider the following similar question, which allows us to drop the assumption “$\|f\|_{H^{2}}=1$” (see Remark 2.6).

Question 3. Suppose that $f$ is an outer function. For $\alpha>1$, denote

[TABLE]

Is it true that

[TABLE]

for all $\alpha>1$?

In Section 2, we will first show that a positive answer to Question 3 implies Conjecture 2. Then we will give a discrete formula of $\frac{\partial}{\partial\alpha}N_{f}(\alpha)$ for a dense set of functions. Based on the formula, in Section 3, we give some sufficient conditions for $\frac{\partial}{\partial\alpha}N_{f}(\alpha)$ to be non-positive. As an application, in Section 4, we obtain the following result.

Theorem 1.1** (Theorem 4.1 ).**

Suppose $f\in H^{2}$ and $f=\eta F$, where $\eta$ is inner and $F$ has no zeros in $\mathbb{D}$. Suppose

[TABLE]

for some $\alpha>1$, and $\mathbf{c}\in\mathbb{C}^{2},\mathbf{w}\in\mathbb{D}^{2}$. Then for any $1\leq\beta\leq\alpha$, we have

[TABLE]

Equality holds if and only if $F^{\alpha}=cK_{w,\alpha}$ for some $c\in\mathbb{C}$ and $w\in\mathbb{D}$. As a consequence, we have

[TABLE]

Equality holds if and only if $f=cK_{w,1}$ for some $c\in\mathbb{C}$ and $w\in\mathbb{D}$.

Some further remarks and numerical evidences are provided in Section 5.

2 A Discrete Formula

It is well-known that

[TABLE]

for $p>0$ and $f\in H^{p}$ [16]. Similarly, we have the following lemma.

Lemma 2.1**.**

Let $\mathcal{O}^{\ast}$ denote the set of holomorphic functions $f$ defined in some open neighborhood of $\overline{\mathbb{D}}$ such that $f(z)\neq 0$ for all $z\in\overline{\mathbb{D}}$. Then for any $f\in\mathcal{O}^{\ast}$ and any $p>0$,

[TABLE]

As a consequence, if $f\in\mathcal{O}^{\ast}$ and $\frac{\partial}{\partial\alpha}N_{f}(\alpha)\leq 0$, $\forall\alpha>1$, then $\|f\|_{A_{\alpha}^{2\alpha}}\leq\|f\|_{H^{2}}$, $\forall\alpha>1$.

Proof.

Without loss of generality, we assume $\|f\|_{H^{p}}=1$. Since $f\in\mathcal{O}^{\ast}$, there exists $C>c>0$ such that $c\leq|f(z)|\leq C$ for $z\in\mathbb{D}$. It is easy to find a constant $M>0$ such that $|x^{\alpha}-x|\leq M(\alpha-1)$ for $x\in[c^{p},C^{p}]$ and $\alpha\in(1,2)$. By (2.1),

[TABLE]

On the other hand, since $|f(z)|^{p}\in[c^{p},C^{p}]$ for all $z\in\mathbb{D}$, we have

[TABLE]

Therefore

[TABLE]

So $\lim_{\alpha\to 1+}\|f\|_{A_{\alpha}^{p\alpha}}=1$. The rest of the lemma is obvious. This completes the proof. ∎

Remark 2.2**.**

From Lemma 2.1, it is easy to see that a positive answer to Question 3 implies Conjecture 2. The statement that $\frac{\partial}{\partial\alpha}N_{f}(\alpha)\leq 0$ looks like a stronger statement than Conjecture 2. However, we are still optimistic enough to expect a positive answer. One of the evidences is the following. In [6], the authors proved a lemma ([6, Lemma 2]) which implies $\|f\|_{A_{k\alpha}^{2k\alpha}}\leq\|f\|_{A_{\alpha}^{2\alpha}}$ for any positive integer $k$, and used this lemma to prove (1.3) in the case when $\alpha>1$ is an integer. In Section 5, we will also provide some numerical evidences that support a positive answer to Question 3.

Denote $\mathrm{Log}z$ the single-valued branch of $\log z$ on $\mathbb{C}\backslash\{z\in\mathbb{R}:z\leq 0\}$ such that $\mathrm{Log}1=0$. By direct computation, we have

Proposition 2.3**.**

If $f\in\mathcal{O}^{\ast}$, then

[TABLE]

where

[TABLE]

Suppose $f$ is an outer function, then $f^{\alpha}$ makes sense and is also an outer function. We have the equation

[TABLE]

Also, from (2.3) and (2.4), we see that the value of $\frac{\partial}{\partial\alpha}N_{f}(\alpha)$ depends only on the function $f^{\alpha}$. This allows us to consider $f^{\alpha}$ instead of $f$ and take advantage of the reproducing kernel Hilbert space structure of $A_{\alpha}^{2}$.

The main goal of this section is to prove the following theorem.

Theorem 2.4**.**

Suppose $f\in\mathcal{O}^{\ast}$ and $\alpha>1$. Suppose there exists $k$ points, $w_{1},\cdots,w_{k}$ in $\mathbb{D}$ and $k$ numbers $c_{1},\cdots,c_{k}\in\mathbb{C}$ such that

[TABLE]

Then

[TABLE]

where

[TABLE]

For $\log f^{\alpha}(w_{i})$, we fix a holomorphic function $g$ such that $f^{\alpha}(z)=e^{g(z)}$ and let $\log f^{\alpha}(w_{i})=g(w_{i})$. (since $f^{\alpha}$ is outer, such function $g$ exists.)

Before proving Theorem 2.4, let us use an example to illustrate our idea.

Example 2.5**.**

It is well-known (and also implied by the proof of [6, Corollary 3]) that for integer-valued $\alpha$, the equation in (1.3) holds if and only if $f(z)=\frac{c}{1-\overline{w}z}$ for some $c\in\mathbb{C}$ and $w\in\mathbb{D}$. Indeed, if $f(z)=\frac{1}{1-\overline{w}z}$, then one can compute directly that $\frac{\partial}{\partial\alpha}N_{f}(\alpha)\equiv 0$.

By Proposition 2.3, in order to compute $\frac{\partial}{\partial\alpha}N_{f}(\alpha)$, one needs to find out $I_{f}(\alpha)$. Applying the Möbius transform $\lambda=\varphi_{w}(z)=\frac{w-z}{1-\overline{w}z}$, we get

[TABLE]

Here $dm$ denotes the Lebesgue measure. Using the polar coordinates and applying an integration by parts, we have

[TABLE]

Similar computations give

[TABLE]

and

[TABLE]

Thus

[TABLE]

From this and Proposition 2.3, it is easy to see that $\frac{\partial}{\partial\alpha}N_{f}(\alpha)\equiv 0$.

By Example 2.5, $\frac{\partial}{\partial\alpha}N_{f}(\alpha)=0$ whenever $f^{\alpha}$ is a constant multiple of a reproducing kernel of $A_{\alpha}^{2}$. The linear span of reproducing kernels form a dense set in $A_{\alpha}^{2}$. This explains our reason of considering such functions in Theorem 2.4.

Next, let us give the proof of Theorem 2.4.

Proof of Theorem 2.4.

Suppose $f\in\mathcal{O}^{\ast}$ and

[TABLE]

As in Example 2.5, in order to calculate $\frac{\partial}{\partial\alpha}N_{f}(\alpha)$, we need to find out $I_{f}(\alpha)$ as defined in (2.4). Compared with Example 2.5, the main difficulty here is that we can not use the Möbius transform. We will get around by applying the Stoke’s Theorem and the Residue Theorem.

Let

[TABLE]

and

[TABLE]

By (2.4), it is easy to see that

[TABLE]

Taking advantage of the fact that $\log f^{\alpha}(z)$ is a holomorphic function in $\mathbb{D}$, we have

[TABLE]

Next, we calculate $III$.

[TABLE]

Here

[TABLE]

For $i,j=1,\cdots,k$, define

[TABLE]

and

[TABLE]

Then

[TABLE]

Since $\overline{\mathrm{Log}z}=\mathrm{Log}\overline{z}$, we have

[TABLE]

It remains to calculate $IV_{ij}$. Let

[TABLE]

and

[TABLE]

By direct computation, we have

[TABLE]

and

[TABLE]

Therefore

[TABLE]

For any $\varepsilon>0$ sufficiently small, define

[TABLE]

Define the one-form $\omega=\varphi_{ij}(z)\psi_{ij}(z)\mathrm{d}z$. Then

[TABLE]

Applying the Stokes’s Theorem on $\mathbb{D}_{\varepsilon,j}$, we get

[TABLE]

The second equality is because $\varphi_{ij}\psi_{ij}=0$ on the unit circle $\mathbb{T}$.

Therefore

[TABLE]

where

[TABLE]

To calculate $VI_{ij}$, notice that

[TABLE]

Standard estimates will give us

[TABLE]

By (2.18) and (2.19), we have

[TABLE]

By (2.15), (2.20) and (2.19), we get

[TABLE]

Then combining (2.9), (2.10), (2.11) and (2.21), we have

[TABLE]

The last equality is because

[TABLE]

Finally, plugging in (2.22) and (2.23) into (2.3), we get (2.5) and (2.6). This completes the proof. ∎

Remark 2.6**.**

In [4], the authors raised the question whether $N_{f}(\alpha)^{2\alpha}=\|f\|_{A_{\alpha}^{2\alpha}}^{2\alpha}$ is non-increasing in $\alpha$ given that $\|f\|_{H^{2}}=1$. Using our method, we can also compute the derivative $\frac{\partial}{\partial\alpha}N_{f}(\alpha)^{2\alpha}$. In fact, by direct computation, we get

[TABLE]

By (2.22), if $f\in\mathcal{O}^{\ast}$ and $f^{\alpha}(z)=\sum_{i=1}^{k}c_{i}\frac{1}{(1-\overline{w_{i}}z)^{\alpha}}$, we have

[TABLE]

Thus

[TABLE]

Using (2.24), one can easily check that if we drop the condition $\|f\|_{H^{2}}=1$, then there exists $f$ such that $N_{f}(\alpha)^{2\alpha}$ is increasing.

An immediate consequence of Theorem 2.4 is the following.

Theorem 2.7**.**

Suppose $f\in\mathcal{O}^{\ast}$, $\alpha>1$ and $f^{\alpha}=\sum_{i=1}^{k}c_{i}\frac{1}{(1-\overline{w_{i}}z)^{\alpha}}$. Suppose further that set of points $\{w_{1},\cdots,w_{k}\}$ belong to a single real line, and that $c_{i}\geq 0,i=1\cdots,k$. Then

[TABLE]

Proof.

The proof simply an application of the Jensen’s inequality. Note that under our assumption,

[TABLE]

Without loss of generality, let us assume that $N_{f}(\alpha)=1$. That is

[TABLE]

Then by the Jensen’s inequality, we have

[TABLE]

The second equality is because

[TABLE]

By (2.23) and (2.25), we have $\frac{\partial}{\partial\alpha}N_{f}(\alpha)\leq 0$. This completes the proof. ∎

From the proof of Theorem 2.7 we know that the inequality $\frac{\partial}{\partial\alpha}N_{f}(\alpha)\leq 0$ holds true if we can “formally” apply the Jensen’s inequality. However, in general, the coefficients involved are not positive, and one needs to find other ways.

3 Sufficient Conditions

In this section, we give some other sufficient conditions for $\frac{\partial}{\partial\alpha}N_{f}(\alpha)$ to be non-positive. We want to consider the right hand side of (2.6) under a suitable general setting. For this, let us first discuss about how Theorem 2.4 can be used to answer Question 3.

In (2.6), the term $\log f^{\alpha}(w_{i})$ depends on the fact that $f$ is an outer function: the imaginary part of $\log f^{\alpha}(w_{i})$ depends on the formula (assuming $f(0)>0$)

[TABLE]

However, it is unclear how this formula could enter the estimates. Things are relatively easy if we are able to apply the single-valued branch $\mathrm{Log}z$ to all $f^{\alpha}(w_{i})$. It turns out that such special cases are enough for our purpose (See Proposition 3.3). Before going into details, let us fix some notations.

Notations:(1) In the rest of this paper, we use $k$ to denote a positive integer. If not otherwise specified, $\mathbf{c}$ denotes a $k$-tuple of complex numbers, and $\mathbf{w}$ denotes a $k$-tuple of points in $\mathbb{D}$. that is, $\mathbf{c}=(c_{1},c_{2},\cdots,c_{k})$, $c_{i}\in\mathbb{C}$, $\mathbf{w}=(w_{1},w_{2},\cdots,w_{k})$, $w_{i}\in\mathbb{D}$. Given $\mathbf{c}$, $\mathbf{w}$ and $\alpha>0$, we use $\mathbf{W}_{\alpha}$ to denote the $k\times k$ matrix with entry $\frac{1}{(1-\overline{w_{i}}w_{j})^{\alpha}}$ in the $i$-th row and $j$-th column. Thinking of $\mathbf{c}$ as a row vector, we reserve the notation $\mathbf{f}_{\alpha}=(f_{1,\alpha},\cdots,f_{k,\alpha})$ for the row vector defined by $\mathbf{f}_{\alpha}=\mathbf{c}\mathbf{W}_{\alpha}$. Denote $N_{\alpha}=\mathbf{c}\mathbf{W}_{\alpha}\mathbf{c}^{*}=\sum_{i,j=1}^{k}c_{i}\overline{c_{j}}\frac{1}{(1-\overline{w_{i}}w_{j})^{\alpha}}$. If $\alpha$ is specified, then we drop the subscription $\alpha$.

(2) It is well-known that for $\alpha>0$ and $w\in\mathbb{D}$, the functions

[TABLE]

define a unique reproducing kernel Hilbert space on $\mathbb{D}$ [1]. If $\alpha>1$, the space is $A_{\alpha}^{2}$; if $\alpha=1$, it is $H^{2}=A_{1}^{2}$. In this paper, for any $\alpha>0$, we use $A_{\alpha}^{2}$ to denote the uniquely defined reproducing kernel Hilbert space determined by $\{K_{w,\alpha}:w\in\mathbb{D}\}$.

We find it convenient to consider the following general setting.

Definition 3.1**.**

(1) Let $\mathbb{H}:=\{z\in\mathbb{C}:\mathrm{Re}z>0\}$ denote the right half plane. Suppose $\alpha>0$, $\mathbf{c}\in\mathbb{C}^{k}$ and $\mathbf{w}\in\mathbb{D}^{k}$ satisfies $f_{i,\alpha}\in\mathbb{H},i=1,\cdots,k$, where $\mathbf{f}_{\alpha}=(f_{1,\alpha},\cdots,f_{k,\alpha})$ is defined as above. Define

[TABLE]

(2) For any $\alpha>0$ and $0<\varepsilon\leq 1$, define

[TABLE]

and

[TABLE]

We use $\Lambda_{\alpha}$, $\mathcal{K}_{\alpha}$ to denote $\Lambda_{\alpha,1}$, $\mathcal{K}_{\alpha,1}$.

(3) Define

[TABLE]

For $(\mathbf{c},\mathbf{w})\in\Gamma$ and $\alpha>0$, define

[TABLE]

Note that since $x\mathrm{Log}x$ tends to [math] as $x$ tends to [math], the definition above makes sense even if $f_{i}=0$ for some $i=1,\cdots,k$. It is also easy to see that (3.4) coincides with (3.2) when $f_{i,\alpha}>0,\forall i=1,\cdots,k$.

Remark 3.2**.**

Suppose $(\mathbf{c},\mathbf{w})\in\Gamma$ and $f^{\alpha}(z)=\sum_{i=1}^{k}c_{i}K_{w,\alpha}(z)\in\mathcal{O}^{\ast}$, then by (2.6) and (3.4), it is easy to see that $D_{f}(\alpha)=D_{\alpha}(\mathbf{c},\mathbf{w})$. If $(\mathbf{c},\mathbf{w})\in\Lambda_{\alpha}$ and $f^{\alpha}=\sum_{i=0}^{k}c_{i}K_{w,\alpha}\in\mathcal{O}^{\ast}$, then it is not necessarily true that $D_{f}(\alpha)=D_{\alpha}(\mathbf{c},\mathbf{w})$. However, if one knows that $\{w_{1},\cdots,w_{k}\}$ is contained in a connected open subset $\Omega$ of $\mathbb{D}$ which is mapped, by $f^{\alpha}$, into $\mathbb{H}$, then by standard argument, the function $\mathrm{Log}f^{\alpha}(z)|_{\Omega}$ differs from the function given in (3.1), by an integer multiple of $2\pi i$. Then from the expression of (2.6) one can see that $D_{f}(\alpha)=D_{\alpha}(\mathbf{c},\mathbf{w})$. We will use this fact later.

It turns out that we only need to consider the case when $f^{\alpha}\in\mathcal{K}_{\alpha,\varepsilon}$ for $\varepsilon$ small enough.

Proposition 3.3**.**

Suppose for any $\alpha>1$ there exists $0<\varepsilon\leq 1$ such that $D_{\alpha}(\mathbf{c},\mathbf{w})\leq 0$ for all $(\mathbf{c},\mathbf{w})\in\Lambda_{\alpha,\varepsilon}$. Then $\frac{\partial}{\partial\alpha}N_{f}(\alpha)\leq 0$ for all $f\in\mathcal{O}^{\ast}$. As a consequence, Conjecture 2 holds.

The proof is based on the following two lemmas.

Lemma 3.4**.**

Suppose $\alpha>1$ and $0<\varepsilon\leq 1$. Then for any $g\in\mathcal{O}^{\ast}$ such that $g(0)=1$, there exists $0<\delta\leq\varepsilon$ and a sequence $\{g_{n}\}\in\mathcal{K}_{\alpha,\delta}$ such that $g_{n}$ converges uniformly on $\overline{\mathbb{D}}$ to $g$. Moreover, $g_{n}(z)\in\mathbb{H}$ for all $n$ and all $z\in\mathbb{D}$ with $|z|<\delta$.

Proof.

Since $g(0)=1$, we can choose $0<\delta\leq\varepsilon$ such that $\mathrm{Re}g(z)>\frac{1}{2}$ for any $z\in\mathbb{D}$ with $|z|<\delta$. Choose $r>1$ such that $g$ is defined on $\{z\in\mathbb{C}:|z|\leq r\}$. Define

[TABLE]

Obviously, $g_{r}\in A_{\alpha}^{2}$. The subspace

[TABLE]

is dense in $A_{\alpha}^{2}$. Choose a sequence $\{\tilde{g}_{n}\}\subset\mathrm{span}\{K_{w,\alpha}:w\in(-\delta,\delta)\}$ such that $\tilde{g}_{n}\to g_{r}$ in $A_{\alpha}^{2}$ norm. Then $\tilde{g}_{n}$ converge uniformly to $g_{r}$ on $\{z\in\mathbb{C}:|z|\leq\frac{1}{r}\}$. Define

[TABLE]

Then $g_{n}$ converge uniformly to $g$ on $\overline{\mathbb{D}}$. By construction, each $\tilde{g}_{n}$ is of form

[TABLE]

where $w_{n,i}\in(-\delta,\delta)$, $\forall i$. Therefore

[TABLE]

Since $r>1$ we have $\frac{w_{n,i}}{r}\in(-\delta,\delta)$. Also, since $g_{n}$ converge uniformly to $g$ and $\mathrm{Re}g(z)>\frac{1}{2}$ if $|z|<\delta$, by passing to a subsequence, we have $g_{n}(z)\in\mathbb{H}$ for any $n$ and any $z$ with $|z|<\delta$. In particular, $g_{n}(\frac{w_{n,i}}{r})\in\mathbb{H}$. Therefore $g_{n}\in\mathcal{K}_{\alpha,\delta}$ for any $n$. This completes the proof. ∎

The following lemma is simply a consequence of the fact that, for $f\in H^{2}$, $f_{r}(z):=f(rz)$ converges to $f$ in $H^{2}$ norm as $r\to 1-$.

Lemma 3.5**.**

Suppose $f$ is an outer function in $H^{2}$. Then there exists a sequence $\{f_{n}\}\subset\mathcal{O}^{\ast}$ such that $f_{n}$ tends to $f$ in the Hardy norm $\|\cdot\|_{H^{2}}$.

Proof of Proposition 3.3.

Assume that for some $\alpha>1$ and $0<\varepsilon\leq 1$ we have $D_{\alpha}(\mathbf{c},\mathbf{w})\leq 0$ for all $(\mathbf{c},\mathbf{w})\in\Lambda_{\alpha,\varepsilon}$. For any $f\in\mathcal{O}^{\ast}$, we want to show $\frac{\partial}{\partial\alpha}N_{f}(\alpha)\leq 0$. Without loss of generality we can assume $f(0)=1$. By Lemma 3.4, there exists $0<\delta\leq\varepsilon$ and a sequence $\{g_{n}\}$ in $\mathcal{K}_{\alpha,\delta}$ such that $g_{n}$ converges uniformly to $f^{\alpha}$ on $\mathbb{D}$ and $g_{n}$ maps $\{z\in\mathbb{D}:|z|<\delta\}$ to $\mathbb{H}$. Also, since $|f^{\alpha}|$ is bounded away from [math] on $\mathbb{D}$, for $n$ large enough, $g_{n}$ is outer and we can define $f_{n}=g_{n}^{1/\alpha}$. By Remark 3.2, we have $\frac{\partial}{\partial\alpha}N_{f_{n}}(\alpha)=D_{\alpha}(\mathbf{c}_{n},\mathbf{w}_{n})\in\Lambda_{\alpha,\delta}$, where $g_{n}$ corresponds to $(\mathbf{c_{n}},\mathbf{w_{n}})$. In particular, we have $\frac{\partial}{\partial\alpha}N_{f_{n}}(\alpha)\leq 0$ for $n$ large enough.

On the other hand, since $|f|$ is bounded away from [math] we also have $\mathrm{Log}|f_{n}|$ converging uniformly to $\mathrm{Log}|f|$ on $\mathbb{D}$. By (2.3) and (2.4) it is easy to see that $\frac{\partial}{\partial\alpha}N_{f_{n}}(\alpha)\to\frac{\partial}{\partial\alpha}N_{f}(\alpha)$ as $n$ tend to infinity. Thus $\frac{\partial}{\partial\alpha}N_{f}(\alpha)\leq 0$. Then by Lemma 2.1, we have $\|f\|_{A_{\alpha}}^{2\alpha}\leq\|f\|_{H^{2}}$ for all $f\in\mathcal{O}^{\ast}$. By Lemma 3.5, the inequality also holds for all outer functions. Suppose $f=\eta g$ where $\eta$ is inner and $g$ is outer. Then $\|f\|_{A_{\alpha}^{2\alpha}}\leq\|g\|_{A_{\alpha}^{2\alpha}}\leq\|g\|_{H^{2}}=\|f\|_{H^{2}}$. This completes the proof. ∎

Now we are ready to give some sufficient conditions for $D_{\alpha}(\mathbf{c},\mathbf{w})$ to be non-positive.

Theorem 3.6**.**

Suppose $\alpha>0$ and $(\mathbf{c},\mathbf{w})\in\Lambda_{\alpha}\cup\Gamma$ satisfy the following conditions.

• (1)

$w_{1}<w_{2}<\cdots<w_{k}$, where $k$ is the number of entries in $\mathbf{w}$;

• (2)

either $\{c_{2},\cdots,c_{k}\}$ or $\{c_{1},\cdots,c_{k-1}\}$ are real and have the same sign.

Then we have $D_{\alpha}(\mathbf{c},\mathbf{w})\leq 0$.

We will need the following lemma in the proof of Theorem 3.6.

Lemma 3.7**.**

Suppose $\alpha>0$, $\mathbf{c}\in\mathbb{C}^{k}$ and $\mathbf{A}=[a_{ij}]$ is semi-positive definite, $a_{ij}>0$. Let $\mathbf{f}=\mathbf{c}\mathbf{A}$ and $N=\mathbf{c}\mathbf{A}\mathbf{c}^{\ast}$. Then for any $x_{1},\cdots,x_{k}\geq 0$ we have

[TABLE]

Proof.

The proof is, again, an application of the Jenson’s Inequality. If some $f_{i}$ equals zero then the left hand side is $-\infty$ and the inequality always holds. Assume $f_{i}$ are all non-zero. Without loss of generality, we can also assume that $\sum_{i,j=1}^{k}x_{i}x_{j}a_{ij}=1$. Applying the Jenson’s Inequality, we get

[TABLE]

Choose $e_{i}\in\mathbb{C}$ such that $|e_{i}|=1$ and $e_{i}f_{i}=|f_{i}|$. Then the right hand side of (3.6) becomes \mathrm{Log}\frac{\big{(}\sum_{i,j=1}^{k}x_{i}e_{i}c_{j}a_{ij}\big{)}^{2}}{\sum_{i,j=1}^{k}c_{i}\overline{c_{j}}a_{ij}}, which is less than or equal to $\mathrm{Log}\sum_{i,j=1}^{k}x_{i}e_{i}x_{j}\overline{e_{j}}a_{ij}$ by the fact that $A$ is semi-positive definite. Since $x_{i}\geq 0$ and $a_{ij}>0$, we have $\sum_{i,j=1}^{k}x_{i}e_{i}x_{j}\overline{e_{j}}a_{ij}\leq\sum_{i,j=1}^{k}x_{i}x_{j}a_{ij}=1$. Therefore the left hand side of (3.5) is less than or equal to [math]. This completes the proof. ∎

Proof of Theorem 3.6.

We will prove the theorem in the case when $(\mathbf{c},\mathbf{w})\in\Lambda_{\alpha}$. The proof when $(\mathbf{c},\mathbf{w})\in\Gamma$ is similar. First, we notice that if we let $-\mathbf{w}=(-w_{1},\cdots,-w_{k})$, then $D_{\alpha}(\mathbf{c},\mathbf{w})=D_{\alpha}(\mathbf{c},\mathbf{-w})$. From this, it is easy to see that it suffices to consider the case when $w_{1}<w_{2}<\cdots<w_{k}$ and $\{c_{2},\cdots,c_{k}\}$ are real and have the same sign.

Let us further reduce the cases. Suppose $w_{1}<w_{2}<\cdots<w_{k}$. Let $z_{i}=-\varphi_{w_{1}}(w_{i})$, $i=1,\cdots,k$. Then it is easy to check that $0=z_{1}<z_{2}<\cdots<z_{k}$. Suppose $(\mathbf{c},\mathbf{w})\in\Lambda_{\alpha}$. Let $d_{i}=\frac{c_{i}(1-z_{i}w_{1})^{\alpha}}{(1-w_{1}^{2})^{\alpha/2}}$ and consider the pair $(\mathbf{d},\mathbf{z})$. Write $\mathbf{W}=[\frac{1}{(1-w_{i}w_{j})^{\alpha}}]$, $\mathbf{\Lambda}=[\frac{1}{(1-z_{i}z_{j})^{\alpha}}]$, $\mathbf{f}=\mathbf{c}\mathbf{W}$ and $\mathbf{g}=\mathbf{d}\mathbf{\Lambda}$. Using the well-known formula

[TABLE]

it is easy to check that

[TABLE]

Then $(\mathbf{d},\mathbf{z})\in\Lambda_{\alpha}$. From the equation above, it is also straight-forward to check that $D_{\alpha}(\mathbf{c},\mathbf{w})=D_{\alpha}(\mathbf{d},\mathbf{z})$. Therefore, in order to prove Theorem 3.6, we only need to consider the case when $0=w_{1}<\cdots<w_{k}$ and $\{c_{2},\cdots,c_{k}\}$ are real and have the same sign.

Assume that $(\mathbf{c},\mathbf{w})\in\Lambda_{\alpha}$, $0=w_{1}<\cdots<w_{k}$ and $c_{i}\geq 0,\forall i=2,\cdots,k$. The case when $c_{2}=\cdots=c_{k}=0$ is trivial. Thus we can assume that $c_{i}>0$ for some $i=2,\cdots,k$. Define $\mathbf{f}$ and $\mathbf{W}$ as before. The idea is to find a non-increasing function that takes value $D_{\alpha}(\mathbf{c},\mathbf{w})$ at $\alpha$ and [math] at [math].

For $0\leq t\leq\alpha$, define $a_{ij,t}=\frac{1}{(1-w_{i}w_{j})^{t}}$, $i,j=1,\cdots,k$. Define $\mathbf{W}_{t}=[a_{ij,t}]$, $N_{t}=\mathbf{c}\mathbf{W}_{t}\mathbf{c}^{*}$ and $\mathbf{f}_{t}=\mathbf{c}\mathbf{W}_{t}$. Since only $c_{1}$ may have imaginary part, the signs of the imaginary part of each $f_{i,t}$ depend only on that of $c_{1}$. Assume, without loss of generality, that $\mathrm{Im}c_{1}\geq 0$. Then $\mathrm{Im}f_{i,t}\geq 0,\forall i=1,\cdots,k$. Define

[TABLE]

Notice that since $w_{1}=0$, we have $\frac{\mathrm{d}}{\mathrm{d}t}f_{i,t}=\sum_{j=2}^{k}c_{j}\frac{1}{(1-w_{i}w_{j})^{t}}\mathrm{Log}\frac{1}{1-w_{i}w_{j}}>0,\forall i=1,\cdots,k$. So the points $t$ such that $f_{i,t}=0$ for some $i$, are isolated. Also, $z\mathrm{Log}z\to 0$ if $z$ tends to [math]. From this we can see that $D_{t}$ is a continuous, piecewise differentiable function.

Next, we show that $D_{t}$ is non-increasing. By the previous argument, it suffices to show that $\frac{\mathrm{d}}{\mathrm{d}t}D_{t}\leq 0$ at the points where each $f_{i,t}$ is non-zero. By direct computation, we get

[TABLE]

Since $w_{1}=0$ and $c_{i}\geq 0,\forall i=2,\cdots,k$, we have

[TABLE]

Here the last inequality is by Lemma 3.7. If $\{c_{2},\cdots,c_{k}\}$ are all non-positive, simply replace $c_{i}$ with $-c_{i}$ in the above argument. Thus in either case we have that $D_{t}$ is non-increasing.

It is obvious that $D_{\alpha}=D_{\alpha}(\mathbf{c},\mathbf{w})$. By straight-forward computation it is also easy to show that $D_{0}=0$. Therefore $D_{\alpha}(\mathbf{c},\mathbf{w})=D_{\alpha}\leq D_{0}=0$. This completes the proof. ∎

Theorem 3.8**.**

Suppose $\alpha>0$, $\mathbf{c}\in\mathbb{C}^{2}$, $\mathbf{w}\in\mathbb{D}^{2}$ and $(\mathbf{c},\mathbf{w})\in\Lambda_{\alpha}\cup\Gamma$. Then

[TABLE]

Moreover, $D_{\alpha}(\mathbf{c},\mathbf{w})=0$ if and only if $c_{1}=0$, or $c_{2}=0$, or $w_{1}=w_{2}$.

Proof.

The proof is similar as that of Theorem 3.6. Define $\mathbf{W}=[a_{ij}]=[\frac{1}{(1-w_{i}w_{j})^{\alpha}}]$ as before. For $0\leq t\leq 1$, define

[TABLE]

Write $\mathbf{W}_{t}=[a_{ij,t}]$, $N_{t}=\mathbf{c}\mathbf{W}_{t}\mathbf{c}^{*}$ and $\mathbf{f}_{t}=\mathbf{c}\mathbf{W}_{t}$. It is easy to check that the following hold.

• (i)

$f_{1,t}=(1-t)f_{1}+t\frac{a_{12}}{a_{22}}f_{2}$, $f_{2,t}\equiv f_{2}$;

• (ii)

$\frac{\mathrm{d}}{\mathrm{d}t}a_{11,t}\leq 0$;

• (iii)

each $\mathbf{W}_{t}$ is semi-positive definite;

• (iv)

$\mathbf{W}_{1}$ has rank $1$.

By (i), the paths $f_{1,t}$ and $f_{2,t}$ stay in $\mathbb{H}$. Define

[TABLE]

Then $D_{t}$ is a differentiable function on $(0,1)$. From (iv) it is easy to compute that $D_{1}=0$. By direct computation, we have

[TABLE]

Since $\mathbf{W}_{t}$ is positive definite, by Lemma 3.7, it is easy to see that $\frac{\mathrm{d}}{\mathrm{d}t}D_{t}\geq 0$. Thus $D_{\alpha}(\mathbf{c},\mathbf{w})=D_{0}\leq D_{1}=0$. Equality holds if and only if $\frac{\mathrm{d}}{\mathrm{d}t}D_{t}\equiv 0$. This always holds when $w_{1}=w_{2}$. If $w_{1}\neq w_{2}$, then $\frac{\mathrm{d}}{\mathrm{d}t}a_{11,t}\neq 0$. Thus either $c_{1}=0$ or $\frac{|f_{1,t}|^{2}}{a_{11,t}N_{t}}\equiv 1$. In particular, if $c_{1}\neq 0$, then

[TABLE]

Here $\mathbf{e}_{1}=(1,0,\cdots,0)$. Since $W_{0}$ is positive definite, this occurs only when $c_{2}=0$. This completes the proof. ∎

In terms of $\frac{\partial}{\partial\alpha}N_{f}(\alpha)$, we summarize our results as follows.

Theorem 3.9**.**

Suppose $f\in\mathcal{O}^{\ast}$, $\alpha>1$, $f^{\alpha}=\sum_{i=1}^{k}c_{i}K_{w_{i},\alpha}$ and $(\mathbf{c},\mathbf{w})\in\Lambda_{\alpha}\cup\Gamma$. Suppose one of the following holds.

• (1)

$c_{i}\geq 0$, $i=1,\cdots,k$.

• (2)

$w_{1}<\cdots<w_{k}$, and either $\{c_{2},\cdots,c_{k}\}$ or $\{c_{1},\cdots,c_{k-1}\}$ are real and have the same sign.

• (3)

$k=2$.

Then we have $\frac{\partial}{\partial\alpha}N_{f}(\alpha)\leq 0$.

4 Norm Inequalities for Linear Combinations of Two Reproducing Kernels

Recall that in Lemma 2.1, we showed that if $\frac{\partial}{\partial\alpha}N_{f}(\alpha)\leq 0,\forall\alpha>1$, for some $f\in\mathcal{O}^{\ast}$, then Conjecture 2 holds for $f$. In this section, we provide an alternative way of proving results on Conjecture 2, using results obtained in Section 3. As a consequence, we prove the following theorem.

Theorem 4.1**.**

Suppose $f\in H^{2}$ and $f=\eta F$, where $\eta$ is inner and $F$ has no zeros in $\mathbb{D}$. Suppose

[TABLE]

for some $\alpha>1$, and $\mathbf{c}\in\mathbb{C}^{2},\mathbf{w}\in\mathbb{D}^{2}$. Then for any $1\leq\beta\leq\alpha$, we have

[TABLE]

Equality holds if and only if $F^{\alpha}=cK_{w,\alpha}$ for some $c\in\mathbb{C}$ and $w\in\mathbb{D}$. As a consequence, we have

[TABLE]

Equality holds if and only if $f=cK_{w,1}$ for some $c\in\mathbb{C}$ and $w\in\mathbb{D}$.

The proof is based on a different way of viewing $D_{\alpha}(\mathbf{c},\mathbf{w})$. Recall that in Definition 3.1, for $\alpha>0$ and $(\mathbf{c},\mathbf{w})\in\Lambda_{\alpha}\cup\Gamma$, we defined $W_{\alpha}=[\frac{1}{(1-\overline{w_{i}}w_{j})^{\alpha}}]$ and $\mathbf{f}_{\alpha}=\mathbf{c}\mathbf{W}_{\alpha}$. Then $D_{\alpha}(\mathbf{c},\mathbf{w})$ is defined using $\mathbf{f}_{\alpha}$ and $\mathbf{W}_{\alpha}$. In the case when $\{w_{i}:i=1,\cdots,k\}$ are distinct points, the matrix $\mathbf{W}_{\alpha}$ is invertible. Therefore we have $\mathbf{c}=\mathbf{f}_{\alpha}\mathbf{W}_{\alpha}^{-1}$. This means we can define $D_{\alpha}(\mathbf{c},\mathbf{w})$ using $\mathbf{f}_{\alpha}$.

Definition 4.2**.**

Suppose $\alpha>0$, $k$ is a positive integer, and $\mathbf{w}\in\mathbb{D}^{k}$ is such that $\{w_{i}:i=1,\cdots,k\}$ are distinct. Define $\mathbf{W}_{\alpha}$ as usual. Suppose either $\mathbf{f}\in\mathbb{H}^{k}$ or $\mathbf{w}\in(-1,1)^{k},\mathbf{f}\in\mathbb{R}^{k}$. Let $\mathbf{c}_{\alpha}=\mathbf{f}\mathbf{W}_{\alpha}^{-1}$. Define

[TABLE]

Definition 4.3**.**

Suppose $\mathbf{w}\in\mathbb{D}^{k}$, $\{w_{1},\cdots,w_{k}\}$ are distinct, and $\alpha>0$. Define

[TABLE]

and $P_{\mathbf{w},\alpha}$ the orthogonal projection from $A_{\alpha}^{2}$ onto $\mathcal{K}_{\mathbf{w},\alpha}$. For $f\in A_{\alpha}^{2}$, if we denote $f(\mathbf{w})=(f(w_{1}),\cdots,f(w_{k}))$, then it is easy to compute that

[TABLE]

Proof of Theorem 4.1.

It is easy to see that (4.2) follows from (4.1). Thus we only need to prove (4.1).

(1) First, we prove (4.1) under the following conditions.

• (i)

$F^{\alpha}=c_{1}+c_{2}K_{w,\alpha}$, $w\geq 0$;

• (ii)

There is a connected open neighborhood $\Omega$, of $\{0,w\}$, such that $F^{\alpha}(\Omega)\subset\mathbb{H}$.

Assume the above, then we can choose $\log F^{\alpha}$ such that $\log F^{\alpha}|_{\Omega}=\mathrm{Log}F^{\alpha}|_{\Omega}$. As a consequence, $F^{\beta}|_{\Omega}\subset\mathbb{H}$ for any $\beta\in[1,\alpha]$.

For $1\leq\beta\leq\alpha$, consider

[TABLE]

where $F^{\beta}(\mathbf{w})=(F^{\beta}(w_{1}),F^{\beta}(w_{2}))$. Then by straight-forward computation, we have

[TABLE]

The last inequality is because of Theorem 3.8. Therefore we have

[TABLE]

If we assume that $w\neq 0$, then by Theorem 3.8, we also know that the equality holds if and only if for each $\beta$, either $c_{1,\beta}=0$ or $c_{2,\beta}=0$. In particular, either $c_{1}=0$ or $c_{2}=0$. On the other hand, if either $c_{1}=0$ or $c_{2}=0$, then it is easy to check that the equality in (4.1) holds. This completes the proof for case (1).

(2) Next, we consider the case when $F^{\alpha}=c_{1}+c_{2}K_{w,\alpha},w\in\mathbb{D},c_{1},c_{2}\in\mathbb{C}$. Choose $\theta\in[0,2\pi]$ so that $e^{i\theta}w\geq 0$. Let $F_{\theta}(z)=F(e^{-i\theta}z)$. Then $F^{\alpha}_{\theta}=c_{1}+c_{2}K_{e^{i\theta}w,\alpha}$. Inequality (4.1) for $F$ follows from (4.1) for $F_{\theta}$. Thus we may assume that $w\geq 0$ in the beginning. Suppose $F$ has no zeros in $\mathbb{D}$ and $F^{\alpha}=c_{1}+c_{2}K_{w,\alpha}$ with $w\geq 0$. We will show that after multiplying $F$ by a non-zero constant, the condition (ii) in case (1) will be satisfied. This will lead to (4.1) for case (2). We may as well assume that $F^{\alpha}=1+cK_{w,\alpha}$ with $w\geq 0$. It is easy to see that $F^{\alpha}$ maps the interval $[0,w]$ onto the (complex valued) interval between $1+c$ and $1+\frac{c}{(1-|w|^{2})^{\alpha}}$, which we denote by $I$. Since $F$ has no zeros in $\mathbb{D}$, $0\notin I$. Thus $I$ must be contained in some half plane $e^{i\theta_{1}}\mathbb{H}$. By standard trick we can find a connected open neighborhood $\Omega$ of $[0,w]$ such that $F^{\alpha}(\Omega)\subset e^{i\theta_{1}}\mathbb{H}$. Therefore \big{(}e^{-i\theta_{1}/\alpha}F\big{)}^{\alpha}(\Omega)\subset\mathbb{H}. This completes the proof for case (2).

(3) In general, suppose $F$ has no zeros in $\mathbb{D}$ and $F^{\alpha}=c_{1}K_{w_{1},\alpha}+c_{2}K_{w_{2},\alpha}$. Denote

[TABLE]

Let $G=F\circ\varphi_{w_{1}}\cdot k_{w_{1},1}$. Then $G$ also has no zeros in $\mathbb{D}$ and $G^{\beta}=F^{\beta}\circ\varphi_{w_{1}}\cdot k_{w_{1},\beta}$. It is standard to check that $\|G^{\beta}\|_{A_{\beta}^{2}}=\|F^{\beta}\|_{A_{\beta}^{2}}$, $\forall\beta\in[1,\alpha]$. Notice that (4.1) is equivalent to

[TABLE]

So it suffices to prove

[TABLE]

which is in turn, equivalent to $\|G\|_{A_{\alpha}^{2\alpha}}\leq\|G\|_{A_{\beta}^{2\beta}},\beta\in[1,\alpha]$. By straight-forward computation we have

[TABLE]

Thus $G$ satisfies case (2). This completes the proof of (4.1) in the general case. Tracing back to the proof of (1), we also see that the equality in (4.1) holds if and only if $F^{\alpha}=cK_{w,\alpha}$ for some $c\in\mathbb{C}$ and $w\in\mathbb{D}$. Then (4.2) follows immediately. This completes the proof. ∎

In terms of Conjecture 1, (4.2) becomes the following.

Corollary 4.4**.**

Suppose $0<p\leq 2$ and $\alpha=\frac{2}{p}$. Suppose $f\in H^{p}$, $f=\eta F$, where $\eta$ is inner and $F$ has no zeros in $\mathbb{D}$. If

[TABLE]

then

[TABLE]

Equality holds if and only if $f=cK_{w,\alpha}$ for some $c\in\mathbb{C}$ and $w\in\mathbb{D}$.

5 Remarks and Numerical Evidences

5.1 Some Further Remarks

(1) In Proposition 3.3, we give a sufficient condition for Conjecture 2 to hold. In its most general form, we list the conjecture below.

Conjecture 4. Suppose $\alpha>0$ and $(\mathbf{c},\mathbf{w})\in\Lambda_{\alpha}$. Then $D_{\alpha}(\mathbf{c},\mathbf{w})$ defined as in (3.2) and (3.4) is non-positive.

Another interesting question to ask is the following.

Question 5. Suppose Conjecture 4 holds. Does it imply that the norms $\|f^{\alpha}\|^{1/\alpha}_{A_{\alpha}^{2}}$ are non-increasing for $\alpha>0$, for some set of functions $f$?

(2) An immediate observation from (3.2) is that $D_{\alpha}(\mathbf{c},\mathbf{w})$ is the bi-linear form $\mathbf{c}\mathbf{W}_{\alpha}\circ\mathbf{B}_{\mathbf{c},\mathbf{w}}\mathbf{c}^{*}$, where \mathbf{B}_{\mathbf{c},\mathbf{w}}=\bigg{[}\overline{\mathrm{Log}f_{i}}+\mathrm{Log}f_{j}-\mathrm{Log}(a_{ij,\alpha}N_{\alpha})\bigg{]}_{i,j=1,\cdots,k} and $A\circ B$ denotes the Hadmadard product of $A$ and $B$. Thus a sufficient condition of Conjecture 4 would be that $-\mathbf{B}_{\mathbf{c},\mathbf{w}}$ is semi-positive definite. However, this is not true, even in the simplest case. Take, for example, $c_{1}=c_{2}=1$, $w_{1}=0$, and $w_{2}\in(0,1)$ such that $\frac{1}{(1-w_{2}^{2})^{\alpha}}=1.1$.

(3) In Section 4, we also defined the function $\widehat{D}_{\alpha}(\mathbf{f},\mathbf{w})$. Then Conjecture 4 is equivalent to the following.

Conjecture 6. Suppose $\alpha>0$, $\mathbf{w}\in(-1,1)^{k}$ and $\mathbf{f}\in\mathbb{H}^{k}$. Then $\widehat{D}_{\alpha}(\mathbf{f},\mathbf{w})\leq 0$.

(4) There is , yet another way of looking at (3.2). Suppose $\mathbf{A}=[a_{ij}]$ is a semi-positive definite $k\times k$ matrix, $a_{ij}\in\mathbb{H}$, and $\mathbf{c}\in\mathbb{C}^{k}$ is such that each entry of $\mathbf{f}:=\mathbf{c}\mathbf{A}$ belongs to $\mathbb{H}$. Then we can define $N(\mathbf{c},\mathbf{A})=\mathbf{c}\mathbf{A}\mathbf{c}^{*}$ and

[TABLE]

Then we can ask whether $\widetilde{D}(\mathbf{c},\mathbf{A})\leq 0$. In the proofs of Theorem 3.6 and Theorem 3.8, we are secretly using this definition: we let the matrix $\mathbf{A}$ to vary from $\mathbf{W}_{\alpha}$ to a rank $1$ matrix. Then we used the fact that if $\mathbf{A}$ has rank $1$, then $\widetilde{D}(\mathbf{c},\mathbf{A})=0$. An interesting observation is the following: if we define $\widetilde{\mathbf{A}}=[\widetilde{a}_{ij}]$ to be the block matrix

[TABLE]

and $\widetilde{\mathbf{c}}=(-1,\mathbf{c})$, then $\widetilde{\mathbf{A}}$ is semi-positive definite and $\widetilde{D}(\mathbf{c},\mathbf{A})=\widetilde{D}(\widetilde{\mathbf{c}},\widetilde{\mathbf{A}})$. Moreover, $\widetilde{D}(\widetilde{\mathbf{c}},\widetilde{\mathbf{A}})$ has the simple expression

[TABLE]

In general, we ask the following question.

Question 7. Suppose $\mathbf{A}=[a_{ij}]$ is semi-positive definite and $\mathrm{Re}a_{ij}\geq 0$, $\forall i,j=1,\cdots,k$. Suppose $\mathbf{c}\in\mathbb{C}^{k}$ is such that $\mathbf{c}\mathbf{A}=\mathbf{0}$. Do we have

[TABLE]

5.2 Numerical Evidences

One of the advantages that Theorem 2.4 offers is that we can now test Question 3 using numerical methods. We have tested for a wide range of values of $\mathbf{c}$ and $\mathbf{w}$. We list a few graphs for the interested readers.

(1) In the proof of Theorem 3.6, we showed that $D_{\alpha}(\mathbf{c},\mathbf{w})$ is non-increasing in $\alpha$ under the given conditions. However, Figure 1 shows that this is not always true.

(2) If we adopt the definition $\widehat{D}_{\alpha}(\mathbf{f},\mathbf{w})$, then by fixing $f_{2}$ and $f_{3}$ and let $f_{1}$ vary, we get Figures 2 and 3.

(3) In the special case when all entries of $\mathbf{c}$ are real, we can use the definition (3.4). By letting one of the coefficients vary, we get Figure 4.

Acknowledgment: The first and second author is partially supported by the Natural Science Foundation of China. The third author is partially supported by the Natural Science Foundation.