# Norm of an integral operator on some analytic function spaces on the unit disk

## Abstract

If f is an analytic function in the unit disc , a class of integral operators is defined as follows:

${I}_{f}\left(h\right)\left(z\right)={âˆ«}_{0}^{z}f\left(w\right){h}^{â€²}\left(w\right)\phantom{\rule{0.2em}{0ex}}dw,\phantom{\rule{1em}{0ex}}hâˆˆH\left(\mathbb{D}\right),zâˆˆ\mathbb{D}.$

The norm of ${I}_{f}$ on some analytic function spaces is computed in this paper.

MSC:47B38, 32A35.

## 1 Introduction

Let $\mathbb{D}=\left\{z:|z|<1\right\}$ be the unit disk of a complex plane â„‚. Denote by $H\left(\mathbb{D}\right)$ the class of functions analytic in . Let dÏƒ denote the normalized Lebesgue area measure in and $g\left(a,z\right)$ the Green function with logarithmic singularity at a, i.e., $g\left(a,z\right)=âˆ’log|{\mathrm{Ï†}}_{a}\left(z\right)|$, where ${\mathrm{Ï†}}_{a}\left(z\right)=\left(aâˆ’z\right)/\left(1âˆ’\stackrel{Â¯}{a}z\right)$ is the MÃ¶bius transformation of .

For $0, the ${Q}_{p}$ is the space of all functions $fâˆˆH\left(\mathbb{D}\right)$, for which

${âˆ¥fâˆ¥}_{{Q}_{p}}^{2}={|f\left(0\right)|}^{2}+\underset{aâˆˆ\mathbb{D}}{sup}{âˆ«}_{\mathbb{D}}{|{f}^{â€²}\left(z\right)|}^{2}{\left(1âˆ’{|{\mathrm{Ï†}}_{a}\left(z\right)|}^{2}\right)}^{p}\phantom{\rule{0.2em}{0ex}}d\mathrm{Ïƒ}\left(z\right)<\mathrm{âˆž}.$
(1.1)

We know that ${Q}_{1}=BMOA$, the space of all analytic functions of bounded mean oscillation [1, 2]. For all $p>1$, the space ${Q}_{p}$ is the same and equal to the Bloch space , consisting of analytic functions f in such that

${âˆ¥fâˆ¥}_{\mathfrak{B}}=|f\left(0\right)|+\underset{zâˆˆ\mathbb{D}}{sup}|{f}^{â€²}\left(z\right)|\left(1âˆ’{|z|}^{2}\right)<\mathrm{âˆž}.$
(1.2)

See [3, 4] for the theory of Bloch functions.

For $\mathrm{Î±}>0$, the Î±-Bloch space, denoted by ${\mathfrak{B}}^{\mathrm{Î±}}$, is the space of all functions f in , for which

${âˆ¥fâˆ¥}_{{\mathfrak{B}}^{\mathrm{Î±}}}=|f\left(0\right)|+\underset{zâˆˆ\mathbb{D}}{sup}|{f}^{â€²}\left(z\right)|{\left(1âˆ’{|z|}^{2}\right)}^{\mathrm{Î±}}<\mathrm{âˆž}.$
(1.3)

Obviously, ${\mathfrak{B}}^{{\mathrm{Î±}}_{1}}â«‹\mathfrak{B}â«‹{\mathfrak{B}}^{{\mathrm{Î±}}_{2}}$ for $0<{\mathrm{Î±}}_{1}<1<{\mathrm{Î±}}_{2}<\mathrm{âˆž}$.

For any $fâˆˆH\left(\mathbb{D}\right)$, the next two integral operators on $H\left(\mathbb{D}\right)$ are induced as follows:

${I}_{f}\left(h\right)\left(z\right)={âˆ«}_{0}^{z}{h}^{â€²}\left(w\right)f\left(w\right)\phantom{\rule{0.2em}{0ex}}dw\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{J}_{f}\left(h\right)\left(z\right)={âˆ«}_{0}^{z}h\left(w\right){f}^{â€²}\left(w\right)\phantom{\rule{0.2em}{0ex}}dw\phantom{\rule{1em}{0ex}}\left(zâˆˆ\mathbb{D}\right).$

Let ${M}_{f}$ denote the multiplication operator, that is, ${M}_{f}\left(h\right)=fh$.

Let $fâˆˆH\left(\mathbb{D}\right)$. Then

$\left({I}_{f}+{J}_{f}\right)h=fhâˆ’f\left(0\right)h\left(0\right)={M}_{f}\left(h\right)âˆ’f\left(0\right)h\left(0\right).$

If f is a constant, then all results about ${I}_{f}$, ${J}_{f}$ or ${M}_{f}$ are trivial. In general, f is assumed to be non-constant. Both integral operators have been studied by many authors. See [5â€“21] and the references therein.

Norm of composition operator, weighted composition operator and some integral operators have been studied extensively by many authors, see [22â€“34] for example. Recently, Liu and Xiong discussed the norm of integral operators ${I}_{f}$ and ${J}_{f}$ on the Bloch space, Dirichlet space, BMOA space and so on in [35].

In this paper, we study the norm of integral operator ${I}_{f}$. The norm of ${I}_{f}$ on several analytic function spaces is computed.

## 2 Main results

In this section, we state and prove our main results. In order to formulate our main results, we need an auxiliary result which is incorporated in the following lemma.

Lemma 2.1 Let $0. For any ${z}_{0}âˆˆ\mathbb{D}$, the function

${g}_{{z}_{0}}\left(z\right)=\frac{{z}_{0}âˆ’z}{1âˆ’{\stackrel{Â¯}{z}}_{0}z}âˆ’{z}_{0}$
(2.1)

is analytic in and ${âˆ¥{g}_{{z}_{0}}âˆ¥}_{{Q}_{p}}=1/{\left(p+1\right)}^{1/2}$.

Proof By (1.1) and [[1], Proposition 1, p.109], we have

$\begin{array}{rl}{âˆ¥{g}_{{z}_{0}}âˆ¥}_{{Q}_{p}}^{2}& =\underset{aâˆˆ\mathbb{D}}{sup}{âˆ«}_{\mathbb{D}}{|{g}_{{z}_{0}}^{â€²}\left(z\right)|}^{2}{\left(1âˆ’{|{\mathrm{Ï†}}_{a}\left(z\right)|}^{2}\right)}^{p}\phantom{\rule{0.2em}{0ex}}d\mathrm{Ïƒ}\left(z\right)\\ =\underset{bâˆˆ\mathbb{D}}{sup}{âˆ«}_{\mathbb{D}}{\left(1âˆ’{|{\mathrm{Ï†}}_{b}\left(z\right)|}^{2}\right)}^{p}\phantom{\rule{0.2em}{0ex}}d\mathrm{Ïƒ}\left(z\right),\end{array}$

where $b={\mathrm{Ï†}}_{{z}_{0}}\left(a\right)$. Taking $w={\mathrm{Ï†}}_{b}\left(z\right)$, we have

${âˆ¥{g}_{{z}_{0}}âˆ¥}_{{Q}_{p}}^{2}=\underset{bâˆˆ\mathbb{D}}{sup}{\left(1âˆ’{|b|}^{2}\right)}^{2}{âˆ«}_{\mathbb{D}}\frac{{\left(1âˆ’{|w|}^{2}\right)}^{p}}{{|1âˆ’\stackrel{Â¯}{b}w|}^{4}}\phantom{\rule{0.2em}{0ex}}d\mathrm{Ïƒ}\left(w\right).$

Since

$\frac{1}{{\left(1âˆ’\stackrel{Â¯}{b}w\right)}^{2}}=\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{\mathrm{Î“}\left(n+2\right)}{n!\mathrm{Î“}\left(2\right)}{\stackrel{Â¯}{b}}^{n}{w}^{n}=\underset{n=0}{\overset{\mathrm{âˆž}}{âˆ‘}}\frac{\mathrm{Î“}\left(n+2\right)}{n!}{\stackrel{Â¯}{b}}^{n}{w}^{n},$

we have

$\begin{array}{rl}{âˆ«}_{\mathbb{D}}\frac{{\left(1âˆ’{|w|}^{2}\right)}^{p}}{{|1âˆ’\stackrel{Â¯}{b}w|}^{4}}\phantom{\rule{0.2em}{0ex}}d\mathrm{Ïƒ}\left(w\right)& =\underset{n=0}{\overset{+\mathrm{âˆž}}{âˆ‘}}\frac{\mathrm{Î“}{\left(n+2\right)}^{2}}{{\left(n!\right)}^{2}}{|b|}^{2n}{âˆ«}_{\mathbb{D}}{\left(1âˆ’{|w|}^{2}\right)}^{p}{|w|}^{2n}\phantom{\rule{0.2em}{0ex}}d\mathrm{Ïƒ}\left(w\right)\\ =\underset{n=0}{\overset{+\mathrm{âˆž}}{âˆ‘}}\frac{\mathrm{Î“}{\left(n+2\right)}^{2}}{{\left(n!\right)}^{2}}{|b|}^{2n}{âˆ«}_{0}^{1}{\left(1âˆ’r\right)}^{p}{r}^{n}\phantom{\rule{0.2em}{0ex}}dr\\ =\underset{n=0}{\overset{+\mathrm{âˆž}}{âˆ‘}}\frac{\mathrm{Î“}{\left(n+2\right)}^{2}}{{\left(n!\right)}^{2}}\frac{\mathrm{Î“}\left(p+1\right)\mathrm{Î“}\left(n+1\right)}{\mathrm{Î“}\left(n+p+2\right)}{|b|}^{2n}\\ =\underset{n=0}{\overset{+\mathrm{âˆž}}{âˆ‘}}\frac{\mathrm{Î“}\left(p+1\right)\mathrm{Î“}{\left(n+2\right)}^{2}}{n!\mathrm{Î“}\left(n+p+2\right)}{|b|}^{2n}.\end{array}$

A simple computation shows

$\frac{\mathrm{Î“}\left(p+1\right)\mathrm{Î“}{\left(n+2\right)}^{2}}{n!\mathrm{Î“}\left(n+p+2\right)}=\frac{\left(n+1\right)!\left(n+1\right)}{\left(p+1\right)\left(p+2\right)â‹¯\left(p+n+1\right)}.$

Also, it is easy to see

$\frac{1}{p+1}â©½\frac{n+1}{p+n+1}â©½\frac{\left(n+1\right)!\left(n+1\right)}{\left(p+1\right)\left(p+2\right)â‹¯\left(p+n+1\right)}â©½\frac{n+1}{p+1}.$

Thus,

${âˆ¥{g}_{{z}_{0}}âˆ¥}_{{Q}_{p}}^{2}â©½\underset{bâˆˆ\mathbb{D}}{sup}\frac{{\left(1âˆ’{|b|}^{2}\right)}^{2}}{p+1}\underset{n=0}{\overset{+\mathrm{âˆž}}{âˆ‘}}\left(n+1\right){|b|}^{2n}=\underset{bâˆˆ\mathbb{D}}{sup}\frac{{\left(1âˆ’{|b|}^{2}\right)}^{2}}{p+1}\frac{1}{{\left(1âˆ’{|b|}^{2}\right)}^{2}}=\frac{1}{p+1},$

and

${âˆ¥{g}_{{z}_{0}}âˆ¥}_{{Q}_{p}}^{2}â©¾\underset{bâˆˆ\mathbb{D}}{sup}\frac{{\left(1âˆ’{|b|}^{2}\right)}^{2}}{p+1}\underset{n=0}{\overset{+\mathrm{âˆž}}{âˆ‘}}{|b|}^{2n}=\underset{bâˆˆ\mathbb{D}}{sup}\frac{{\left(1âˆ’{|b|}^{2}\right)}^{2}}{p+1}\frac{1}{1âˆ’{|b|}^{2}}=\frac{1}{p+1}.$

Then the proof is complete.â€ƒâ–¡

First, we consider the norm of ${I}_{f}$ on ${Q}_{p}$, $0.

Theorem 2.2 Let $0. If $fâˆˆH\left(\mathbb{D}\right)$, then ${I}_{f}$ is bounded on ${Q}_{p}$ if and only if $fâˆˆ{H}^{\mathrm{âˆž}}$. Moreover,

$âˆ¥{I}_{f}âˆ¥={âˆ¥fâˆ¥}_{{H}^{\mathrm{âˆž}}}.$

Proof For any $hâˆˆ{Q}_{p}$ with ${âˆ¥hâˆ¥}_{{Q}_{p}}=1$, it is trivial that $âˆ¥{I}_{f}âˆ¥â©½{âˆ¥fâˆ¥}_{{H}^{\mathrm{âˆž}}}$. To prove the converse, define $c={sup}_{zâˆˆ\mathbb{D}}|f\left(z\right)|$. Given any $\mathrm{Ïµ}>0$, there exists ${z}_{1}âˆˆ\mathbb{D}$ such that $|f\left({z}_{1}\right)|>câˆ’\mathrm{Ïµ}$. Let $h\left(z\right)={g}_{{z}_{1}}\left(z\right)/{âˆ¥{g}_{{z}_{1}}âˆ¥}_{{Q}_{p}}$, where

${g}_{{z}_{1}}\left(z\right)=\frac{{z}_{1}âˆ’z}{1âˆ’{\stackrel{Â¯}{z}}_{1}z}âˆ’{z}_{1}.$

It is easy to see that

${âˆ¥hâˆ¥}_{{Q}_{p}}=1,\phantom{\rule{2em}{0ex}}|{h}^{â€²}\left({z}_{1}\right)|\left(1âˆ’{|{z}_{1}|}^{2}\right)=1/{âˆ¥{g}_{{z}_{1}}âˆ¥}_{{Q}_{p}}.$

Henceforth,

$\begin{array}{rl}{âˆ¥{I}_{f}âˆ¥}^{2}& â©¾{âˆ¥{I}_{f}hâˆ¥}_{{Q}_{p}}^{2}=\underset{aâˆˆ\mathbb{D}}{sup}{âˆ«}_{\mathbb{D}}{|{h}^{â€²}\left(z\right)f\left(z\right)|}^{2}{\left(1âˆ’{|{\mathrm{Ï†}}_{a}\left(z\right)|}^{2}\right)}^{p}\phantom{\rule{0.2em}{0ex}}d\mathrm{Ïƒ}\left(z\right)\\ =\underset{aâˆˆ\mathbb{D}}{sup}{âˆ«}_{\mathbb{D}}{|{h}^{â€²}\left({\mathrm{Ï†}}_{a}\left(w\right)\right)f\left({\mathrm{Ï†}}_{a}\left(w\right)\right){\mathrm{Ï†}}_{a}^{â€²}\left(w\right)|}^{2}{\left(1âˆ’{|w|}^{2}\right)}^{p}\phantom{\rule{0.2em}{0ex}}d\mathrm{Ïƒ}\left(w\right).\end{array}$

Taking $w={\mathit{re}}^{i\mathrm{Î¸}}$ and by the subharmonicity of ${|{h}^{â€²}\left({\mathrm{Ï†}}_{a}\left(w\right)\right)f\left({\mathrm{Ï†}}_{a}\left(w\right)\right){\mathrm{Ï†}}_{a}^{â€²}\left(w\right)|}^{2}$, we obtain

$\begin{array}{rl}{âˆ¥{I}_{f}âˆ¥}^{2}& â©¾\underset{aâˆˆ\mathbb{D}}{sup}{âˆ«}_{\mathbb{D}}{|{h}^{â€²}\left(z\right)f\left(z\right)|}^{2}{\left(1âˆ’{|{\mathrm{Ï†}}_{a}\left(z\right)|}^{2}\right)}^{p}\phantom{\rule{0.2em}{0ex}}d\mathrm{Ïƒ}\left(z\right)\\ =\underset{aâˆˆ\mathbb{D}}{sup}{âˆ«}_{0}^{1}\frac{1}{\mathrm{Ï€}}{âˆ«}_{0}^{2\mathrm{Ï€}}{|{h}^{â€²}\left({\mathrm{Ï†}}_{a}\left({\mathit{re}}^{i\mathrm{Î¸}}\right)\right)f\left({\mathrm{Ï†}}_{a}\left({\mathit{re}}^{i\mathrm{Î¸}}\right)\right){\mathrm{Ï†}}_{a}^{â€²}\left({\mathit{re}}^{i\mathrm{Î¸}}\right)|}^{2}{\left(1âˆ’{r}^{2}\right)}^{p}r\phantom{\rule{0.2em}{0ex}}dr\phantom{\rule{0.2em}{0ex}}d\mathrm{Î¸}\\ â©¾\underset{aâˆˆ\mathbb{D}}{sup}{|{h}^{â€²}\left(a\right)f\left(a\right)|}^{2}{\left(1âˆ’{|a|}^{2}\right)}^{2}2{âˆ«}_{0}^{1}{\left(1âˆ’{r}^{2}\right)}^{p}r\phantom{\rule{0.2em}{0ex}}dr\\ =\frac{1}{p+1}\underset{aâˆˆ\mathbb{D}}{sup}{|{h}^{â€²}\left(a\right)f\left(a\right)|}^{2}{\left(1âˆ’{|a|}^{2}\right)}^{2}â©¾\frac{1}{p+1}{|{h}^{â€²}\left({z}_{1}\right)f\left({z}_{1}\right)|}^{2}{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{2}\\ â©¾\frac{1}{p+1}\frac{{|f\left({z}_{1}\right)|}^{2}}{{âˆ¥{g}_{{z}_{1}}âˆ¥}_{{Q}_{p}}^{2}}.\end{array}$
(2.2)

By Lemma 2.1 we have

$âˆ¥{I}_{f}âˆ¥â©¾|f\left({z}_{1}\right)|>câˆ’\mathrm{Ïµ}.$

Since Ïµ is arbitrary, we have $âˆ¥{I}_{f}âˆ¥â©¾{sup}_{zâˆˆ\mathbb{D}}|f\left(z\right)|$ and the proof is complete.â€ƒâ–¡

Next, we consider the norm of ${I}_{f}$ from ${Q}_{p}$ ($0) to .

Theorem 2.3 Let $0. If $fâˆˆH\left(\mathbb{D}\right)$, then ${I}_{f}$ is bounded from ${Q}_{p}$ space to space if and only if $fâˆˆ{H}^{\mathrm{âˆž}}$. Moreover, we have

$âˆ¥{I}_{f}âˆ¥={\left(p+1\right)}^{1/2}{âˆ¥fâˆ¥}_{{H}^{\mathrm{âˆž}}}.$

Proof If $fâˆˆ{H}^{\mathrm{âˆž}}$, then (1.2) gives

${âˆ¥{I}_{f}hâˆ¥}_{\mathfrak{B}}=\underset{zâˆˆ\mathbb{D}}{sup}|f\left(z\right){h}^{â€²}\left(z\right)|\left(1âˆ’{|z|}^{2}\right)â©½{âˆ¥fâˆ¥}_{{H}^{\mathrm{âˆž}}}\underset{zâˆˆ\mathbb{D}}{sup}|{h}^{â€²}\left(z\right)|\left(1âˆ’{|z|}^{2}\right).$

From a part of the proof of estimate (2.2) for $fâ‰¡1$, we see that

$\underset{zâˆˆ\mathbb{D}}{sup}|{h}^{â€²}\left(z\right)|\left(1âˆ’{|z|}^{2}\right)â©½{\left(p+1\right)}^{1/2}{âˆ¥hâˆ¥}_{{Q}_{p}},$

and so

${âˆ¥{I}_{f}hâˆ¥}_{\mathfrak{B}}â©½{âˆ¥fâˆ¥}_{{H}^{\mathrm{âˆž}}}{\left(p+1\right)}^{1/2}{âˆ¥hâˆ¥}_{{Q}_{p}}.$

$âˆ¥{I}_{f}âˆ¥â©½{\left(p+1\right)}^{1/2}{âˆ¥fâˆ¥}_{{H}^{\mathrm{âˆž}}}.$

On the other hand, define $c={sup}_{zâˆˆ\mathbb{D}}|f\left(z\right)|$. Given any $\mathrm{Ïµ}>0$, there exists ${z}_{1}âˆˆ\mathbb{D}$ such that $|f\left({z}_{1}\right)|>câˆ’\mathrm{Ïµ}$. Let $h\left(z\right)={g}_{{z}_{1}}\left(z\right)/{âˆ¥{g}_{{z}_{1}}âˆ¥}_{{Q}_{p}}$, where

${g}_{{z}_{1}}\left(z\right)=\frac{{z}_{1}âˆ’z}{1âˆ’{\stackrel{Â¯}{z}}_{1}z}âˆ’{z}_{1}.$

This together with Lemma 2.1 gives the following:

$\begin{array}{rl}âˆ¥{I}_{f}âˆ¥& â©¾{âˆ¥{I}_{f}hâˆ¥}_{\mathfrak{B}}=\underset{zâˆˆ\mathbb{D}}{sup}|f\left(z\right){h}^{â€²}\left(z\right)|\left(1âˆ’{|z|}^{2}\right)â©¾|f\left({z}_{1}\right){h}^{â€²}\left({z}_{1}\right)|\left(1âˆ’{|{z}_{1}|}^{2}\right)\\ =|f\left({z}_{1}\right)|/{âˆ¥{g}_{{z}_{1}}âˆ¥}_{{Q}_{p}}>{\left(p+1\right)}^{1/2}\left(câˆ’\mathrm{Ïµ}\right).\end{array}$

Since Ïµ is arbitrary, we have

$âˆ¥{I}_{f}âˆ¥â©¾{\left(p+1\right)}^{1/2}\underset{zâˆˆ\mathbb{D}}{sup}|f\left(z\right)|={\left(p+1\right)}^{1/2}{âˆ¥fâˆ¥}_{{H}^{\mathrm{âˆž}}}.$

The proof is complete.â€ƒâ–¡

Finally, we consider the norm of the integral operator ${I}_{f}$ on ${\mathfrak{B}}^{\mathrm{Î±}}$, $0<\mathrm{Î±}<1$.

Theorem 2.4 Let $0<\mathrm{Î±}<1$ and $fâˆˆH\left(\mathbb{D}\right)$. Then the integral operator ${I}_{f}$ is bounded on ${\mathfrak{B}}^{\mathrm{Î±}}$ if and only if $fâˆˆ{H}^{\mathrm{âˆž}}$. Moreover,

$âˆ¥{I}_{f}âˆ¥={âˆ¥fâˆ¥}_{{H}^{\mathrm{âˆž}}}.$

Proof For any $hâˆˆ{\mathfrak{B}}^{\mathrm{Î±}}$ with ${âˆ¥hâˆ¥}_{{\mathfrak{B}}^{\mathrm{Î±}}}=1$, by (1.3) we have

${âˆ¥{I}_{f}hâˆ¥}_{{\mathfrak{B}}^{\mathrm{Î±}}}=\underset{zâˆˆ\mathbb{D}}{sup}{\left(1âˆ’{|z|}^{2}\right)}^{\mathrm{Î±}}|f\left(z\right)||{h}^{â€²}\left(z\right)|â©½{âˆ¥hâˆ¥}_{{\mathfrak{B}}^{\mathrm{Î±}}}â‹\dots {âˆ¥fâˆ¥}_{{H}^{\mathrm{âˆž}}}.$

This implies $âˆ¥{I}_{f}âˆ¥â©½{âˆ¥fâˆ¥}_{{H}^{\mathrm{âˆž}}}$.

Now we need to show the reverse inequality. Define $c={sup}_{zâˆˆ\mathbb{D}}|f\left(z\right)|$. Given any $\mathrm{Ïµ}>0$, there exists ${z}_{1}âˆˆ\mathbb{D}$ such that $|f\left({z}_{1}\right)|>câˆ’\mathrm{Ïµ}$. Put

$h\left(z\right)={âˆ«}_{\mathrm{Î“}\left(z\right)}\frac{{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{\mathrm{Î±}}}{{\left(1âˆ’{\stackrel{Â¯}{z}}_{1}\mathrm{Î¶}\right)}^{2\mathrm{Î±}}}\phantom{\rule{0.2em}{0ex}}d\mathrm{Î¶},$
(2.3)

where $\mathrm{Î“}\left(z\right)$ is any path in from 0 to z, and a single-valued analytic branch is specified. By Theorem 13.11 in [[36], p.274], we know h is an analytic function in and ${h}^{â€²}\left(z\right)={\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{\mathrm{Î±}}/{\left(1âˆ’{\stackrel{Â¯}{z}}_{1}z\right)}^{2\mathrm{Î±}}$. Also, it is easy to check ${âˆ¥hâˆ¥}_{{\mathfrak{B}}^{\mathrm{Î±}}}=1$. In fact,

$\begin{array}{rl}{âˆ¥hâˆ¥}_{{\mathfrak{B}}^{\mathrm{Î±}}}& =\underset{zâˆˆ\mathbb{D}}{sup}|{h}^{â€²}\left(z\right)|{\left(1âˆ’{|z|}^{2}\right)}^{\mathrm{Î±}}=\underset{zâˆˆ\mathbb{D}}{sup}\frac{{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{\mathrm{Î±}}}{{|1âˆ’{\stackrel{Â¯}{z}}_{1}z|}^{2\mathrm{Î±}}}{\left(1âˆ’{|z|}^{2}\right)}^{\mathrm{Î±}}\\ â©½\underset{zâˆˆ\mathbb{D}}{sup}\frac{{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{\mathrm{Î±}}{\left(1âˆ’{|z|}^{2}\right)}^{\mathrm{Î±}}}{{\left(1âˆ’|{z}_{1}||z|\right)}^{2\mathrm{Î±}}}â©½1.\end{array}$
(2.4)

On the other hand, we have

$\begin{array}{rl}{âˆ¥hâˆ¥}_{{\mathfrak{B}}^{\mathrm{Î±}}}& =\underset{zâˆˆ\mathbb{D}}{sup}|{h}^{â€²}\left(z\right)|{\left(1âˆ’{|z|}^{2}\right)}^{\mathrm{Î±}}=\underset{zâˆˆ\mathbb{D}}{sup}\frac{{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{\mathrm{Î±}}}{{|1âˆ’{\stackrel{Â¯}{z}}_{1}z|}^{2\mathrm{Î±}}}{\left(1âˆ’{|z|}^{2}\right)}^{\mathrm{Î±}}\\ â©¾\frac{{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{\mathrm{Î±}}}{{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{2\mathrm{Î±}}}{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{\mathrm{Î±}}=1.\end{array}$
(2.5)

Hence, the assertion follows by (2.4) and (2.5). Thus

$\begin{array}{rl}âˆ¥{I}_{f}âˆ¥& â©¾{âˆ¥{I}_{f}hâˆ¥}_{{\mathfrak{B}}^{\mathrm{Î±}}}=\underset{zâˆˆ\mathbb{D}}{sup}|f\left(z\right){h}^{â€²}\left(z\right)|{\left(1âˆ’{|z|}^{2}\right)}^{\mathrm{Î±}}â©¾|f\left({z}_{1}\right){h}^{â€²}\left({z}_{1}\right)|{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{\mathrm{Î±}}\\ â©¾|f\left({z}_{1}\right)|\frac{{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{\mathrm{Î±}}}{{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{2\mathrm{Î±}}}{\left(1âˆ’{|{z}_{1}|}^{2}\right)}^{\mathrm{Î±}}=|f\left({z}_{1}\right)|>câˆ’\mathrm{Ïµ}.\end{array}$

Since the Ïµ is arbitrary, the proof is complete.â€ƒâ–¡

## References

1. Aulaskari R, Xiao J, Zhao R: On subspaces and subsets of BMOA and UBC. Analysis 1995, 15: 101â€“121.

2. Xiao J Lecture Notes in Math. 1767. In Holomorphic $\mathcal{Q}$ Classes. Springer, Berlin; 2001.

3. Zhu K: Operator Theory in Function Spaces. Dekker, New York; 1990.

4. Zhu K: Bloch type spaces of analytic functions. Rocky Mt. J. Math. 1993, 23: 1143â€“1177. 10.1216/rmjm/1181072549

5. Aleman A, Siskakis A:An integral operator on ${H}^{p}$. Complex Var. Theory Appl. 1995, 28: 140â€“158.

6. Aleman A, Siskakis A: Integral operators on Bergman spaces. Indiana Univ. Math. J. 1997, 46: 337â€“356.

7. Austin, A: Multiplication and integral operators on Banach spaces of analytic functions. Ph.D. thesis, University of Hawai (2010)

8. Krantz S, SteviÄ‡ S: On the iterated logarithmic Bloch space on the unit ball. Nonlinear Anal. TMA 2009, 71: 1772â€“1795. 10.1016/j.na.2009.01.013

9. Li S, SteviÄ‡ S: Volterra-type operators on Zygmund spaces. J. Inequal. Appl. 2007., 2007: Article ID 32124

10. Li S, SteviÄ‡ S: Integral type operators from mixed-norm spaces to Î± -Bloch spaces. Integral Transforms Spec. Funct. 2007, 18(7):485â€“493. 10.1080/10652460701320703

11. Li S, SteviÄ‡ S: Riemann-Stieltjes operators between different weighted Bergman spaces. Bull. Belg. Math. Soc. Simon Stevin 2008, 15(4):677â€“686.

12. Li S, SteviÄ‡ S:Products of composition and integral type operators from ${H}^{\mathrm{âˆž}}$ to the Bloch space. Complex Var. Elliptic Equ. 2008, 53(5):463â€“474. 10.1080/17476930701754118

13. Li S, SteviÄ‡ S: Generalized composition operators on Zygmund spaces and Bloch type spaces. J. Math. Anal. Appl. 2008, 338: 1282â€“1295. 10.1016/j.jmaa.2007.06.013

14. Li S, SteviÄ‡ S: Products of integral-type operators and composition operators between Bloch-type spaces. J. Math. Anal. Appl. 2009, 349: 596â€“610. 10.1016/j.jmaa.2008.09.014

15. Li S:On an integral-type operator from the Bloch space into the ${Q}_{K}\left(p,q\right)$ space. Filomat 2012, 26: 125â€“133.

16. Pan C: On an integral-type operator from ${Q}_{K}\left(p,q\right)$spaces to Î± -Bloch space. Filomat 2011, 25: 163â€“173.

17. Pommerenke C: Schlichte funktionen und analytische funktionen vonÂ beschrÃ¤nkter mittlerer oszillation. Comment. Math. Helv. 1977, 52: 591â€“602. 10.1007/BF02567392

18. SteviÄ‡ S: On a new operator from the logarithmic Bloch space to the Bloch-type space on the unit ball. Appl. Math. Comput. 2008, 206: 313â€“320. 10.1016/j.amc.2008.09.002

19. SteviÄ‡ S: On a new integral-type operator from the weighted Bergman space to the Bloch-type space on the unit ball. Discrete Dyn. Nat. Soc. 2008., 2008: Article ID 154263

20. SteviÄ‡ S: On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball. J. Math. Anal. Appl. 2009, 354: 426â€“434. 10.1016/j.jmaa.2008.12.059

21. Zhu X:An integral-type operator from ${H}^{\mathrm{âˆž}}$ to Zygmund-type spaces. Bull. Malays. Math. Soc. 2012, 35: 679â€“686.

22. Bourdon P, Fry E, Hammond C, Spofford C: Norms of linear-fractional composition operators. Trans. Am. Math. Soc. 2003, 356: 2459â€“2480.

23. Colonna F, Easley G, Singman D:Norm of the multiplication operators from ${H}^{\mathrm{âˆž}}$ to the Bloch space of a bounded symmetric domain. J. Math. Anal. Appl. 2011, 382: 621â€“630. 10.1016/j.jmaa.2011.04.064

24. Hammond C: The norm of a composition operator with linear symbol acting on the Dirichlet space. J. Math. Anal. Appl. 2005, 303: 499â€“508. 10.1016/j.jmaa.2004.08.049

25. Liu J, Lou Z, Xiong C: Essential norms of integral operators on spaces of analytic functions. Nonlinear Anal. 2012, 75: 5145â€“5156. 10.1016/j.na.2012.04.030

26. MartÃ­n M: Norm-attaining composition operators on the Bloch spaces. J. Math. Anal. Appl. 2010, 369: 15â€“21. 10.1016/j.jmaa.2010.02.028

27. SteviÄ‡ S:Norm of weighted composition operators from Bloch space to ${H}_{\mathrm{Î¼}}^{\mathrm{âˆž}}$ on the unit ball. Ars Comb. 2008, 88: 125â€“127.

28. SteviÄ‡ S: Norms of some operators from Bergman spaces to weighted and Bloch-type spaces. Util. Math. 2008, 76: 59â€“64.

29. SteviÄ‡ S: Norm of weighted composition operators from Î± -Bloch spaces to weighted-type spaces. Appl. Math. Comput. 2009, 215: 818â€“820. 10.1016/j.amc.2009.06.005

30. SteviÄ‡ S: Norm and essential norm of composition followed by differentiation from Î± -Bloch spaces to ${H}_{\mathrm{Î¼}}^{\mathrm{âˆž}}$. Appl. Math. Comput. 2009, 207: 225â€“229. 10.1016/j.amc.2008.10.032

31. SteviÄ‡ S: Norms of some operators on bounded symmetric domains. Appl. Math. Comput. 2010, 215: 187â€“191.

32. SteviÄ‡ S: Norm of an integral-type operator from Dirichlet to Bloch space on the unit disk. Util. Math. 2010, 83: 301â€“303.

33. SteviÄ‡ S: On an integral operator between Bloch-type spaces on the unit ball. Bull. Sci. Math. 2010, 134: 329â€“339. 10.1016/j.bulsci.2008.10.005

34. Yang W: On an integral-type operator between Bloch-type spaces. Appl. Math. Comput. 2009, 215: 954â€“960. 10.1016/j.amc.2009.06.016

35. Liu J, Xiong C: Norm-attaining integral operators on analytic function spaces. J. Math. Anal. Appl. 2013, 399: 108â€“115. 10.1016/j.jmaa.2012.09.044

36. Rudin W: Real and Complex Analysis. 3rd edition. McGraw-Hill, New York; 1987.

## Acknowledgements

The first author is supported by the National Natural Science Foundation of China (No. 11126284). The second author is supported by the project of Department of Education of Guangdong Province (No. 2012KJCX0096).

## Author information

Authors

### Corresponding author

Correspondence to Songxiao Li.

### Competing interests

The authors declare that they have no competing interests.

### Authorsâ€™ contributions

All the authors contributed to the writing of the present article. They also read and approved the final manuscript.

## Rights and permissions

Reprints and permissions

Li, H., Li, S. Norm of an integral operator on some analytic function spaces on the unit disk. J Inequal Appl 2013, 342 (2013). https://doi.org/10.1186/1029-242X-2013-342