Table of Contents
ISRN Mathematical Analysis
Volume 2011, Article ID 149830, 11 pages
http://dx.doi.org/10.5402/2011/149830
Research Article

General Properties for Volterra-Type Operators in the Unit Disk

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Selangor Darul Ehsan 43600, Malaysia

Received 26 October 2010; Accepted 5 December 2010

Academic Editors: A. L. Sasu, B. Kaltenbacher, and J. Colliander

Copyright © 2011 Rabha W. Ibrahim and Maslina Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The object of this paper is to study general properties such as boundedness, compactness, and geometric properties for two integral operators of Volterra-Type in the unit disk.

1. Introduction

Let be the class of analytic functions in 𝑈={𝑧|𝑧|<1}. Suppose that 𝑔𝑈 is a holomorphic map, 𝑓. The integral operator, called Volterra-type operator,𝐽𝑔𝑓(𝑧)=𝑧0𝑓𝑑𝑔=10𝑓(𝑡𝑧)𝑧𝑔(𝑡𝑧)𝑑𝑡=𝑧0𝑓(𝜉)𝑔(𝜉)𝑑𝜉,𝑧𝑈,(1.1) was introduced by Pommerenke in [1]. Another natural integral operator is defined as follows:𝐼𝑔𝑓(𝑧)=𝑧0𝑓(𝜉)𝑔(𝜉)𝑑𝜉,𝑧𝑈.(1.2) The importance of the operators 𝐽𝑔 and 𝐼𝑔 comes from the fact that𝐽𝑔𝑓+𝐼𝑔𝑓=𝑀𝑔𝑓𝑓(0)𝑔(0),(1.3) where the multiplication operator 𝑀𝑔 is defined by𝑀𝑔𝑓(𝑧)=𝑔(𝑧)𝑓(𝑧),𝑓,𝑧𝑈.(1.4) Furthermore, Volterra integral equations arise in many physical applications (see [24]).

In the past few years, many authors focused on the boundedness and compactness of Volterra-type integral operator between several spaces of holomorphic functions. In [1], Pommerenke showed that 𝐽𝑔 is a bounded operator on the Hardy space 𝐻2. The boundedness and compactness of 𝐽𝑔𝑓 and 𝐼𝑔𝑓 between some spaces of analytic functions, as well as their 𝑛-dimensional extensions, were investigated in [511].

For functions 𝑓, the integral operators 𝐽𝑔𝑓 and 𝐼𝑔𝑓 contain well-known integral operators in the analytic function theory and geometric function theory such as the generalized Bernardi-Libera-Livingston linear integral operator (cf. [1214]) and the Srivastava-Owa fractional derivative operators (cf. [15, 16]). Recently, Breaz and Breaz introduced two integral operators of analytic functions taking the form (1.1) and (1.2) (see [17]). Further, the integral operators of Volterra-Type involving the integral operators were studied in [1822]. Finally, these operators are involving the Cesáro integral operator (see [2325]).

A function 𝑓 is called in the class Σ if and only if it has the norm (see [26])𝑓=sup𝑧𝑈1|𝑧|2||||𝑓(𝑧)𝑓||||(𝑧)<,(𝑧𝑈).(1.5) Note that the fraction 𝑇𝑓=𝑓(𝑧)/𝑓(𝑧) is called pre-Schwarzian derivative which is usually used to discuss the univalency of analytic functions (see [27]). Moreover, the norm in (1.5) is a modification to one defined in [28].

The purpose of this paper is to study the boundedness, compactness, and some geometric properties of the integral operators 𝐽𝑔𝑓 and 𝐼𝑔𝑓 for the functions 𝑓Σ and 𝑔 is an analytic function on the open unit disk.

2. The Boundedness and Compactness

In this section, we consider the boundedness and compactness of the operators 𝐽𝑔𝑓 and 𝐼𝑔𝑓 on the classes Σ.

Consider the space log of all functions 𝑓 which are satisfying𝑓log=sup𝑧𝑈1|𝑧|2||||𝑓(𝑧)||||1𝑓(𝑧)ln1|𝑧|2<,(𝑧𝑈).(2.1)

Theorem 2.1. Assume that 𝑔 is an analytic function on 𝑈. Then, for functions 𝑓log, 𝐽𝑔 is bounded if and only if 𝑔Σ.

Proof. Assume that 𝐽𝑔 is bounded. Taking the function given by 𝑓(𝑧)=1, we see that 𝑔Σ.
Conversely, assume that 𝑔Σ, we have1|𝑧|2|||||𝐽𝑔𝑓(𝑧)𝐽𝑔𝑓|||||=(𝑧)1|𝑧|2||||𝑓(𝑧)𝑔(𝑧)+𝑓(𝑧)𝑔(𝑧)𝑓(𝑧)𝑔||||=(𝑧)1|𝑧|2||||𝑔(𝑧)𝑔+𝑓(𝑧)(𝑧)||||𝑓(𝑧)𝑔+1|𝑧|2||𝑓||(𝑧)/𝑓(𝑧)ln1/1|𝑧|2ln1/1|𝑧|2𝑔+𝑓logln1/1|𝑧|2𝑔+𝑓logln1/1|𝑧|21|𝑧|2.(2.2) By taking the supremum for the last assertion over 𝑈 and using the fact that the quantity sup]𝑥(0,1𝑥1ln𝑥(2.3) is finite, the boundedness of the operator 𝐽𝑔 follows.

Theorem 2.2. Assume that 𝑔 is an analytic function on 𝑈. Then, 𝐼𝑔ΣΣ is bounded if and only if 𝑔log, where 𝑔log=sup𝑧𝑈1|𝑧|2||||𝑔(𝑧)||||1𝑔(𝑧)ln1|𝑧|2.(2.4)

Proof. Assume that 𝑔log. Then, we obtain 1|𝑧|2|||||𝐼𝑔𝑓(𝑧)𝐼𝑔𝑓|||||=(𝑧)1|𝑧|2||||𝑓(𝑧)𝑔(𝑧)+𝑓(𝑧)𝑔(𝑧)𝑓||||=(𝑧)𝑔(𝑧)1|𝑧|2||||𝑓(𝑧)𝑓+𝑔(𝑧)(𝑧)||||𝑔(𝑧)𝑓+1|𝑧|2||𝑔||(𝑧)/𝑔(𝑧)ln1/1|𝑧|2ln1/1|𝑧|2𝑓+𝑔logln1/1|𝑧|2,(𝑧𝑈)𝑓+𝑔logln1/1|𝑧|21|𝑧|2.(2.5) By taking the supremum for the last assertion over 𝑈, the boundedness of the operator 𝐼𝑔 follows.
Conversely, assume that 𝐼𝑔ΣΣ is bounded, then there is a positive constant 𝐶 such that𝐼𝑔𝑓𝐶𝑓(2.6) for every 𝑓Σ. Set 𝑎(𝑧)=𝑎𝑧1𝑎11+ln1𝑎𝑧21+1ln1|𝑎|21,(2.7) for 𝑎𝑈 such that 1(1/𝑒)<|𝑎|<1. Then, we have 𝑎1(𝑧)=ln1𝑎𝑧21ln1|𝑎|21,𝑎2(𝑧)=𝑎11𝑎𝑧ln11𝑎𝑧ln1|𝑎|21.(2.8) Thus, 𝑎(𝑧)𝑎=2(𝑧)𝑎11𝑎𝑧ln1𝑎𝑧1,(2.9) and then 𝑎(𝑎)𝑎=2(𝑎)𝑎1|𝑎|21ln1|𝑎|21.(2.10) It is clear that the relation (2.9) is finite when |𝑧|<1, hence 𝑎(𝑧)<. Setting 𝑀=sup1(1/𝑒)<|𝑎|<1𝑎(𝑧)<,(2.11) therefore, we have 𝐼>𝑔𝑎𝐼𝑔𝑎sup𝑧𝑈1|𝑧|2||||𝑎(𝑧)𝑎+𝑔(𝑧)(𝑧)||||𝑔(𝑧)1|𝑎|2||||𝑎(𝑎)𝑎+𝑔(𝑎)(𝑎)||||||||2𝑔(𝑎)𝑎ln1/1|𝑎|2+1|𝑎|2𝑔(𝑎)||||𝑔(𝑎)2|𝑎|+1|𝑎|2||𝑔||(𝑎)/𝑔(𝑎)ln1/1|𝑎|2ln1/1|𝑎|2.(2.12)
Now letting𝑓𝑎(𝑧)=2𝑎𝑧1𝑎11+ln1𝑎𝑧21+1ln1|𝑎|21𝑧01ln1𝑎𝑥𝑑𝑥(2.13) for 𝑎𝑈 such that 1(1/𝑒)<|𝑎|<1. Then, we obtain 𝑓𝑎1(𝑧)=2ln1𝑎𝑧21ln1|𝑎|211ln1,𝑓𝑎𝑧𝑎4(𝑧)=𝑎11𝑎𝑧ln11𝑎𝑧ln1|𝑎|21𝑎1.𝑎𝑧(2.14) Thus, we conclude that 𝑓𝑎(𝑎)𝑓𝑎=(𝑎)3|𝑎|/1|𝑎|2ln1/1|𝑎|2.(2.15) In the same manner of the previous case, we have 𝑁=sup1(1/𝑒)<|𝑎|<1𝑓𝑎<.(2.16) Consequently, we have 𝐼>𝑔𝑓𝑎𝐼𝑔𝑓𝑎sup𝑧𝑈1|𝑧|2||||𝑓𝑎(𝑧)𝑓𝑎+𝑔(𝑧)(𝑧)||||𝑔(𝑧)1|𝑎|2||||𝑓𝑎(𝑎)𝑓𝑎+𝑔(𝑎)(𝑎)||||𝑔(𝑎)1|𝑎|2||||3|𝑎|/1|𝑎|2ln1/1|𝑎|2+𝑔(𝑎)||||𝑔(𝑎)3|𝑎|+1|𝑎|2||𝑔||(𝑎)/𝑔(𝑎)ln1/1|𝑎|2ln1/1|𝑎|2.(2.17)
From (2.12) and (2.17), we have1|𝑎|2||||𝑔(𝑎)||||1𝑔(𝑎)ln1|𝑎|2<(2.18) for all 1(1/𝑒)<|𝑎|<1. Also, we have sup|𝑎|1(1/𝑒)1|𝑎|2||||𝑔(𝑎)||||1𝑔(𝑎)ln1|𝑎|2sup1(1/𝑒)|𝑎|<11|𝑎|2||||𝑔(𝑎)||||1𝑔(𝑎)ln1|𝑎|2.(2.19) From (2.18) and (2.19), we obtain 𝑔log, as desired.
In the following results, we study the compactness of the integral operators 𝐽𝑔 and 𝐼𝑔 in an open disc.

Theorem 2.3. Assume that 𝑔 is an analytic function on 𝑈. Then, for functions 𝑓log, the integral operator 𝐽𝑔 is compact if and only if 𝑔Σ.

Proof. If 𝐽𝑔 is compact, then it is bounded, and by Theorem 2.1 it follows that 𝑔Σ.
Now assume that 𝑔Σ, that (𝑓𝑛)𝑛 is a sequence in log, and 𝑓𝑛0 uniformly on 𝑈 as 𝑛. Now for every 𝜀>0, there is 𝛿(0,1) such that11|𝑧|2<𝜀,(2.20) where 𝛿<|𝑧|<1. Since 𝛿 is arbitrary, then we can chose ln(1/(1|𝑧|2))>1 for 𝛿<|𝑧|<1 and 𝐽𝑔𝑓𝑛=sup𝑧𝑈1|𝑧|2|||||𝐽𝑔𝑓𝑛(𝑧)𝐽𝑔𝑓𝑛|||||(𝑧)=sup𝑧𝑈1|𝑧|2||||𝑓𝑛(𝑧)𝑔(𝑧)+𝑓𝑛(𝑧)𝑔(𝑧)𝑓𝑛(𝑧)𝑔||||(𝑧)sup𝑧𝑈1|𝑧|2||||𝑔(𝑧)𝑔||||(𝑧)+sup𝑧𝑈1|𝑧|2||||𝑓𝑛(𝑧)𝑓𝑛||||1(𝑧)ln1|𝑧|2𝑔1|𝑧|2+𝑓𝑛log𝑓<𝜀𝑔+𝑛log.(2.21) Since for 𝑓𝑛0 on 𝑈 we have 𝑓𝑛log0, and that 𝜀 is an arbitrary positive number, by letting 𝑛 in the last inequality, we obtain that lim𝑛𝐽𝑔𝑓𝑛=0. Therefore, 𝐽𝑔 is compact.

Theorem 2.4. Assume that 𝑔 is an analytic function on 𝑈. Then, the integral operator 𝐼𝑔ΣΣ is compact if and only if 𝑔 is a constant defer from zero.

Proof. Assume that 𝑔 is a constant without loss of generality and assume that 𝑓(𝑧)=𝑧. Then, it is clear that 𝐼𝑔 is compact.
Conversely, assume that 𝐼𝑔ΣΣ is compact. Let (𝑧𝑛)𝑛, be a sequence in 𝑈 such that |𝑧𝑛|1 as 𝑛. Our aim is to show that 𝑔(𝑧𝑛)0 as 𝑛, then by the maximum modulus theorem, we have 𝑔 is a constant. In fact, setting𝑓𝑛(𝑧)=2𝑧𝑛𝑧1𝑧𝑛11+ln1𝑧𝑛𝑧21+1ln1|𝑧|214𝑧01ln1𝑧𝑛𝑤𝑑𝑤.(2.22) Then, we obtain 𝑓𝑛1(𝑧)=2ln1𝑧𝑛𝑧21ln1|𝑧|2114ln1𝑧𝑛𝑧,𝑓𝑛4(𝑧)=𝑧𝑛1𝑧𝑛𝑧1ln1𝑧𝑛𝑧1ln1|𝑧|214𝑧𝑛1𝑧𝑛𝑧.(2.23) Consequently, we have 𝑓𝑛𝑧𝑛𝑓𝑛𝑧𝑛=0.(2.24) Similar to the proof of Theorem 2.2, we see that 𝑓𝑛0 uniformly on 𝑈. Since 𝐼𝑔ΣΣ is compact, then we get 𝐼𝑔𝑓𝑛0,𝑛.(2.25) Thus, ||||𝑔𝑧𝑛𝑔𝑧𝑛||||sup𝑧𝑈||||𝑔(𝑧)||||𝑔(𝑧)+sup𝑧𝑈||||𝑓𝑛(𝑧)𝑓𝑛||||𝐼(𝑧)𝑔𝑓𝑛0(2.26) Implies that 𝑔𝑛(𝑧)0 and consequently 𝑔 is a constant as desired.

3. Some Geometric Properties

In this section, we introduce some geometric properties for analytic function 𝑓Σ. A function 𝑓 which normalized as 𝑓(0)=𝑓(0)1=0 denoted this class by 𝒜. Recall that a function 𝑓𝒜 is said to be star-like of order 𝜇[0,1) in 𝑈 if it satisfies𝑓𝒮𝜇𝑧𝑓(𝑧)𝑓(𝑧)>𝜇,(𝑧𝑈).(3.1) Also, a function 𝑓𝒜 is called convex in 𝑈 if it satisfies𝑓𝒦𝜇1+𝑧𝑓(𝑧)𝑓(𝑧)>𝜇,(𝑧𝑈).(3.2) It follows that𝑓𝒦𝜇𝑧𝑓𝒮𝜇.(3.3)

In the next result, we discuss the convexity of the integral operators 𝐽𝑔 and 𝐼𝑔.

Theorem 3.1. Assume that 𝑓,𝑔𝒜. If 𝑓𝒮𝜇 and 𝑔𝒦𝜈 such that 0𝜇+𝜈<1, then the function 𝐽𝑔𝑓 is convex of order 𝜇+𝜈.

Proof. Assume that 𝑓,𝑔𝒜. Then, we obtain 𝑧𝐽𝑔𝑓(𝑧)𝐽𝑔𝑓=(𝑧)𝑧𝑔(𝑧)𝑔+(𝑧)𝑧𝑓(𝑧)𝑓(𝑧).(3.4) Consequently, we get 𝑧𝐽1+𝑔𝑓(𝑧)𝐽𝑔𝑓(𝑧)=𝑧𝑔(𝑧)𝑔(𝑧)+1+𝑧𝑓(𝑧)𝑓(𝑧)>𝜇+𝜈.(3.5) Hence, 𝐽𝑔𝒦𝜇+𝜈.

Theorem 3.2. Assume that 𝑓,𝑔𝒜. If 𝑓𝒦𝜇 and 𝑔𝒮𝜈 such that 0𝜇+𝜈<1, then the function 𝐼𝑔𝑓 is convex of order 𝜇+𝜈.

Proof. Assume that 𝑓,𝑔𝒜. Then, we have 𝑧𝐼𝑔𝑓(𝑧)𝐼𝑔𝑓=(𝑧)𝑧𝑓(𝑧)𝑓+(𝑧)𝑧𝑔(𝑧)𝑔(𝑧).(3.6) Consequently, we get 𝑧𝐼1+𝑔𝑓(𝑧)𝐼𝑔𝑓(𝑧)=𝑧𝑓(𝑧)𝑓(𝑧)+1+𝑧𝑔(𝑧)𝑔(𝑧)>𝜇+𝜈.(3.7) Hence, 𝐼𝑔𝒦𝜇+𝜈.

Theorem 3.3. Assume that 𝑓,𝑔𝒜. If 𝑓𝒮𝜇 and 𝑔𝒮𝜈 such that 0𝜇+𝜈<1, then the multiplication operator 𝑀𝑔𝑓 is star-like of order 𝜇+𝜈.

Proof. Assume that 𝑓,𝑔𝒜. Then, we obtain 𝑧𝑀𝑔𝑓(𝑧)𝑀𝑔𝑓(𝑧)=𝑧𝑓(𝑧)𝑓(𝑧)+𝑧𝑔(𝑧)𝑔(𝑧)>𝜇+𝜈.(3.8) Hence, 𝑀𝑔𝑓𝒮𝜇+𝜈.
The next result comes directly from the definition of the class Σ and the fact that 𝑇𝑓< if and only if 𝑓 is uniformly locally univalent (see [23]).

Theorem 3.4. Assume that 𝑔 is an analytic function on 𝑈 and 𝑓𝒜. Then, the functions 𝐼𝑔𝑓 and 𝐽𝑔𝑓 are in the class Σ if and only if 𝑓 is locally univalent in 𝑈.

Acknowledgment

The work presented here was supported by the MOHE Grant no. UKM-ST-06-FRGS0107-2009, Malaysia.

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