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 โ„‹ be the class of analytic functions in ๐‘ˆโˆถ={๐‘งโˆˆโ„‚โˆถ|๐‘ง|<1}. Suppose that ๐‘”โˆถ๐‘ˆโ†’โ„‚ is a holomorphic map, ๐‘“โˆˆโ„‹. The integral operator, called Volterra-type operator,๐ฝ๐‘”๎€œ๐‘“(๐‘ง)=๐‘ง0๎€œ๐‘“๐‘‘๐‘”=10๐‘“(๐‘ก๐‘ง)๐‘ง๐‘”๎…ž๎€œ(๐‘ก๐‘ง)๐‘‘๐‘ก=๐‘ง0๐‘“(๐œ‰)๐‘”๎…ž(๐œ‰)๐‘‘๐œ‰,๐‘งโˆˆ๐‘ˆ,(1.1) was introduced by Pommerenke in [1]. Another natural integral operator is defined as follows:๐ผ๐‘”๎€œ๐‘“(๐‘ง)=๐‘ง0๐‘“๎…ž(๐œ‰)๐‘”(๐œ‰)๐‘‘๐œ‰,๐‘งโˆˆ๐‘ˆ.(1.2) The importance of the operators ๐ฝ๐‘” and ๐ผ๐‘” comes from the fact that๐ฝ๐‘”๐‘“+๐ผ๐‘”๐‘“=๐‘€๐‘”๐‘“โˆ’๐‘“(0)๐‘”(0),(1.3) where the multiplication operator ๐‘€๐‘” is defined by๎€ท๐‘€๐‘”๐‘“๎€ธ(๐‘ง)=๐‘”(๐‘ง)๐‘“(๐‘ง),๐‘“โˆˆโ„‹,๐‘งโˆˆ๐‘ˆ.(1.4) Furthermore, Volterra integral equations arise in many physical applications (see [2โ€“4]).

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 ๐ฝ๐‘” is a bounded operator on the Hardy space ๐ป2. The boundedness and compactness of ๐ฝ๐‘”๐‘“ and ๐ผ๐‘”๐‘“ between some spaces of analytic functions, as well as their ๐‘›-dimensional extensions, were investigated in [5โ€“11].

For functions ๐‘“โˆˆโ„‹, the integral operators ๐ฝ๐‘”๐‘“ and ๐ผ๐‘”๐‘“ 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. [12โ€“14]) 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 [18โ€“22]. Finally, these operators are involving the Cesรกro integral operator (see [23โ€“25]).

A function ๐‘“โˆˆโ„‹ is called in the class ฮฃ if and only if it has the norm (see [26])โ€–๐‘“โ€–=sup๐‘งโˆˆ๐‘ˆ๎€ท1โˆ’|๐‘ง|2๎€ธ||||๐‘“๎…ž๎…ž(๐‘ง)๐‘“๎…ž||||(๐‘ง)<โˆž,(๐‘งโˆˆ๐‘ˆ).(1.5) Note that the fraction ๐‘‡๐‘“โˆถ=๐‘“๎…ž๎…ž(๐‘ง)/๐‘“๎…ž(๐‘ง) 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 ๐ฝ๐‘”๐‘“ and ๐ผ๐‘”๐‘“ for the functions ๐‘“โˆˆฮฃ and ๐‘” 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 ๐ฝ๐‘”๐‘“ and ๐ผ๐‘”๐‘“ on the classes ฮฃ.

Consider the space โ„ฌlog of all functions ๐‘“โˆˆโ„‹ which are satisfyingโ€–๐‘“โ€–โ„ฌlog=sup๐‘งโˆˆ๐‘ˆ๎€ท1โˆ’|๐‘ง|2๎€ธ||||๐‘“๎…ž(๐‘ง)||||1๐‘“(๐‘ง)ln๎€ท1โˆ’|๐‘ง|2๎€ธ<โˆž,(๐‘งโˆˆ๐‘ˆ).(2.1)

Theorem 2.1. Assume that ๐‘” is an analytic function on ๐‘ˆ. Then, for functions ๐‘“โˆˆโ„ฌlog, ๐ฝ๐‘” is bounded if and only if ๐‘”โˆˆฮฃ.

Proof. Assume that ๐ฝ๐‘” is bounded. Taking the function given by ๐‘“(๐‘ง)=1, we see that ๐‘”โˆˆฮฃ.
Conversely, assume that ๐‘”โˆˆฮฃ, we have๎€ท1โˆ’|๐‘ง|2๎€ธ|||||๎€ท๐ฝ๐‘”๐‘“๎€ธ๎…ž๎…ž(๐‘ง)๎€ท๐ฝ๐‘”๐‘“๎€ธ๎…ž|||||=๎€ท(๐‘ง)1โˆ’|๐‘ง|2๎€ธ||||๐‘“(๐‘ง)๐‘”๎…ž๎…ž(๐‘ง)+๐‘“๎…ž(๐‘ง)๐‘”๎…ž(๐‘ง)๐‘“(๐‘ง)๐‘”๎…ž||||=๎€ท(๐‘ง)1โˆ’|๐‘ง|2๎€ธ๎ƒฌ||||๐‘”๎…ž๎…ž(๐‘ง)๐‘”๎…ž+๐‘“(๐‘ง)๎…ž(๐‘ง)||||๎ƒญโ‰ค๎€ท๐‘“(๐‘ง)โ€–๐‘”โ€–+1โˆ’|๐‘ง|2๎€ธ||๐‘“๎…ž||๎€บ๎€ท(๐‘ง)/๐‘“(๐‘ง)ln1/1โˆ’|๐‘ง|2๎€ธ๎€ป๎€บ๎€ทln1/1โˆ’|๐‘ง|2โ€–๎€ธ๎€ปโ‰คโ€–๐‘”โ€–+๐‘“โ€–โ„ฌlog๎€บ๎€ทln1/1โˆ’|๐‘ง|2๎€ธ๎€ปโ‰คโ€–๐‘”โ€–+โ€–๐‘“โ€–โ„ฌlog๎€บ๎€ทln1/1โˆ’|๐‘ง|2๎€ธ๎€ป๎€ท1โˆ’|๐‘ง|2๎€ธ.(2.2) By taking the supremum for the last assertion over ๐‘ˆ and using the fact that the quantity sup]๐‘ฅโˆˆ(0,1๐‘ฅ๎‚€1ln๐‘ฅ๎‚(2.3) is finite, the boundedness of the operator ๐ฝ๐‘” follows.

Theorem 2.2. Assume that ๐‘” is an analytic function on ๐‘ˆ. Then, ๐ผ๐‘”โˆถฮฃโ†’ฮฃ is bounded if and only if ๐‘”โˆˆโ„ฌlog, where โ€–๐‘”โ€–โ„ฌlog=sup๐‘งโˆˆ๐‘ˆ๎€ท1โˆ’|๐‘ง|2๎€ธ||||๐‘”๎…ž(๐‘ง)||||1๐‘”(๐‘ง)ln๎€ท1โˆ’|๐‘ง|2๎€ธ.(2.4)

Proof. Assume that ๐‘”โˆˆโ„ฌlog. Then, we obtain ๎€ท1โˆ’|๐‘ง|2๎€ธ|||||๎€ท๐ผ๐‘”๐‘“๎€ธ๎…ž๎…ž(๐‘ง)๎€ท๐ผ๐‘”๐‘“๎€ธ๎…ž|||||=๎€ท(๐‘ง)1โˆ’|๐‘ง|2๎€ธ||||๐‘“๎…ž๎…ž(๐‘ง)๐‘”(๐‘ง)+๐‘“๎…ž(๐‘ง)๐‘”๎…ž(๐‘ง)๐‘“๎…ž||||=๎€ท(๐‘ง)๐‘”(๐‘ง)1โˆ’|๐‘ง|2๎€ธ๎ƒฌ||||๐‘“๎…ž๎…ž(๐‘ง)๐‘“๎…ž+๐‘”(๐‘ง)๎…ž(๐‘ง)||||๎ƒญโ‰ค๎€ท๐‘”(๐‘ง)โ€–๐‘“โ€–+1โˆ’|๐‘ง|2๎€ธ||๐‘”๎…ž||๎€บ๎€ท(๐‘ง)/๐‘”(๐‘ง)ln1/1โˆ’|๐‘ง|2๎€ธ๎€ป๎€บ๎€ทln1/1โˆ’|๐‘ง|2โ€–๎€ธ๎€ปโ‰คโ€–๐‘“โ€–+๐‘”โ€–โ„ฌlog๎€บ๎€ทln1/1โˆ’|๐‘ง|2๎€ธ๎€ป,(๐‘งโˆˆ๐‘ˆ)โ‰คโ€–๐‘“โ€–+โ€–๐‘”โ€–โ„ฌlog๎€บ๎€ทln1/1โˆ’|๐‘ง|2๎€ธ๎€ป๎€ท1โˆ’|๐‘ง|2๎€ธ.(2.5) By taking the supremum for the last assertion over ๐‘ˆ, the boundedness of the operator ๐ผ๐‘” follows.
Conversely, assume that ๐ผ๐‘”โˆถฮฃโ†’ฮฃ is bounded, then there is a positive constant ๐ถ such thatโ€–โ€–๐ผ๐‘”๐‘“โ€–โ€–โ‰ค๐ถโ€–๐‘“โ€–(2.6) for every ๐‘“โˆˆฮฃ. Set โ„Ž๐‘Ž๎€ท(๐‘ง)=๎€ธ๐‘Ž๐‘งโˆ’1๐‘Ž๎‚ธ๎‚€11+ln1โˆ’๎‚๐‘Ž๐‘ง21+1๎‚น๎‚ธln1โˆ’|๐‘Ž|2๎‚นโˆ’1,(2.7) for ๐‘Žโˆˆ๐‘ˆ such that โˆš1โˆ’(1/๐‘’)<|๐‘Ž|<1. Then, we have โ„Ž๎…ž๐‘Ž๎‚€1(๐‘ง)=ln1โˆ’๎‚๐‘Ž๐‘ง2๎‚ธ1ln1โˆ’|๐‘Ž|2๎‚นโˆ’1,โ„Ž๐‘Ž๎…ž๎…ž2(๐‘ง)=๐‘Ž1โˆ’๎‚€1๐‘Ž๐‘งln1โˆ’๎‚๎‚ธ1๐‘Ž๐‘งln1โˆ’|๐‘Ž|2๎‚นโˆ’1.(2.8) Thus, โ„Ž๐‘Ž๎…ž๎…ž(๐‘ง)โ„Ž๎…ž๐‘Ž=2(๐‘ง)๐‘Ž1โˆ’๎‚ƒ1๐‘Ž๐‘งln1โˆ’๎‚„๐‘Ž๐‘งโˆ’1,(2.9) and then โ„Ž๐‘Ž๎…ž๎…ž(๐‘Ž)โ„Ž๎…ž๐‘Ž=2(๐‘Ž)๐‘Ž1โˆ’|๐‘Ž|2๎‚ธ1ln1โˆ’|๐‘Ž|2๎‚นโˆ’1.(2.10) It is clear that the relation (2.9) is finite when |๐‘ง|<1, hence โ€–โ„Ž๐‘Ž(๐‘ง)โ€–<โˆž. Setting ๐‘€โˆถ=supโˆš1โˆ’(1/๐‘’)<|๐‘Ž|<1โ€–โ€–โ„Ž๐‘Žโ€–โ€–(๐‘ง)<โˆž,(2.11) therefore, we have โ€–โ€–๐ผโˆž>๐‘”โ€–โ€–โ€–โ€–โ„Ž๐‘Žโ€–โ€–โ‰ฅโ€–โ€–๐ผ๐‘”โ„Ž๐‘Žโ€–โ€–โ‰ฅsup๐‘งโˆˆ๐‘ˆ๎€ท1โˆ’|๐‘ง|2๎€ธ||||โ„Ž๐‘Ž๎…ž๎…ž(๐‘ง)โ„Ž๎…ž๐‘Ž+๐‘”(๐‘ง)๎…ž(๐‘ง)||||โ‰ฅ๎€ท๐‘”(๐‘ง)1โˆ’|๐‘Ž|2๎€ธ||||โ„Ž๐‘Ž๎…ž๎…ž(๐‘Ž)โ„Ž๎…ž๐‘Ž+๐‘”(๐‘Ž)๎…ž(๐‘Ž)||||โ‰ฅ||||2๐‘”(๐‘Ž)๐‘Ž๎€ท๎€ทln1/1โˆ’|๐‘Ž|2+๎€ท๎€ธ๎€ธ1โˆ’|๐‘Ž|2๎€ธ๐‘”๎…ž(๐‘Ž)||||โ‰ฅ๎€ท๐‘”(๐‘Ž)โˆ’2|๐‘Ž|+1โˆ’|๐‘Ž|2๎€ธ||๐‘”๎…ž||๎€ท๎€ท(๐‘Ž)/๐‘”(๐‘Ž)ln1/1โˆ’|๐‘Ž|2๎€ธ๎€ธ๎€ท๎€ทln1/1โˆ’|๐‘Ž|2.๎€ธ๎€ธ(2.12)
Now letting๐‘“๐‘Ž๎€ท(๐‘ง)โˆถ=2๎€ธ๐‘Ž๐‘งโˆ’1๐‘Ž๎‚ธ๎‚€11+ln1โˆ’๎‚๐‘Ž๐‘ง21+1๎‚น๎‚ธln1โˆ’|๐‘Ž|2๎‚นโˆ’1โˆ’๎€œ๐‘ง01ln1โˆ’๐‘Ž๐‘ฅ๐‘‘๐‘ฅ(2.13) for ๐‘Žโˆˆ๐‘ˆ such that โˆš1โˆ’(1/๐‘’)<|๐‘Ž|<1. Then, we obtain ๐‘“๎…ž๐‘Ž๎‚€1(๐‘ง)=2ln1โˆ’๎‚๐‘Ž๐‘ง2๎‚ธ1ln1โˆ’|๐‘Ž|2๎‚นโˆ’11โˆ’ln1โˆ’,๐‘“๐‘Ž๐‘ง๐‘Ž๎…ž๎…ž4(๐‘ง)=๐‘Ž1โˆ’๎‚€1๐‘Ž๐‘งln1โˆ’๎‚๎‚ธ1๐‘Ž๐‘งln1โˆ’|๐‘Ž|2๎‚นโˆ’1โˆ’๐‘Ž1โˆ’.๐‘Ž๐‘ง(2.14) Thus, we conclude that ๐‘“๐‘Ž๎…ž๎…ž(๐‘Ž)๐‘“๎…ž๐‘Ž=๎€ท(๐‘Ž)3|๐‘Ž|/1โˆ’|๐‘Ž|2๎€ธ๎€ท๎€ทln1/1โˆ’|๐‘Ž|2๎€ธ๎€ธ.(2.15) In the same manner of the previous case, we have ๐‘โˆถ=supโˆš1โˆ’(1/๐‘’)<|๐‘Ž|<1โ€–โ€–๐‘“๐‘Žโ€–โ€–<โˆž.(2.16) Consequently, we have โ€–โ€–๐ผโˆž>๐‘”โ€–โ€–โ€–โ€–๐‘“๐‘Žโ€–โ€–โ‰ฅโ€–โ€–๐ผ๐‘”๐‘“๐‘Žโ€–โ€–โ‰ฅsup๐‘งโˆˆ๐‘ˆ๎€ท1โˆ’|๐‘ง|2๎€ธ||||๐‘“๐‘Ž๎…ž๎…ž(๐‘ง)๐‘“๎…ž๐‘Ž+๐‘”(๐‘ง)๎…ž(๐‘ง)||||โ‰ฅ๎€ท๐‘”(๐‘ง)1โˆ’|๐‘Ž|2๎€ธ||||๐‘“๐‘Ž๎…ž๎…ž(๐‘Ž)๐‘“๎…ž๐‘Ž+๐‘”(๐‘Ž)๎…ž(๐‘Ž)||||โ‰ฅ๎€ท๐‘”(๐‘Ž)1โˆ’|๐‘Ž|2๎€ธ||||๎€ท3|๐‘Ž|/1โˆ’|๐‘Ž|2๎€ธ๎€ท๎€ทln1/1โˆ’|๐‘Ž|2+๐‘”๎€ธ๎€ธ๎…ž(๐‘Ž)||||โ‰ฅ๎€ท๐‘”(๐‘Ž)โˆ’3|๐‘Ž|+1โˆ’|๐‘Ž|2๎€ธ||๐‘”๎…ž||๎€ท๎€ท(๐‘Ž)/๐‘”(๐‘Ž)ln1/1โˆ’|๐‘Ž|2๎€ธ๎€ธ๎€ท๎€ทln1/1โˆ’|๐‘Ž|2.๎€ธ๎€ธ(2.17)
From (2.12) and (2.17), we have๎€ท1โˆ’|๐‘Ž|2๎€ธ||||๐‘”๎…ž(๐‘Ž)||||1๐‘”(๐‘Ž)ln๎€ท1โˆ’|๐‘Ž|2๎€ธ<โˆž(2.18) for all โˆš1โˆ’(1/๐‘’)<|๐‘Ž|<1. Also, we have supโˆš|๐‘Ž|โ‰ค1โˆ’(1/๐‘’)๎€ท1โˆ’|๐‘Ž|2๎€ธ||||๐‘”๎…ž(๐‘Ž)||||1๐‘”(๐‘Ž)ln๎€ท1โˆ’|๐‘Ž|2๎€ธโ‰คsupโˆš1โˆ’(1/๐‘’)โ‰ค|๐‘Ž|<1๎€ท1โˆ’|๐‘Ž|2๎€ธ||||๐‘”๎…ž(๐‘Ž)||||1๐‘”(๐‘Ž)ln๎€ท1โˆ’|๐‘Ž|2๎€ธ.(2.19) From (2.18) and (2.19), we obtain ๐‘”โˆˆโ„ฌlog, as desired.
In the following results, we study the compactness of the integral operators ๐ฝ๐‘” and ๐ผ๐‘” in an open disc.

Theorem 2.3. Assume that ๐‘” is an analytic function on ๐‘ˆ. Then, for functions ๐‘“โˆˆโ„ฌlog, the integral operator ๐ฝ๐‘” is compact if and only if ๐‘”โˆˆฮฃ.

Proof. If ๐ฝ๐‘” is compact, then it is bounded, and by Theorem 2.1 it follows that ๐‘”โˆˆฮฃ.
Now assume that ๐‘”โˆˆฮฃ, that (๐‘“๐‘›)๐‘›โˆˆโ„• is a sequence in โ„ฌlog, and ๐‘“๐‘›โ†’0 uniformly on ๐‘ˆ as ๐‘›โ†’โˆž. Now for every ๐œ€>0, there is ๐›ฟโˆˆ(0,1) such that11โˆ’|๐‘ง|2<๐œ€,(2.20) where ๐›ฟ<|๐‘ง|<1. Since ๐›ฟ is arbitrary, then we can chose ln(1/(1โˆ’|๐‘ง|2))>1 for ๐›ฟ<|๐‘ง|<1 and โ€–โ€–๐ฝ๐‘”๐‘“๐‘›โ€–โ€–=sup๐‘งโˆˆ๐‘ˆ๎€ท1โˆ’|๐‘ง|2๎€ธ|||||๎€ท๐ฝ๐‘”๐‘“๐‘›๎€ธ๎…ž๎…ž(๐‘ง)๎€ท๐ฝ๐‘”๐‘“๐‘›๎€ธ๎…ž|||||(๐‘ง)=sup๐‘งโˆˆ๐‘ˆ๎€ท1โˆ’|๐‘ง|2๎€ธ||||๐‘“๐‘›(๐‘ง)๐‘”๎…ž๎…ž(๐‘ง)+๐‘“๎…ž๐‘›(๐‘ง)๐‘”๎…ž(๐‘ง)๐‘“๐‘›(๐‘ง)๐‘”๎…ž||||(๐‘ง)โ‰คsup๐‘งโˆˆ๐‘ˆ๎€ท1โˆ’|๐‘ง|2๎€ธ||||๐‘”๎…ž๎…ž(๐‘ง)๐‘”๎…ž||||(๐‘ง)+sup๐‘งโˆˆ๐‘ˆ๎€ท1โˆ’|๐‘ง|2๎€ธ||||๐‘“๎…ž๐‘›(๐‘ง)๐‘“๐‘›||||๎‚ต1(๐‘ง)ln1โˆ’|๐‘ง|2๎‚ถโ‰คโ€–๐‘”โ€–1โˆ’|๐‘ง|2+โ€–โ€–๐‘“๐‘›โ€–โ€–โ„ฌlogโ€–โ€–๐‘“<๐œ€โ€–๐‘”โ€–+๐‘›โ€–โ€–โ„ฌlog.(2.21) Since for ๐‘“๐‘›โ†’0 on ๐‘ˆ we have โ€–๐‘“๐‘›โ€–โ„ฌlogโ†’0, and that ๐œ€ is an arbitrary positive number, by letting ๐‘›โ†’โˆž in the last inequality, we obtain that lim๐‘›โ†’โˆžโ€–๐ฝ๐‘”๐‘“๐‘›โ€–=0. Therefore, ๐ฝ๐‘” is compact.

Theorem 2.4. Assume that ๐‘” is an analytic function on ๐‘ˆ. Then, the integral operator ๐ผ๐‘”โˆถฮฃโ†’ฮฃ is compact if and only if ๐‘” is a constant defer from zero.

Proof. Assume that ๐‘” is a constant without loss of generality and assume that ๐‘“(๐‘ง)=๐‘ง. Then, it is clear that ๐ผ๐‘” is compact.
Conversely, assume that ๐ผ๐‘”โˆถฮฃโ†’ฮฃ is compact. Let (๐‘ง๐‘›)๐‘›โˆˆโ„•, be a sequence in ๐‘ˆ such that |๐‘ง๐‘›|โ†’1 as ๐‘›โ†’โˆž. Our aim is to show that ๐‘”๎…ž(๐‘ง๐‘›)โ†’0 as ๐‘›โ†’โˆž, then by the maximum modulus theorem, we have ๐‘” is a constant. In fact, setting๐‘“๐‘›๎€ท(๐‘ง)=2๐‘ง๐‘›๎€ธ๐‘งโˆ’1๐‘ง๐‘›๎ƒฌ๎‚ต11+ln1โˆ’๐‘ง๐‘›๐‘ง๎‚ถ2๎ƒญ๎‚ธ1+1ln1โˆ’|๐‘ง|2๎‚นโˆ’1๎€œโˆ’4๐‘ง01ln1โˆ’๐‘ง๐‘›๐‘ค๐‘‘๐‘ค.(2.22) Then, we obtain ๐‘“๎…ž๐‘›๎‚ต1(๐‘ง)=2ln1โˆ’๐‘ง๐‘›๐‘ง๎‚ถ2๎‚ธ1ln1โˆ’|๐‘ง|2๎‚นโˆ’1๎‚ธ1โˆ’4ln1โˆ’๐‘ง๐‘›๐‘ง๎‚น,๐‘“๐‘›๎…ž๎…ž4(๐‘ง)=๐‘ง๐‘›1โˆ’๐‘ง๐‘›๐‘ง๎‚ต1ln1โˆ’๐‘ง๐‘›๐‘ง1๎‚ถ๎‚ธln1โˆ’|๐‘ง|2๎‚นโˆ’1โˆ’4๐‘ง๐‘›1โˆ’๐‘ง๐‘›๐‘ง.(2.23) Consequently, we have ๐‘“๐‘›๎…ž๎…ž๎€ท๐‘ง๐‘›๎€ธ๐‘“๎…ž๐‘›๎€ท๐‘ง๐‘›๎€ธ=0.(2.24) Similar to the proof of Theorem 2.2, we see that ๐‘“๐‘›โ†’0 uniformly on ๐‘ˆ. Since ๐ผ๐‘”โˆถฮฃโ†’ฮฃ is compact, then we get โ€–โ€–๐ผ๐‘”๐‘“๐‘›โ€–โ€–โŸถ0,๐‘›โŸถโˆž.(2.25) Thus, ||||๐‘”๎…ž๎€ท๐‘ง๐‘›๎€ธ๐‘”๎€ท๐‘ง๐‘›๎€ธ||||โ‰คsup๐‘งโˆˆ๐‘ˆ||||๐‘”๎…ž(๐‘ง)||||๐‘”(๐‘ง)+sup๐‘งโˆˆ๐‘ˆ||||๐‘“๐‘›๎…ž๎…ž(๐‘ง)๐‘“๎…ž๐‘›||||โ‰คโ€–โ€–๐ผ(๐‘ง)๐‘”๐‘“๐‘›โ€–โ€–โŸถ0(2.26) Implies that ๐‘”๎…ž๐‘›(๐‘ง)โ†’0 and consequently ๐‘” is a constant as desired.

3. Some Geometric Properties

In this section, we introduce some geometric properties for analytic function ๐‘“โˆˆฮฃ. A function ๐‘“โˆˆโ„‹ which normalized as ๐‘“(0)=๐‘“๎…ž(0)โˆ’1=0 denoted this class by ๐’œ. Recall that a function ๐‘“โˆˆ๐’œ is said to be star-like of order ๐œ‡โˆˆ[0,1) in ๐‘ˆ if it satisfies๐‘“โˆˆ๐’ฎ๐œ‡๎‚ปโŸบโ„œ๐‘ง๐‘“๎…ž(๐‘ง)๎‚ผ๐‘“(๐‘ง)>๐œ‡,(๐‘งโˆˆ๐‘ˆ).(3.1) Also, a function ๐‘“โˆˆ๐’œ is called convex in ๐‘ˆ if it satisfies๐‘“โˆˆ๐’ฆ๐œ‡๎‚ปโŸบโ„œ1+๐‘ง๐‘“๎…ž๎…ž(๐‘ง)๐‘“๎…ž๎‚ผ(๐‘ง)>๐œ‡,(๐‘งโˆˆ๐‘ˆ).(3.2) It follows that๐‘“โˆˆ๐’ฆ๐œ‡โŸบ๐‘ง๐‘“๎…žโˆˆ๐’ฎ๐œ‡.(3.3)

In the next result, we discuss the convexity of the integral operators ๐ฝ๐‘” and ๐ผ๐‘”.

Theorem 3.1. Assume that ๐‘“,๐‘”โˆˆ๐’œ. If ๐‘“โˆˆ๐’ฎ๐œ‡ and ๐‘”โˆˆ๐’ฆ๐œˆ such that 0โ‰ค๐œ‡+๐œˆ<1, then the function ๐ฝ๐‘”๐‘“ is convex of order ๐œ‡+๐œˆ.

Proof. Assume that ๐‘“,๐‘”โˆˆ๐’œ. Then, we obtain ๐‘ง๎€ท๐ฝ๐‘”๐‘“๎€ธ๎…ž๎…ž(๐‘ง)๎€ท๐ฝ๐‘”๐‘“๎€ธ๎…ž=(๐‘ง)๐‘ง๐‘”๎…ž๎…ž(๐‘ง)๐‘”๎…ž+(๐‘ง)๐‘ง๐‘“๎…ž(๐‘ง)๐‘“(๐‘ง).(3.4) Consequently, we get โ„œ๎ƒฏ๐‘ง๎€ท๐ฝ1+๐‘”๐‘“๎€ธ๎…ž๎…ž(๐‘ง)๎€ท๐ฝ๐‘”๐‘“๎€ธ๎…ž๎ƒฐ๎‚ป(๐‘ง)=โ„œ๐‘ง๐‘”๎…ž๎…ž(๐‘ง)๐‘”๎…ž๎‚ผ๎‚ป(๐‘ง)+1+โ„œ๐‘ง๐‘“๎…ž(๐‘ง)๎‚ผ๐‘“(๐‘ง)>๐œ‡+๐œˆ.(3.5) Hence, ๐ฝ๐‘”โˆˆ๐’ฆ๐œ‡+๐œˆ.

Theorem 3.2. Assume that ๐‘“,๐‘”โˆˆ๐’œ. If ๐‘“โˆˆ๐’ฆ๐œ‡ and ๐‘”โˆˆ๐’ฎ๐œˆ such that 0โ‰ค๐œ‡+๐œˆ<1, then the function ๐ผ๐‘”๐‘“ is convex of order ๐œ‡+๐œˆ.

Proof. Assume that ๐‘“,๐‘”โˆˆ๐’œ. Then, we have ๐‘ง๎€ท๐ผ๐‘”๐‘“๎€ธ๎…ž๎…ž(๐‘ง)๎€ท๐ผ๐‘”๐‘“๎€ธ๎…ž=(๐‘ง)๐‘ง๐‘“๎…ž๎…ž(๐‘ง)๐‘“๎…ž+(๐‘ง)๐‘ง๐‘”๎…ž(๐‘ง)๐‘”(๐‘ง).(3.6) Consequently, we get โ„œ๎ƒฏ๐‘ง๎€ท๐ผ1+๐‘”๐‘“๎€ธ๎…ž๎…ž(๐‘ง)๎€ท๐ผ๐‘”๐‘“๎€ธ๎…ž๎ƒฐ๎‚ป(๐‘ง)=โ„œ๐‘ง๐‘“๎…ž๎…ž(๐‘ง)๐‘“๎…ž๎‚ผ๎‚ป(๐‘ง)+1+โ„œ๐‘ง๐‘”๎…ž(๐‘ง)๎‚ผ๐‘”(๐‘ง)>๐œ‡+๐œˆ.(3.7) Hence, ๐ผ๐‘”โˆˆ๐’ฆ๐œ‡+๐œˆ.

Theorem 3.3. Assume that ๐‘“,๐‘”โˆˆ๐’œ. If ๐‘“โˆˆ๐’ฎ๐œ‡ and ๐‘”โˆˆ๐’ฎ๐œˆ such that 0โ‰ค๐œ‡+๐œˆ<1, then the multiplication operator ๐‘€๐‘”๐‘“ is star-like of order ๐œ‡+๐œˆ.

Proof. Assume that ๐‘“,๐‘”โˆˆ๐’œ. Then, we obtain โ„œ๎ƒฏ๐‘ง๎€ท๐‘€๐‘”๐‘“๎€ธ๎…ž(๐‘ง)๎€ท๐‘€๐‘”๐‘“๎€ธ๎ƒฐ๎‚ป(๐‘ง)=โ„œ๐‘ง๐‘“๎…ž(๐‘ง)๎‚ผ๎‚ป๐‘“(๐‘ง)+โ„œ๐‘ง๐‘”๎…ž(๐‘ง)๎‚ผ๐‘”(๐‘ง)>๐œ‡+๐œˆ.(3.8) Hence, ๐‘€๐‘”๐‘“โˆˆ๐’ฎ๐œ‡+๐œˆ.
The next result comes directly from the definition of the class ฮฃ and the fact that โ€–๐‘‡๐‘“โ€–<โˆž if and only if ๐‘“ is uniformly locally univalent (see [23]).

Theorem 3.4. Assume that ๐‘” is an analytic function on ๐‘ˆ and ๐‘“โˆˆ๐’œ. Then, the functions ๐ผ๐‘”๐‘“ and ๐ฝ๐‘”๐‘“ are in the class ฮฃ if and only if ๐‘“ is locally univalent in ๐‘ˆ.

Acknowledgment

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