# General Properties for Volterra-Type Operators in the Unit Disk

Authors: Rabha W. Ibrahim; Maslina Darus
Journal: ISRN Mathematical Analysis
 (2011)

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## 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.


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## Body

## 1. Introduction

Letℋ be the class of analytic functions in U:={z∈ℂ:|z|<1}. Suppose that g:U→ℂ is a holomorphic map, f∈ℋ. The integral operator, called Volterra-type operator,(1.1)Jgf(z)=∫0zfdg=∫01f(tz)zg′(tz)dt=∫0zf(ξ)g′(ξ)dξ,z∈U,
was introduced by Pommerenke in [1]. Another natural integral operator is defined as follows:(1.2)Igf(z)=∫0zf′(ξ)g(ξ)dξ,z∈U.
The importance of the operators Jg and Ig comes from the fact that(1.3)Jgf+Igf=Mgf-f(0)g(0),
where the multiplication operator Mg is defined by(1.4)(Mgf)(z)=g(z)f(z),f∈H,z∈U.
Furthermore, Volterra integral equations arise in many physical applications (see [2–4]).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 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 [5–11].For functionsf∈ℋ, 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. [12–14]) 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 [18–22]. Finally, these operators are involving the Cesáro integral operator (see [23–25]).A functionf∈ℋ is called in the class Σ if and only if it has the norm (see [26])(1.5)‖f‖=sup⁡z∈U(1-|z|2)|f′′(z)f′(z)|<∞,(z∈U).
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 operatorsJgf and Igf for the functions f∈Σ 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 operatorsJgf and Igf on the classes Σ.Consider the spaceℬlog of all functions f∈ℋ which are satisfying(2.1)‖f‖Blog⁡=sup⁡z∈U(1-|z|2)|f′(z)f(z)|ln⁡1(1-|z|2)<∞,(z∈U).Theorem 2.1.
Assume thatg is an analytic function on U. Then, for functions f∈ℬlog,  Jg is bounded if and only if g∈Σ.Proof.
Assume thatJg is bounded. Taking the function given by f(z)=1, we see that g∈Σ.
Conversely, assume thatg∈Σ, we have(2.2)(1-|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′(z)+f′(z)f(z)|]≤‖g‖+(1-|z|2)|f′(z)/f(z)|ln⁡[1/(1-|z|2)]ln⁡[1/(1-|z|2)]≤‖g‖+‖f‖Blog⁡ln⁡[1/(1-|z|2)]≤‖g‖+‖f‖Blog⁡ln⁡[1/(1-|z|2)](1-|z|2).
By taking the supremum for the last assertion over U and using the fact that the quantity
(2.3)sup⁡x∈(0,1]x(ln⁡1x)
is finite, the boundedness of the operator Jg follows.Theorem 2.2.
Assume thatg is an analytic function on U. Then, Ig:Σ→Σ is bounded if and only if g∈ℬlog, where
(2.4)‖g‖Blog⁡=sup⁡z∈U(1-|z|2)|g′(z)g(z)|ln⁡1(1-|z|2).Proof.
Assume thatg∈ℬlog. Then, we obtain
(2.5)(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′(z)+g′(z)g(z)|]≤‖f‖+(1-|z|2)|g′(z)/g(z)|ln⁡[1/(1-|z|2)]ln⁡[1/(1-|z|2)]≤‖f‖+‖g‖Blog⁡ln⁡[1/(1-|z|2)],(z∈U)≤‖f‖+‖g‖Blog⁡ln⁡[1/(1-|z|2)](1-|z|2).
By taking the supremum for the last assertion over U, the boundedness of the operator Ig follows.
Conversely, assume thatIg:Σ→Σ is bounded, then there is a positive constant C such that(2.6)‖Igf‖≤C‖f‖
for every f∈Σ. Set
(2.7)ha(z)=(a¯z-1)a¯[(1+ln⁡11-a¯z)2+1][ln⁡11-|a|2]-1,
for a∈U such that 1-(1/e)<|a|<1. Then, we have
(2.8)ha′(z)=(ln⁡11-a¯z)2[ln⁡11-|a|2]-1,ha′′(z)=2a¯1-a¯z(ln⁡11-a¯z)[ln⁡11-|a|2]-1.
Thus,
(2.9)ha′′(z)ha′(z)=2a¯1-a¯z[ln⁡11-a¯z]-1,
and then
(2.10)ha′′(a)ha′(a)=2a¯1-|a|2[ln⁡11-|a|2]-1.
It is clear that the relation (2.9) is finite when |z|<1, hence ∥ha(z)∥<∞. Setting
(2.11)M:=sup⁡1-(1/e)<|a|<1‖ha(z)‖<∞,
therefore, we have
(2.12)∞>‖Ig‖‖ha‖≥‖Igha‖≥sup⁡z∈U(1-|z|2)|ha′′(z)ha′(z)+g′(z)g(z)|≥(1-|a|2)|ha′′(a)ha′(a)+g′(a)g(a)|≥|2a¯ln⁡(1/(1-|a|2))+(1-|a|2)g′(a)g(a)|≥-2|a|+(1-|a|2)|g′(a)/g(a)|ln⁡(1/(1-|a|2))ln⁡(1/(1-|a|2)).
Now letting(2.13)fa(z):=2(a¯z-1)a¯[(1+ln⁡11-a¯z)2+1][ln⁡11-|a|2]-1-∫0zln⁡11-a¯xdx
for a∈U such that 1-(1/e)<|a|<1. Then, we obtain
(2.14)fa′(z)=2(ln⁡11-a¯z)2[ln⁡11-|a|2]-1-ln⁡11-a¯z,fa′′(z)=4a¯1-a¯z(ln⁡11-a¯z)[ln⁡11-|a|2]-1-a¯1-a¯z.
Thus, we conclude that
(2.15)fa′′(a)fa′(a)=3|a|/(1-|a|2)ln⁡(1/(1-|a|2)).
In the same manner of the previous case, we have
(2.16)N:=sup⁡1-(1/e)<|a|<1‖fa‖<∞.
Consequently, we have
(2.17)∞>‖Ig‖‖fa‖≥‖Igfa‖≥sup⁡z∈U(1-|z|2)|fa′′(z)fa′(z)+g′(z)g(z)|≥(1-|a|2)|fa′′(a)fa′(a)+g′(a)g(a)|≥(1-|a|2)|3|a|/(1-|a|2)ln⁡(1/(1-|a|2))+g′(a)g(a)|≥-3|a|+(1-|a|2)|g′(a)/g(a)|ln⁡(1/(1-|a|2))ln⁡(1/(1-|a|2)).
From (2.12) and (2.17), we have(2.18)(1-|a|2)|g′(a)g(a)|ln⁡1(1-|a|2)<∞
for all 1-(1/e)<|a|<1. Also, we have
(2.19)sup⁡|a|≤1-(1/e)(1-|a|2)|g′(a)g(a)|ln⁡1(1-|a|2)≤sup⁡1-(1/e)≤|a|<1(1-|a|2)|g′(a)g(a)|ln⁡1(1-|a|2).
From (2.18) and (2.19), we obtain g∈ℬlog, as desired.
In the following results, we study the compactness of the integral operatorsJg and Ig in an open disc.Theorem 2.3.
Assume thatg is an analytic function on U. Then, for functions f∈ℬlog, the integral operator Jg is compact if and only if g∈Σ.Proof.
IfJg is compact, then it is bounded, and by Theorem 2.1 it follows that g∈Σ.
Now assume thatg∈Σ, that (fn)n∈ℕ is a sequence in ℬlog, and fn→0 uniformly on U¯ as n→∞. Now for every ε>0, there is δ∈(0,1) such that(2.20)11-|z|2<ε,
where δ<|z|<1. Since δ is arbitrary, then we can chose ln(1/(1-|z|2))>1 for δ<|z|<1 and
(2.21)‖Jgfn‖=sup⁡z∈U(1-|z|2)|(Jgfn)′′(z)(Jgfn)′(z)|=sup⁡z∈U(1-|z|2)|fn(z)g′′(z)+fn′(z)g′(z)fn(z)g′(z)|≤sup⁡z∈U(1-|z|2)|g′′(z)g′(z)|+sup⁡z∈U(1-|z|2)|fn′(z)fn(z)|(ln⁡11-|z|2)≤‖g‖1-|z|2+‖fn‖Blog⁡<ε‖g‖+‖fn‖Blog⁡.
Since for fn→0 on U¯ we have ∥fn∥ℬlog→0, and that ε is an arbitrary positive number, by letting n→∞ in the last inequality, we obtain that limn→∞∥Jgfn∥=0. Therefore, Jg is compact.Theorem 2.4.
Assume thatg is an analytic function on U. Then, the integral operator Ig:Σ→Σ is compact if and only if g is a constant defer from zero.Proof.
Assume thatg is a constant without loss of generality and assume that f(z)=z. Then, it is clear that Ig is compact.
Conversely, assume thatIg:Σ→Σ is compact. Let (zn)n∈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, setting(2.22)fn(z)=2(z¯nz-1)z¯n[(1+ln⁡11-z¯nz)2+1][ln⁡11-|z|2]-1-4∫0zln⁡11-z¯nwdw.
Then, we obtain
(2.23)fn′(z)=2(ln⁡11-z¯nz)2[ln⁡11-|z|2]-1-4[ln⁡11-z¯nz],fn′′(z)=4z¯n1-z¯nz(ln⁡11-z¯nz)[ln⁡11-|z|2]-1-4z¯n1-z¯nz.
Consequently, we have
(2.24)fn′′(zn)fn′(zn)=0.
Similar to the proof of Theorem 2.2, we see that fn→0 uniformly on U¯. Since Ig:Σ→Σ is compact, then we get
(2.25)‖Igfn‖⟶0,n⟶∞.
Thus,
(2.26)|g′(zn)g(zn)|≤sup⁡z∈U|g′(z)g(z)|+sup⁡z∈U|fn′′(z)fn′(z)|≤‖Igfn‖⟶0
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 functionf∈Σ. A function f∈ℋ which normalized as f(0)=f′(0)-1=0 denoted this class by 𝒜. Recall that a function f∈𝒜 is said to be star-like of order μ∈[0,1) in U if it satisfies(3.1)f∈Sμ⟺R{zf′(z)f(z)}>μ,(z∈U).
Also, a function f∈𝒜 is called convex in U if it satisfies(3.2)f∈Kμ⟺R{1+zf′′(z)f′(z)}>μ,(z∈U).
It follows that(3.3)f∈Kμ⟺zf′∈Sμ.In the next result, we discuss the convexity of the integral operatorsJg and Ig.Theorem 3.1.
Assume thatf,g∈𝒜. If f∈𝒮μ and g∈𝒦ν such that 0≤μ+ν<1, then the function Jgf is convex of order μ+ν.Proof.
Assume thatf,g∈𝒜. Then, we obtain
(3.4)z(Jgf)′′(z)(Jgf)′(z)=zg′′(z)g′(z)+zf′(z)f(z).
Consequently, we get
(3.5)R{1+z(Jgf)′′(z)(Jgf)′(z)}=R{zg′′(z)g′(z)+1}+R{zf′(z)f(z)}>μ+ν.
Hence, Jg∈𝒦μ+ν.Theorem 3.2.
Assume thatf,g∈𝒜. If f∈𝒦μ and g∈𝒮ν such that 0≤μ+ν<1, then the function Igf is convex of order μ+ν.Proof.
Assume thatf,g∈𝒜. Then, we have
(3.6)z(Igf)′′(z)(Igf)′(z)=zf′′(z)f′(z)+zg′(z)g(z).
Consequently, we get
(3.7)R{1+z(Igf)′′(z)(Igf)′(z)}=R{zf′′(z)f′(z)+1}+R{zg′(z)g(z)}>μ+ν.
Hence, Ig∈𝒦μ+ν.Theorem 3.3.
Assume thatf,g∈𝒜. If f∈𝒮μ and g∈𝒮ν such that 0≤μ+ν<1, then the multiplication operator Mgf is star-like of order μ+ν.Proof.
Assume thatf,g∈𝒜. Then, we obtain
(3.8)R{z(Mgf)′(z)(Mgf)(z)}=R{zf′(z)f(z)}+R{zg′(z)g(z)}>μ+ν.
Hence, Mgf∈𝒮μ+ν.
The next result comes directly from the definition of the classΣ and the fact that ∥Tf∥<∞ if and only if f is uniformly locally univalent (see [23]).Theorem 3.4.
Assume thatg is an analytic function on U and f∈𝒜. Then, the functions Igf and Jgf are in the class Σ if and only if f is locally univalent in U.



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