# Geometric Properties of a New Integral Operator

Authors: Roberta Bucur; Loriana Andrei; Daniel Breaz
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

We obtain sufficient conditions for the univalence, starlikeness, and convexity of a new integral operator defined on the space of normalized analytic functions in the open unit disk. Some subordination results for the new integral operator are also given. Several corollaries follow as special cases.


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## Body

## 1. Introduction

LetU=z:z<1 be the open unit disk and A the class of all functions of the form(1)fz=z+∑n=2∞anzn,z∈U,which are analytic in U and satisfy the condition f(0)=f′(0)-1=0. Consider S the class of functions f∈A which are univalent in U.A domainD⊂C is convex if the line segment joining any two points in D lies entirely in D, while a domain is starlike with respect to a point w0∈D if the line segment joining any point of D to w0 lies inside D.A functionf∈A is starlike if f(U) is a starlike domain with respect to origin and convex if f(U) is convex.Analytically,f∈A is starlike if and only if(2)Rezf′zfz>0,∀z∈Uand f∈A is convex if and only if(3)Re1+zf′′zf′z>0,∀z∈U.The classes consisting of starlike and convex functions are denoted by S∗ and K, respectively. We denote by S∗(δ) and K(δ) the classes consisting of starlike and convex functions of order δ, 0≤δ<1, characterized, respectively, by (4)Rezf′zfz>δ,Re1+zf′′zf′z>δ.Iff and g are analytic functions in U, we say that f is subordinate to g, written as f≺g, if there is a function w analytic in U, with w(0)=0, |w(z)|<1, for all z∈U, such that f(z)=g(w(z)) for all z∈U. If g is univalent, then f≺g if and only if f(0)=g(0) and f(U)⊆g(U) (see Miller and Mocanu [1]).Using subordinations, Owa et al. [2] have defined the following subclass Sb(a) of A.A functionf∈A is said to be in the class Sb(a) if it satisfies(5)f′zb≺a1-za-z,z∈Ufor some real a>1 and b>0.For the classSb(a), Owa et al. proved the following.Theorem 1 (see [2]).
Iff∈A satisfies(6)Rezf′′zf′z<a-12ba+1,z∈U,for some real a>1 and b>0, then f∈Sb(a).Breaz et al. [3] have introduced the following subclasses Sb∗(a) and Cb∗(a) of A. A function f∈A is said to be in the class Sb∗(a) if it satisfies inequality (6) for some real a>1 and b>0.A functionf∈A is said to be a member of the class Cb∗(a) if it satisfies(7)Rezf′zfz>1-a2ba+1+1,z∈U,for some real a>1 and b>0.Recently, Frasin and Jahangiri [4] defined the family B(μ,λ), μ≥0, 0≤λ<1 consisting of functions f∈A satisfying the condition(8)f′zzfzμ-1<1-λ,z∈U.We note that B(1,λ)=S∗(λ) and B(2,λ)=B(λ) (see [5]).In 1975, Pfaltzgraff [6] introduced the operator(9)Iαfz=∫0zf′tαdtand proved that Iα(S)⊂S if α is a complex number and α≤1/4.During the last several years many authors have employed different methods to study generalizations of Pfaltzgraff [6] integral operator which maps subsets of A into S.Pascu and Pescar [7] studied the operator(10)Iα,nfz=∫0zf′tuαdt,where n is a positive integer.The next operator was defined by Breaz et al. in [8] and has the form(11)Iαf1,…,fnz=∫0zf1′tα1⋯fn′tαndt,where fi∈A and αi∈C for all i∈{1,…,n}.Further extensions of these operators were obtained in [9–11].Recently, starlikeness and convexity properties for certain general families of integral operators in the open unit disk were given in [12–14].In the present paper, we introduce the new integral operator(12)Iα:A×A⟶Adefined by(13)Iαf,gz=∫0ztf′tgtαdt,for the functions f,g∈A and the complex number α.We observe that if we takeg(z)=z, we obtain Pfaltzgraff [6] integral operator.In the first section of this paper, our purpose is to derive univalence conditions, starlikeness properties, and the order of convexity for the integral operator introduced in (13). The object of the second section of this paper is to discuss some properties for the integral operator Iα(f,g) with the above classes Sb(a), Sb∗(a) and Cb∗(a).The following results will be required in our investigation.Lemma 2 (Mocanu and Şerb [15]).
LetM0=1,5936…, the positive solution of equation(14)2-MeM=2.If f∈A and (15)f′′zf′z≤M0,∀z∈U,then(16)zf′zfz-1<1,∀z∈U.The edge M0 is sharp.Theorem 3 (Becker [16]).
If the function f is regular in the unit disk U,fz=z+a2z2+⋯, and(17)1-z2·zf′′zf′z≤1,for all z∈U, then the function f is univalent in U.Lemma 4 (the general Schwarz Lemma [17]).
Let the functionf be regular function in the disk UR={z∈C:z<R}, with fz<M for fixed M. If f has one zero with multiplicity order bigger than m for z=0, then (18)fz≤MRmzm,z∈UR.The equality case holds only if f(z)=eiθM/Rmzm, where θ is constant.Lemma 5 (see [18]).
Let the functionsp and q be analytic in U with (19)p0=q0=0,and let δ be a real number. If the function q maps the unit disk U onto a region which is starlike with respect to the origin, the inequality(20)Rep′zq′z>δ,∀z∈Uimplies that(21)Repzqz>δ,∀z∈U.


## 2. Univalence, Starlikeness, and Convexity Properties of the Integral OperatorIα(f,g)

The univalence condition for the operatorIα(f,g) defined in (13) is proved in the next theorem, by using the Becker univalence criterion.Theorem 6.
Letα be a complex number, M0 the positive solution of (14), M0=1,5936…, and f,g∈A. If(22)f′′zf′z≤M0,g′′zg′z≤M0,z∈U,(23)α≤332M0+33,then the integral operator (24)Iαf,gz=∫0ztf′tgtαdtis in the class S.Proof.
Let the function be as follows:(25)hz≔Iαf,gz.Then, we have(26)zh′′zh′z=αzf′′zf′z-zg′zgz-1.From (26) we get (27)1-z2·zh′′zh′z≤1-z2·z·α·f′′zf′z+1-z2·α·zg′zgz-1.By using (22) and applying Lemma 2, we have (28)zg′zgz-1<1,∀z∈U,which implies that(29)1-z2·zh′′zh′z≤1-z2·z·α·M0+1-z2·α.Let us consider the function(30)T:0,1⟶R,Tx=x1-x2,x=z.Then we have(31)Tx≤233,∀x∈0,1.From (29), (31), and (23) we obtain(32)1-z2·zh′′zh′z≤α·233M0+1≤1.From Theorem 3 and (32), we yield that the function Iα(f,g)(z) is in the class S.If we putg(z)=z in Theorem 6, we obtain the following.Example 7.
Letα be a complex number, M0 the positive solution of (14), M0=1,5936…, and f,g∈A. If(33)f′′zf′z≤M0,z∈U,α≤332M0,then the integral operator (34)Iαfz=∫0zf′tαdtis in the class S.In the following theorem we give sufficient conditions such that the integral operatorIα(f,g)(z)∈S∗.Theorem 8.
Letf∈A with zf′′z/f′z<1, z∈U, and g in the class B(μ,λ). Let M be a positive real number such that(35)gz<M,M≥1,z∈U.If(36)α≤12+2-λMμ-1,then the integral operator Iα(f,g)(z) is in the class S∗.Proof.
For the functionh defined in (25) we have(37)zh′zhz=zα+1f′z/gzα∫0ztf′t/gtαdt.Letting(38)pz=zh′z,qz=hz,we find that(39)p′zq′z=1+zh′′zh′z=1+αzf′′zf′z+1-zg′zgz.Thus,(40)p′zq′z-1≤α·zf′′zf′z+α+α·g′zzgzμ·gzzμ-1.Since gz<M, z∈U, by applying the Schwarz Lemma, we have(41)gzz≤M,∀z∈U.By using the hypothesis and (41), we obtain (42)p′zq′z-1≤2α+α·g′zzgzμ-1+1·Mμ-1≤α2+2-λ·Mμ-1≤1;that is,(43)Rep′zq′z>0,z∈U.Therefore, applying Lemma 5, we find that(44)Repzqz>0,z∈U.This completes the proof of the theorem.Lettingμ=1 in Theorem 8, we have the following.Corollary 9.
Letf∈A with zf′′z/f′z<1, z∈U, and g in the class S∗(λ). Let M be a positive real number such that(45)gz<M,M≥1,z∈U.If(46)α≤14-λ,then the integral operator Iα(f,g)(z) is in the class S∗.Lettingλ=0 in Corollary 9, we obtain the following.Corollary 10.
Letf∈A with zf′′z/f′z<1, z∈U, and g∈S∗. Let M be a positive real number such that(47)gz<M,M≥1,z∈U.If(48)α≤14,then the integral operator Iα(f,g)(z) is in the class S∗.Sinceg(z)=z is a starlike function, from Corollary 10, we obtain the following.Example 11.
Letf∈A with zf′′z/f′z<1, z∈U. If(49)α≤14,then the integral operator Iα(f)(z) is in the class S∗.Theorem 12.
Letα be a complex number, f∈A, and g∈B(μ,λ). If (50)gz<M,f′′zf′z<N,for M≥1, N≥1, and z∈U, then the integral operator (51)Iαf,gz=∫0ztf′tgtαdtis in the class K(δ), where (52)δ=1-αN+1+2-λMμ-1,0<αN+1+2-λMμ-1≤1.Proof.
Let the functionh be defined in (25). Then, we obtain (53)zh′′zh′z=αzf′′zf′z+1-zg′zgz.Therefore,(54)zh′′zh′z≤z·α·f′′zf′z+α+α·g′zzgzμ·gzzμ-1.Since |g(z)|<M, z∈U, applying the Schwarz Lemma, we have (55)gzz≤M,∀z∈U.
From (54) we obtain (56)zh′′zh′z≤αN+1+α·g′zzgzμ-1+1·Mμ-1≤αN+1+2-λ·Mμ-1=1-δ.This evidently completes the proof.Lettingμ=1 in Theorem 12, we have the following.Corollary 13.
Letα be a complex number, f,g∈A, with g in the class S∗(λ), 0≤λ<1. If gz≤M, for M≥1, z∈U, and (57)f′′zf′z<N,N≥1,z∈U,then the integral operator is as follows: (58)Iαf,gz=∫0ztf′tgtαdt∈Kδ,where δ=1-|α|(3+N-λ) and 0<|α|(3+N-λ)≤1.Lettingδ=λ=0 in Corollary 13, we obtain the following.Corollary 14.
Letα be a complex number, f,g∈A, with g a starlike function in U. If |g(z)|≤M, for M≥1, z∈U, and (59)f′′zf′z<N,N≥1,z∈U,then the integral operator (60)Iαf,gz=∫0ztf′tgtαdtis convex in U, where α=1/3+N.Takingg(z)=z in Corollary 14, we obtain the following.Example 15.
Letα be a complex number and function f∈A. If (61)f′′zf′z<N,N≥1,z∈U,then the integral operator Iα(f)(z) is convex in U, where |α|=1/3+N.


## 3. Subordination Results

In view of the results due to Breaz et al. [3], we obtain some subordination results of the above integral operator Iαf,g.Theorem 16.
Iff∈Sb∗(a) and g∈Cb∗(a), a>1, and b>0, then for α>0(62)RezIα′′f,gzIα′f,gz<αa-1ba+1,z∈U.This implies that Iαf,g(z)∈Sb/2α(a).Proof.
We have(63)zIα′′f,gzIα′f,gz=αzf′′zf′z-zg′zgz+1.
Noting thatf∈Sb∗(a) and g∈Cb∗(a), we see that (64)RezIα′′f,gzIα′f,gz=αRezf′′zf′z-zg′zgz+1=αRezf′′zf′z-Rezg′zgz+1<αa-12ba+1+a-12ba+1=a-12b/2αa+1.
This means thatIαf,g(z)∈Sb/2α(a).By using the definition of the classSb(a) (5), we obtain the following.Corollary 17.
Iff∈Sb∗(a) and g∈Cb∗(a), for a>1, b>0, and α>0,(65)tf′tgtb/2≺a1-za-z,z∈U.Example 18.
Let us consider the functionsf and g which satisfy (66)tf′tgtα=1-zp-1,where p=21-aα/ba+1+1. We have (67)Iαf,gz=1p1-1-zp,z∈U.Thus, (68)RezIα′′f,gzIα′f,gz=Re1-pz1-z<αa-1ba+1,z∈U.Example 19.
Let us consider the functionsf which satisfy (69)f′zα=1-zp-1,where z∈U and p=21-aα/ba+1+1. We have (70)Iαfz=1p1-1-zp,z∈U.Thus, (71)RezIα′′fzIα′fz=Re1-pz1-z<αa-1ba+1,z∈U.



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