# A Connection between Basic Univalence Criteria

Authors: Horiana Tudor
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We obtain a sufficient condition for the analyticity and the univalence of a class of functions defined by an integral operator. The well-known univalence criteria of Alexander, Noshiro-Warschawski, Nehari, Goluzin, Ozaki-Nunokawa, Becker, and Lewandowski would follow upon specializing the functions and the parameters involved in the main result. The results obtained not only reduce to those earlier works, but they also extend the previous results.


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## Body

## 1. Introduction

Let𝒰r={z∈ℂ:|z|<r},0<r≤1, be the disk of radius r centered at 0, let 𝒰=𝒰1 be the unit disk, and let I=[0,∞).Denote by𝒜 the class of analytic functions in 𝒰 which satisfy the usual normalization f(0)=f′(0)-1=0.The first results concerning univalence criteria are related to the univalence of an analytic function in the unit disk. Among the most important sufficient conditions for univalence we mention those obtained by Alexander [1], Noshiro [2] and Warschawski [3], Nehari [4], Goluzin [5], Ozaki and Nunokawa [6], Becker [7], and Lewandowski [8].Furthermore, the first extension of univalence criteria was obtained by Pascu in [9]. In his paper, starting from an analytic function f in the unit disk he established not only the univalence of f but also the analyticity and the univalence of a whole class of functions defined by an integral operator.Other extensions of the univalence criteria, for an integral operator, were obtained in the papers [10–14]. From the main result of this paper, we found all the univalence criteria mentioned earlier and at the same time other new ones.


## 2. Loewner Chains

Before proving our main result we need a brief summary of theory of Loewner chains.A functionL(z,t):𝒰×I→ℂ is said to be a Loewner chain or a subordination chain if(i)
L(z,t) is analytic and univalent in 𝒰 for all t∈I;(ii)
L(z,t)≺L(z,s) for all 0≤t≤s<∞, where the symbol ‘‘≺’’ stands for subordination.The following result due to Pommerenke is often used to obtain univalence criteria.Theorem 1 (see [15, 16]).
LetL(z,t)=a1(t)z+a2(t)z2+⋯, a1(t)≠0, be an analytic function in 𝒰r for all t∈I, locally absolutely continuous in I, locally uniform with respect to 𝒰r. For almost all t∈I, suppose that
(1)z∂L(z,t)∂z=p(z,t)∂L(z,t)∂t,z∈𝒰r,
where p(z,t) is analytic in 𝒰 and satisfying ℜp(z,t)>0 for all z∈𝒰, t∈I. If limt→∞|a1(t)|=∞ and {L(z,t)/a1(t)}t≥0 forms a normal family in 𝒰r, then for each t∈I, the function L(z,t) has an analytic and univalent extension to the whole disk 𝒰.


## 3. Main Result

Making use of Theorem1, the essence of which is the construction of suitable Loewner chain, we can prove our main result.Theorem 2.
Letα,β, and c be complex numbers such that ℜα>0,|β|<ℜ(α+β),ℜc>-1/2, and
(2)|αc-βα(1+c)|≤1.
For f∈𝒜, if there exist two analytic functions in 𝒰,g(z)=1+b1z+⋯,h(z)=c0+c1z+⋯ such that the inequalities
(3)|f′(z)(1+c)g(z)-1|<1,(4)|(f′(z)(1+c)g(z)-1)|z|2(α+β)n+1-|z|2(α+β)α+β(2α+β1+czf′(z)h(z)g(z)+zg′(z)g(z)-β)n+(1-|z|2(α+β))2(α+β)|z|2(α+β)z2n×(α+β1+cf′(z)h2(z)g(z)+g′(z)h(z)g(z)nnnn+(α-1)h(z)z-h′(z))|≤1
are true for all z∈𝒰∖{0}, then the function Fα,
(5)Fα(z)=(α∫0z‍uα-1f′(u)du)1/α,
is analytic and univalent in 𝒰, where the principal branch is intended.Proof.
We consider the functionρ1(z,t) defined by
(6)ρ1(z,t)=1+(e2(α+β)t-1)·e-tzh(e-tz).
For all t≥0 and z∈𝒰 we have e-tz∈𝒰, and from the analyticity of h in 𝒰 it follows that ρ1(z,t) is also analytic in 𝒰. Since ρ1(0,t)=1, there exists a disk 𝒰r1, 0<r1≤1, in which ρ1(z,t)≠0 for all t≥0. Since f∈𝒜, it is easy to see that the function
(7)ρ2(z,t)=(α+β)∫0e-tz‍uα-1f′(u)du
can be written as ρ2(z,t)=zα·ρ3(z,t), where ρ3(z,t) is analytic in 𝒰r1, for all t≥0, and ρ3(0,t)=((α+β)/α)e-αt. It follows that the function
(8)ρ4(z,t)=ρ3(z,t)+(1+c)(e(α+2β)t-e-αt)g(e-tz)ρ1(z,t)
is also analytic in a disk 𝒰r2,0<r2≤r1, and
(9)ρ4(0,t)=e(α+2β)t[(1+c)+(βα-c)e-2(α+β)t].
Let us prove thatρ4(0,t)≠0 for all t≥0. We have ρ4(0,0)=1+β/α. From |β|<ℜ(α+β) and since ℜ(α+β)≤|α+β|, we see that |β|<|α+β| which is equivalent to Re(β/α)>-1/2. It follows that ρ4(0,0)≠0. Assume now that there exists t0>0 such that ρ4(0,t0)=0. Then e2(α+β)t0=(αc-β)/α(1+c). From ℜ(α+β)>0, t0>0, it results that e2ℜ(α+β)t0>1, and from inequality (2), we conclude that ρ4(0,t)≠0 for all t≥0. Therefore, there is a disk 𝒰r3,0<r3≤r2, in which ρ4(z,t)≠0, for all t≥0, and we can choose an analytic branch of [ρ4(z,t)]1/α, denoted by ρ(z,t). We fix a determination of (1+β/α)1/α, denoted by δ. For δ(t) we fix, for t=0, the determination equal to δ, where
(10)δ(t)=e(1+2(β/α))t[(1+c)+(βα-c)e-2(α+β)t]1/α.
From these considerations it follows that the function L(z,t)=z·ρ(z,t) is analytic in 𝒰r3, for all t≥0, and can be written as follows:
(11)L(z,t)=[(e(α+2β)t-e-αt)zαg(e-tz)1+(e2(α+β)t-1)·e-tzh(e-tz)(α+β)∫0e-tz‍uα-1f′(u)du+(1+c)n×(e(α+2β)t-e-αt)zαg(e-tz)1+(e2(α+β)t-1)·e-tzh(e-tz)]1/α.
If L(z,t)=a1(t)z+a2(t)z2+⋯ is the Taylor expansion of L(z,t) in 𝒰r3, we have a1(t)=δ(t). Since ℜ(α+β)>0,ℜ(β/α)>-1/2, we have limt→∞|a1(t)|=∞. We saw also that a1(t)≠0 for all t∈I.
From the analyticity ofL(z,t) in 𝒰r3, it follows that there exists a number r4,0<r4≤r3, and a constant K=K(r4) such that
(12)|L(z,t)a1(t)|<K,∀z∈𝒰r4,t∈I,
and thus {L(z,t)/a1(t)} is a normal family in 𝒰r4. From the analyticity of ∂L(z,t)/∂t, for all fixed numbers T>0 and r5,0<r5≤r4, there exists a constant K1>0 (that depends on T and r5) such that
(13)|∂L(z,t)∂t|<K1,∀z∈𝒰r5,t∈[0,T].
It follows that the function L(z,t) is locally absolutely continuous in [0,∞), locally uniform with respect to z∈𝒰r5. The function p(z,t) defined by (1) is analytic in a disk 𝒰r,0<r≤r5, for all t≥0. In order to prove that the function p(z,t) is analytic and has positive real part in 𝒰, we will show that the function w(z,t)=(p(z,t)-1)/(p(z,t)+1),z∈𝒰r,t∈I, is analytic in 𝒰, and
(14)|w(z,t)|<1∀z∈𝒰,t∈I.
Elementary calculation gives
(15)w(z,t)=(f′(e-tz)(1+c)g(e-tz)-1)e-2(α+β)t+1-e-2(α+β)tα+β[2α+β1+ce-tzf′(e-tz)h(e-tz)g(e-tz)+1-e-2(α+β)tα+β=+e-tzg′(e-tz)g(e-tz)-β]+(1-e-2(α+β)t)2(α+β)e-2(α+β)te-2tz2×[α+β1+cf′(e-tz)h2(e-tz)g(e-tz)+g′(e-tz)h(e-tz)g(e-tz)==+(α-1)h(e-tz)e-tz-h′(e-tz)].
From (3) and (4) we deduce that g(z)≠0, for all z∈𝒰, and then the function w(z,t) is analytic in the unit disk 𝒰. For t=0, in view of (3), we have
(16)|w(z,0)|=|f′(z)(1+c)g(z)-1|<1.
In order to evaluate |w(0,t)|, we will use the following inequality (see [17]):
(17)|1-|c|2αα|≤1-|c|2ℜαℜα,c∈𝒰,c≠0,ℜα>0.
For z=0 and t>0, from (15), we have
(18)|w(0,t)|=|-c1+ce-2(α+β)t+1-e-2(α+β)tα+β(-β)|≤|c1+c|e-2ℜ(α+β)t+1-e-2ℜ(α+β)tℜ(α+β)|β|.
From ℜc>-1/2 which is equivalent to |c|<|c+1| and since |β|<ℜ(α+β), we have |w(0,t)|<1.
Lett be a fixed number, t>0, and let z∈𝒰,z≠0. Since |e-tz|≤e-t<1 for all z∈𝒰¯={z∈ℂ:|z|≤1}, the function w(z,t) is analytic in 𝒰¯. Using the maximum modulus principle it follows that for each t>0, arbitrary fixed, there exists θ=θ(t)∈ℝ such that
(19)|w(z,t)|<max|ξ|=1|w(ξ,t)|=|w(eiθ,t)|.
Denote that u=e-t·eiθ. Then |u|=e-t<1, and from (15) we obtain
(20)w(eiθ,t)=(f′(u)(1+c)g(u)-1)|u|2(α+β)+1-|u|2(α+β)α+β×(2α+β1+cuf′(u)h(u)g(u)+ug′(u)g(u)-β)+(1-|u|2(α+β))2(α+β)|u|2(α+β)u2×[α+β1+cf′(u)h2(u)g(u)+g′(u)h(u)g(u)==+(α-1)h(u)u-h′(u)].
Since u∈𝒰, inequality (4) implies that |w(eiθ,t)|≤1, and from (16), (18), and (19) we conclude that inequality (14) holds true for all z∈𝒰 and t≥0. Since all the conditions of Theorem 1 are satisfied, it follows that L(z,t) is a Loewner chain, for each t≥0. For t=0 it results that the function
(21)L(z,0)=[(α+β)∫0z‍uα-1f′(u)du]1/α
is analytic and univalent in 𝒰, and then the function Fα defined by (5) is analytic and univalent in 𝒰.


## 4. Specific Cases and Examples

Suitable choices of the functionsg and h and special values of the parameter c yield various types of univalence criteria. So, if in Theorem 2 we take c=0 and h(z)≡0, we get the following result.Theorem 3.
Letα and β be complex numbers such that ℜα>0,|β|<ℜ(α+β), and |β|≤|α|. For f∈𝒜, if there exists an analytic function in 𝒰,g(z)=1+b1z+⋯, such that the inequalities
(22)|f′(z)g(z)-1|<1,|(f′(z)g(z)-1)|z|2(α+β)+1-|z|2(α+β)α+β(zg′(z)g(z)-β)|≤1
are true for all z∈𝒰∖{0}, then the function Fα defined by (5) is analytic and univalent in 𝒰, where the principal branch is intended.Theorem3 gives us a ‘‘continuous’’ passage from Becker’s criterion to Lewandowski’s criterion. Indeed, for g(z)≡f′(z), we have the following.Corollary 4.
Letα and β be complex numbers, ℜα>0, |β|<ℜ(α+β), |β|≤|α|, and f∈𝒜. If for all z∈𝒰∖{0}(23)|1-|z|2(α+β)α+β(zf′′(z)f′(z)-β)|≤1,
then the function Fα defined by (5) is analytic and univalent in 𝒰.Remark 5.
Corollary4 generalizes the well-known univalence criterion due to Becker. For β=0 we found the result from [9]. In the case when β=0 and α=1, the previous corollary reduces to Becker’s criterion [7].Forg(z)≡f′(z)·(p(z)+1)/2, where p is analytic in 𝒰, p(0)=1, from Theorem 3 we have the following.Corollary 6.
Letα and β be complex numbers, ℜα>0, |β|<ℜ(α+β), |β|≤|α|, and f∈𝒜. If there exists an analytic function p with positive real part in 𝒰, p(0)=1, such that the inequality
(24)|1-p(z)1+p(z)|z|2(α+β)+1-|z|2(α+β)α+βn×(zf′′(z)f′(z)+zp′(z)p(z)+1-β)|≤1
is true for all z∈𝒰∖{0}, then the function Fα defined by (5) is analytic and univalent in 𝒰.Remark 7.
Corollary6 represents a generalization of the univalence criterion due to Lewandowski. For β=0 we found the result from [12]. In the case when β=0 and α=1, the previous corollary reduces to Lewandowski’s criterion [8].Forc=β and h(z)≡0, from Theorem 2 we can derived some results from paper [18].Theorem 8.
Letα and β be complex numbers such that ℜα≥1/2, |β|<ℜ(α+β). For f∈𝒜, if there exists an analytic function in 𝒰, g(z)=1+b1z+⋯, such that the inequalities
(25)|f′(z)(1+β)g(z)-1|<1,(26)|(f′(z)(1+β)g(z)-1)|z|2(α+β)=+1-|z|2(α+β)α+β(zg′(z)g(z)-β)|≤1
are true for all z∈𝒰∖{0}, then the function Fα defined by (5) is analytic and univalent in 𝒰.Proof.
In view of assumptionℜα≥1/2 and since ℜ(α+β)>0, it follows that ℜβ>-1/2. But ℜα≥1/2 is equivalent to |α-1|≤|α| and ℜβ>-1/2 with |β|<|β+1|. It results that inequality (2) is true. From (3) and (4) we get immediately inequalities (25) and (26).Forα=1 and g(z)≡f(z)/z, from Theorem 8 we obtain the following.Corollary 9.
Letβ be a complex number, |β|<ℜ(1+β). If for all z∈𝒰 the function f∈𝒜 satisfies
(27)|zf′(z)f(z)-(β+1)|<|β+1|,
then the function f is univalent in 𝒰. Moreover, it is a spiral-like function.Proof.
Forα=1, we have F1=f, and in view of (27), inequality (25) of Theorem 8 is verified and inequality (26) is also reduced to (25). It follows that f is univalent in 𝒰. The condition (27) of the corollary can be written as |(1/(β+1))(zf′(z)/f(z))-1|<1. It follows that ℜ((1/(β+1))(zf′(z)/f(z)))>0. If we put β+1=|β+1|eıφ, where from ℜ(1+β)>0 we have |φ|<π/2, then for all z∈𝒰 we have ℜ(e-ıφ(zf′(z)/f(z)))>0, which shows that f is spiral-like in 𝒰.Takingg(z)≡f(z)/z, we get the following useful corollary which generalizes the result from [19].Corollary 10.
Letα and β be complex numbers such that ℜα≥1/2, |β+1|≤ℜ(α+β), and f∈𝒜. If the inequality
(28)|zf′(z)f(z)-(β+1)|<|β+1|
holds true for all z∈𝒰, then the function Fα defined by (5) is analytic and univalent in 𝒰.Proof.
It is easy to check that inequality (28) implies inequality (26) of Theorem 8. Indeed, for |β+1|≤ℜ(α+β) and making use of (17), we have
(29)|1β+1(zf′(z)f(z)-(β+1))|z|2(α+β)=+1-|z|2(α+β)α+β(zf′(z)f(z)-(β+1))|≤|z|2ℜ(α+β)+|β+1|ℜ(α+β)(1-|z|2ℜ(α+β))≤1.For the functiong(z)≡1, from Theorem 8 we get the following.Corollary 11.
Letα and β be complex numbers such that ℜα≥1/2, |β|<ℜ(α+β), and f∈𝒜. If the inequality
(30)|f′(z)-(β+1)|<|β+1|
holds true for all z∈𝒰, then the function Fα defined by (5) is analytic and univalent in 𝒰. In particular, the function f is univalent in 𝒰, where |β|<ℜ(1+β).Remark 12.
From inequality (30), only for β real number, β>-1/2, we get ℜf′(z)>0. For β complex number, if we put β+1=|β+1|eıφ, where from ℜ(1+β)>0 we have |φ|<π/2, then from inequality (28) we obtain ℜe-ıφf′(z)>0. So, in both cases, we can also conclude that f is univalent in 𝒰 from Alexander’s theorem [1], and respectively, from Noshiro-Warschawski’s theorem [2, 3].Example 1.
Consider the functionf(z)=z+(β/4)z2+(β/6)z3, where β∈ℂ, |β-1/3|≤2/3. The condition (30) of Corollary 11 is satisfied. Indeed, since |β-1/3|≤2/3 is equivalent with 2|β|≤|β+1|, we get
(31)|f′(z)-(β+1)|=|β2z+β2z2-β|<2|β|≤|β+1|.
Then, for all complex numbers α, ℜα≥1/2, and |β|<ℜ(α+β), by using (5), we obtain that
(32)Fα(z)=(α∫0z‍uα-1f′(u)du)1/α=z(1+αβ2(α+1)z+αβ2(α+2)z2)1/α
is analytic and univalent in 𝒰.If in Theorem2 we take c=0 and g(z)≡f′(z), then we have the following.Theorem 13.
Letα and β be complex numbers such that ℜα>0, |β|<ℜ(α+β), and |β|≤|α|. For f∈𝒜, if there exists an analytic function in 𝒰, h(z)=c0+c1z+⋯, such that the inequality
(33)|1-|z|2(α+β)α+β(2(α+β)zh(z)+zf′′(z)f′(z)-β)=+(1-|z|2(α+β))2(α+β)|z|2(α+β)·z2=×((α+β)h2(z)+f′′(z)h(z)f′(z)+(α-1)h(z)z-h′(z))|≤1
is true for all z∈𝒰∖{0}, then the function Fα defined by (5) is analytic and univalent in 𝒰.Forh(z)≡-(1/2(α+β))(f′′(z)/f′(z)) the following results.Corollary 14.
Letα and β be complex numbers, ℜα>0, |β|<ℜ(α+β), |β|≤|α|, and f∈𝒜. If for all z∈𝒰∖{0}(34)|(1-|z|2(α+β))22(α+β)2|z|2(α+β)(z2{f;z}+(1-α)zf′′(z)f′(z))=-βα+β(1-|z|2(α+β))(1-|z|2(α+β))22(α+β)2|z|2(α+β)(z2{f;z}+(1-α)zf′′(z)f′(z))|≤1,
where
(35){f;z}=(f′′(z)f′(z))′-12(f′′(z)f′(z))2,
then the function Fα defined by (5) is analytic and univalent in 𝒰.Remark 15.
For special values of the parametersα and β, from Corollary 14 we get some known results. For β=0, we get the result given in [13]. For α=1, since F1(z)=f(z), Corollary 14 generalizes the criterion of univalence due to Nehari, and for α=1 and β=0 we obtain the univalence criterion due to Nehari [4].Forh(z)≡(1/(α+β))(1/z-f′(z)/f(z)) we have the following.Corollary 16.
Letα and β be complex numbers, ℜα>0, |β|<ℜ(α+β), |β|≤|α|, and f∈𝒜. If for all z∈𝒰∖{0}(36)|+(1-|z|2(α+β))2(α+β)2|z|2(α+β)·zddzlog(z1+αf′(z)f1+α(z))1-|z|2(α+β)α+β[zddzlog(z2f′(z)f2(z))-β]=+(1-|z|2(α+β))2(α+β)2|z|2(α+β)·zddzlog(z1+αf′(z)f1+α(z))|≤1,
then the function Fα defined by (5) is analytic and univalent in 𝒰.Remark 17.
Corollary16 represents a generalization of the univalence criterion due to Goluzin. For β=0 we obtain the results from paper [11]. For α=1 and β=0 we get Goluzin’s criterion [5].Forc=0,g(z)≡(f(z)/z)2, and h(z)≡(1/(α+β))(1/z-f(z)/z2), from Theorem 2 we get the following.Corollary 18.
Letα and β be complex numbers, ℜα>0, |β|<ℜ(α+β), |β|≤|α|, and f∈𝒜. If f satisfies the inequalities
(37)|z2f′(z)f2(z)-1|<1,(38)|(z2f′(z)f2(z)-1)|z|2(α+β)+1-|z|2(α+β)α+β=×[2(z2f′(z)f2(z)-1)-β]+(1-|z|2(α+β))2(α+β)2|z|2(α+β)=×[(z2f′(z)f2(z)-1)+(1-α)(f(z)z-1)]|≤1,
then the function Fα defined by (5) is analytic and univalent in 𝒰.Remark 19.
Corollary18 represents a generalization of the univalence criterion due to Ozaki and Nunokawa. For β=0 we found the result from [14]. In the case when β=0 and α=1, Corollary 18 reduces to the univalence criterion of Ozaki and Nunokawa [6].Example 2.
Letn be a natural number, n≥3. We consider the function
(39)f(z)=z1-zn+1/n.
We note that
(40)z2f′(z)f2(z)-1=zn+1,f(z)z-1=zn+1n-zn+1.
The condition (37) of Corollary 18 is verified, and it assures the univalence of the function f. Taking into account (40), for α=n and β=(1-n)/2, from (38) we have that
(41)|+4(1-|z|n+1)2(n+1)2|z|n+1(zn+1+(1-n)zn+1n-zn+1)zn+1|z|n+1+21-|z|n+1n+1(2zn+1+n-12)=+4(1-|z|n+1)2(n+1)2|z|n+1(zn+1+(1-n)zn+1n-zn+1)|≤|z|2(n+1)+1-|z|n+1n+1(4|z|n+1+n-1)=+8(n+1)2(1-|z|n+1)2=1(n+1)2[(n2-2n+5)|z|2(n+1)========-(n2-4n+11)|z|n+1+(n2+7)]≤1,
because the greatest value of the function
(42)g(x)=(n2-2n+5)x2-(n2-4n+11)x+(n2+7),
for x∈[0,1],n≥3, is taken for x=1 and is g(1)=(n+1)2. It follows that all the conditions of Corollary 18 are satisfied, and therefore the function Fn defined by (5) is analytic and univalent in 𝒰.Remark 20.
Theorem2 gives us a connection between Alexander’s theorem, Noshiro-Warschawski’s theorem, and the univalence criteria of Becker, Lewandowski, Nehari, Goluzin, and Ozaki and Nunokawa as well as their generalizations.



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