A Connection between Basic Univalence Criteria

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.

1. Introduction

Let 𝒰_𝑟={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 by 𝒜 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 function 𝐿(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]). Let 𝐿(𝑧,𝑡)=𝑎₁(𝑡)𝑧+𝑎₂(𝑡)𝑧²+⋯, 𝑎₁(𝑡)≠0, be an analytic function in 𝒰_𝑟 for all 𝑡∈𝐼, locally absolutely continuous in 𝐼, locally uniform with respect to 𝒰_𝑟. For almost all 𝑡∈𝐼, suppose that 𝑧𝜕𝐿(𝑧,𝑡)()()𝜕𝑧=𝑝𝑧,𝑡𝜕𝐿𝑧,𝑡𝜕𝑡,𝑧∈𝒰_𝑟, (1) where 𝑝(𝑧,𝑡) is analytic in 𝒰 and satisfying ℜ𝑝(𝑧,𝑡)>0 for all 𝑧∈𝒰, 𝑡∈𝐼. If lim_𝑡→∞|𝑎₁(𝑡)|=∞ and {𝐿(𝑧,𝑡)/𝑎₁(𝑡)}_𝑡≥0 forms a normal family in 𝒰_𝑟, 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 Theorem 1, the essence of which is the construction of suitable Loewner chain, we can prove our main result.

Theorem 2. Let 𝛼,𝛽, and 𝑐 be complex numbers such that ℜ𝛼>0,|𝛽|<ℜ(𝛼+𝛽),ℜ𝑐>−1/2, and 𝛼𝑐−𝛽𝛼(1+𝑐)≤1. (2) For 𝑓∈𝒜, if there exist two analytic functions in 𝒰,𝑔(𝑧)=1+𝑏₁𝑧+⋯,ℎ(𝑧)=𝑐₀+𝑐₁𝑧+⋯ such that the inequalities 𝑓^′(𝑧)()𝑔(𝑧)𝑓1+𝑐−1<1,_′(𝑧)()𝑔(𝑧)|𝑧|1+𝑐−1^2(𝛼+𝛽)+1−|𝑧|^2(𝛼+𝛽)2𝛼+𝛽𝛼+𝛽1+𝑐𝑧𝑓^′(𝑧)ℎ(𝑧)𝑔(𝑧)+𝑧𝑔^′(𝑧)𝑔(𝑧)+|𝑧|−𝛽1−^{2(𝛼+𝛽)2}𝛼+𝛽|𝑧|^2(𝛼+𝛽)𝑧²×𝛼+𝛽𝑓1+𝑐^′(𝑧)ℎ²(𝑧)𝑔(𝑧)+𝑔^′(𝑧)ℎ(𝑧)𝑔(𝑧)+()ℎ(𝑧)𝛼−1𝑧−ℎ^′(𝑧)≤1 (3)(4) are true for all 𝑧∈𝒰⧵{0}, then the function 𝐹_𝛼, 𝐹_𝛼(𝑧)=𝛼^𝑧₀𝑢^𝛼−1𝑓^′(𝑢)𝑑𝑢^1/𝛼, (5) is analytic and univalent in 𝒰, where the principal branch is intended.

Proof. We consider the function 𝜌₁(𝑧,𝑡) defined by 𝜌₁(𝑧,𝑡)=1+𝑒^{2(𝛼+𝛽)𝑡}−1⋅𝑒^−𝑡𝑧ℎ𝑒^−𝑡𝑧. (6) 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 𝒰_𝑟1, 0<𝑟₁≤1, in which 𝜌₁(𝑧,𝑡)≠0 for all 𝑡≥0. Since 𝑓∈𝒜, it is easy to see that the function 𝜌₂(𝑧,𝑡)=𝛼+𝛽^{𝑒−𝑡𝑧}₀𝑢^𝛼−1𝑓^′(𝑢)𝑑𝑢 (7) can be written as 𝜌₂(𝑧,𝑡)=𝑧^𝛼⋅𝜌₃(𝑧,𝑡), where 𝜌₃(𝑧,𝑡) is analytic in 𝒰_𝑟1, for all 𝑡≥0, and 𝜌₃(0,𝑡)=((𝛼+𝛽)/𝛼)𝑒^−𝛼𝑡. It follows that the function 𝜌₄(𝑧,𝑡)=𝜌₃(𝑧,𝑡)+(1+𝑐)𝑒^{(𝛼+2𝛽)𝑡}−𝑒^−𝛼𝑡𝑔𝑒^−𝑡𝑧𝜌₁(𝑧,𝑡) (8) is also analytic in a disk 𝒰_𝑟2,0<𝑟₂≤𝑟₁, and 𝜌₄(0,𝑡)=𝑒^{(𝛼+2𝛽)𝑡})+𝛽(1+𝑐𝛼−𝑐𝑒^{−2(𝛼+𝛽)𝑡}. (9)
Let us prove that 𝜌₄(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 𝑒^{2(𝛼+𝛽)𝑡0}=(𝛼𝑐−𝛽)/𝛼(1+𝑐). From ℜ(𝛼+𝛽)>0, 𝑡₀>0, it results that 𝑒^{2ℜ(𝛼+𝛽)𝑡0}>1, and from inequality (2), we conclude that 𝜌₄(0,𝑡)≠0 for all 𝑡≥0. Therefore, there is a disk 𝒰_𝑟3,0<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 𝛿(𝑡)=𝑒^{(1+2(𝛽/𝛼))𝑡})+𝛽(1+𝑐𝛼−𝑐𝑒^{−2(𝛼+𝛽)𝑡1/𝛼}. (10) From these considerations it follows that the function 𝐿(𝑧,𝑡)=𝑧⋅𝜌(𝑧,𝑡) is analytic in 𝒰_𝑟3, for all 𝑡≥0, and can be written as follows: 𝐿(𝑧,𝑡)=𝛼+𝛽^{𝑒−𝑡𝑧}₀𝑢^𝛼−1𝑓^′(𝑢)()×𝑒𝑑𝑢+1+𝑐_{(𝛼+2𝛽)𝑡}−𝑒^−𝛼𝑡𝑧^𝛼𝑔𝑒^−𝑡𝑧1+𝑒^{2(𝛼+𝛽)𝑡}−1⋅𝑒^−𝑡𝑧ℎ(𝑒^−𝑡𝑧)^1/𝛼. (11) If 𝐿(𝑧,𝑡)=𝑎₁(𝑡)𝑧+𝑎₂(𝑡)𝑧²+⋯ is the Taylor expansion of 𝐿(𝑧,𝑡) in 𝒰_𝑟3, we have 𝑎₁(𝑡)=𝛿(𝑡). Since ℜ(𝛼+𝛽)>0,ℜ(𝛽/𝛼)>−1/2, we have lim_𝑡→∞|𝑎₁(𝑡)|=∞. We saw also that 𝑎₁(𝑡)≠0 for all 𝑡∈𝐼.
From the analyticity of 𝐿(𝑧,𝑡) in 𝒰_𝑟3, it follows that there exists a number 𝑟₄,0<𝑟₄≤𝑟₃, and a constant 𝐾=𝐾(𝑟₄) such that 𝐿(𝑧,𝑡)𝑎₁(𝑡)<𝐾,∀𝑧∈𝒰_𝑟4,𝑡∈𝐼, (12) and thus {𝐿(𝑧,𝑡)/𝑎₁(𝑡)} is a normal family in 𝒰_𝑟4. 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 𝜕𝐿(𝑧,𝑡)𝜕𝑡<𝐾₁,∀𝑧∈𝒰_𝑟5,𝑡∈[0,𝑇]. (13) It follows that the function 𝐿(𝑧,𝑡) is locally absolutely continuous in [0,∞), locally uniform with respect to 𝑧∈𝒰_𝑟5. The function 𝑝(𝑧,𝑡) defined by (1) is analytic in a disk 𝒰_𝑟,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∈𝒰_𝑟,𝑡∈𝐼, is analytic in 𝒰, and |𝑤(𝑧,𝑡)|<1∀𝑧∈𝒰,𝑡∈𝐼. (14) Elementary calculation gives 𝑤()=𝑓𝑧,𝑡^′𝑒^−𝑡𝑧(1+𝑐)𝑔(𝑒^−𝑡𝑧)−1𝑒^{−2(𝛼+𝛽)𝑡}+1−𝑒^{−2(𝛼+𝛽)𝑡}2𝛼+𝛽𝛼+𝛽𝑒1+𝑐^−𝑡𝑧𝑓^′𝑒^−𝑡𝑧ℎ𝑒^−𝑡𝑧𝑔(𝑒^−𝑡𝑧)+𝑒^−𝑡𝑧𝑔^′𝑒^−𝑡𝑧𝑔(𝑒^−𝑡𝑧)+−𝛽1−𝑒^{−2(𝛼+𝛽)𝑡2}𝛼+𝛽𝑒^{−2(𝛼+𝛽)𝑡}𝑒^−2𝑡𝑧²×𝛼+𝛽𝑓1+𝑐^′𝑒^−𝑡𝑧ℎ²𝑒^−𝑡𝑧𝑔(𝑒^−𝑡𝑧)+𝑔^′𝑒^−𝑡𝑧ℎ𝑒^−𝑡𝑧𝑔(𝑒^−𝑡𝑧)+()ℎ𝑒𝛼−1^−𝑡𝑧𝑒^−𝑡𝑧−ℎ^′𝑒^−𝑡𝑧. (15) 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 |𝑤(=𝑓𝑧,0)|^′(𝑧)(1+𝑐)𝑔(𝑧)−1<1. (16) In order to evaluate |𝑤(0,𝑡)|, we will use the following inequality (see [17]): 1−|𝑐|^2𝛼𝛼≤1−|𝑐|^2ℜ𝛼ℜ𝛼,𝑐∈𝒰,𝑐≠0,ℜ𝛼>0. (17) For 𝑧=0 and 𝑡>0, from (15), we have |𝑤(0,𝑡)|=−𝑐𝑒1+𝑐^{−2(𝛼+𝛽)𝑡}+1−𝑒^{−2(𝛼+𝛽)𝑡}≤𝑐𝛼+𝛽−𝛽𝑒1+𝑐^{−2ℜ(𝛼+𝛽)𝑡}+1−𝑒^{−2ℜ(𝛼+𝛽)𝑡}ℜ𝛽.𝛼+𝛽 (18) From ℜ𝑐>−1/2 which is equivalent to |𝑐|<|𝑐+1| and since |𝛽|<ℜ(𝛼+𝛽), we have |𝑤(0,𝑡)|<1.
Let 𝑡 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 |𝑤(𝑧,𝑡)|<max_𝜉=1𝑤(𝜉,𝑡)=𝑤𝑒^𝑖𝜃,𝑡. (19) Denote that 𝑢=𝑒^−𝑡⋅𝑒^𝑖𝜃. Then |𝑢|=𝑒^−𝑡<1, and from (15) we obtain 𝑤𝑒^𝑖𝜃=𝑓,𝑡^′(𝑢)()𝑔(𝑢)|𝑢|1+𝑐−1^2(𝛼+𝛽)+1−|𝑢|^2(𝛼+𝛽)×2𝛼+𝛽𝛼+𝛽1+𝑐𝑢𝑓^′(𝑢)ℎ(𝑢)𝑔(𝑢)+𝑢𝑔^′(𝑢)𝑔(𝑢)+|𝑢|−𝛽1−^{2(𝛼+𝛽)2}𝛼+𝛽|𝑢|^2(𝛼+𝛽)𝑢²×𝛼+𝛽𝑓1+𝑐^′(𝑢)ℎ²(𝑢)𝑔(𝑢)+𝑔^′(𝑢)ℎ(𝑢)𝑔(𝑢)+()ℎ(𝑢)𝛼−1𝑢−ℎ^′(𝑢). (20) Since 𝑢∈𝒰, inequality (4) implies that |𝑤(𝑒^𝑖𝜃,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 𝐿(𝑧,0)=𝛼+𝛽^𝑧₀𝑢^𝛼−1𝑓^′(𝑢)𝑑𝑢^1/𝛼 (21) is analytic and univalent in 𝒰, and then the function 𝐹_𝛼 defined by (5) is analytic and univalent in 𝒰.

4. Specific Cases and Examples

Suitable choices of the functions 𝑔 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. Let 𝛼 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 𝑓^′(𝑧)𝑔(𝑧)𝑓−1<1,_′(𝑧)𝑔(𝑧)−1|𝑧|^2(𝛼+𝛽)+1−|𝑧|^2(𝛼+𝛽)𝛼+𝛽𝑧𝑔^′(𝑧)𝑔(𝑧)−𝛽≤1 (22) are true for all 𝑧∈𝒰⧵{0}, then the function 𝐹_𝛼 defined by (5) is analytic and univalent in 𝒰, where the principal branch is intended.

Theorem 3 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. Let 𝛼 and 𝛽 be complex numbers, ℜ𝛼>0, |𝛽|<ℜ(𝛼+𝛽), |𝛽|≤|𝛼|, and 𝑓∈𝒜. If for all 𝑧∈𝒰⧵{0}1−|𝑧|^2(𝛼+𝛽)𝛼+𝛽𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝛽≤1, (23) then the function 𝐹_𝛼 defined by (5) is analytic and univalent in 𝒰.

Remark 5. Corollary 4 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].

For 𝑔(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. Let 𝛼 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 1−𝑝(𝑧)(𝑧)|𝑧|1+𝑝^2(𝛼+𝛽)+1−|𝑧|^2(𝛼+𝛽)×𝛼+𝛽𝑧𝑓^′′(𝑧)𝑓^′(𝑧)+𝑧𝑝^′(𝑧)𝑝(𝑧)+1−𝛽≤1 (24) is true for all 𝑧∈𝒰⧵{0}, then the function 𝐹_𝛼 defined by (5) is analytic and univalent in 𝒰.

Remark 7. Corollary 6 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].

For 𝑐=𝛽 and ℎ(𝑧)≡0, from Theorem 2 we can derived some results from paper [18].

Theorem 8. Let 𝛼 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 𝑓^′(𝑧)𝑔(𝑧)𝑓1+𝛽−1<1,_′(𝑧)𝑔(𝑧)|𝑧|1+𝛽−1^2(𝛼+𝛽)+1−|𝑧|^2(𝛼+𝛽)𝛼+𝛽𝑧𝑔^′(𝑧)𝑔(𝑧)−𝛽≤1 (25)(26) are true for all 𝑧∈𝒰⧵{0}, then the function 𝐹_𝛼 defined by (5) is analytic and univalent in 𝒰.

Proof. In view of assumption ℜ𝛼≥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).

For 𝛼=1 and 𝑔(𝑧)≡𝑓(𝑧)/𝑧, from Theorem 8 we obtain the following.

Corollary 9. Let 𝛽 be a complex number, |𝛽|<ℜ(1+𝛽). If for all 𝑧∈𝒰 the function 𝑓∈𝒜 satisfies 𝑧𝑓^′(𝑧)𝑓(𝑧)−𝛽+1<𝛽+1, (27) then the function 𝑓 is univalent in 𝒰. Moreover, it is a spiral-like function.

Proof. For 𝛼=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|𝑒^𝚤𝜑, 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.

Taking 𝑔(𝑧)≡𝑓(𝑧)/𝑧, we get the following useful corollary which generalizes the result from [19].

Corollary 10. Let 𝛼 and 𝛽 be complex numbers such that ℜ𝛼≥1/2, |𝛽+1|≤ℜ(𝛼+𝛽), and 𝑓∈𝒜. If the inequality 𝑧𝑓^′(𝑧)𝑓(𝑧)−𝛽+1<𝛽+1 (28) holds true for all 𝑧∈𝒰, then the function 𝐹_𝛼 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 1𝛽+1𝑧𝑓^′(𝑧)𝑓(𝑧)−𝛽+1|𝑧|^2(𝛼+𝛽)+1−|𝑧|^2(𝛼+𝛽)𝛼+𝛽𝑧𝑓^′(𝑧)𝑓(𝑧)−≤|𝑧|𝛽+1_{2ℜ(𝛼+𝛽)}+𝛽+1ℜ|𝑧|𝛼+𝛽1−^{2ℜ(𝛼+𝛽)}≤1. (29)

For the function 𝑔(𝑧)≡1, from Theorem 8 we get the following.

Corollary 11. Let 𝛼 and 𝛽 be complex numbers such that ℜ𝛼≥1/2, |𝛽|<ℜ(𝛼+𝛽), and 𝑓∈𝒜. If the inequality 𝑓^′(𝑧)−𝛽+1<𝛽+1 (30) holds true for all 𝑧∈𝒰, then the function 𝐹_𝛼 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|𝑒^𝚤𝜑, 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 function 𝑓(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 𝑓^′(𝑧)−=𝛽𝛽+12𝛽𝑧+2𝑧²−𝛽<2𝛽≤𝛽+1. (31) Then, for all complex numbers 𝛼, ℜ𝛼≥1/2, and |𝛽|<ℜ(𝛼+𝛽), by using (5), we obtain that 𝐹_𝛼(𝑧)=𝛼^𝑧₀𝑢^𝛼−1𝑓^′(𝑢)𝑑𝑢^1/𝛼=𝑧1+𝛼𝛽2(𝛼+1)𝑧+𝛼𝛽2()𝑧𝛼+2^21/𝛼 (32) is analytic and univalent in 𝒰.

If in Theorem 2 we take 𝑐=0 and 𝑔(𝑧)≡𝑓^′(𝑧), then we have the following.

Theorem 13. Let 𝛼 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 1−|𝑧|^2(𝛼+𝛽)2(𝑧)+𝛼+𝛽𝛼+𝛽𝑧ℎ𝑧𝑓^′′(𝑧)𝑓^′(𝑧)+|𝑧|−𝛽1−^{2(𝛼+𝛽)2}𝛼+𝛽|𝑧|^2(𝛼+𝛽)⋅𝑧²×𝛼+𝛽ℎ²(𝑧)+𝑓^′′(𝑧)ℎ(𝑧)𝑓^′(𝑧)+()ℎ(𝑧)𝛼−1𝑧−ℎ^′(𝑧)≤1 (33) is true for all 𝑧∈𝒰⧵{0}, then the function 𝐹_𝛼 defined by (5) is analytic and univalent in 𝒰.

For ℎ(𝑧)≡−(1/2(𝛼+𝛽))(𝑓^′′(𝑧)/𝑓^′(𝑧)) the following results.

Corollary 14. Let 𝛼 and 𝛽 be complex numbers, ℜ𝛼>0, |𝛽|<ℜ(𝛼+𝛽), |𝛽|≤|𝛼|, and 𝑓∈𝒜. If for all 𝑧∈𝒰⧵{0}1−|𝑧|^{2(𝛼+𝛽)2}2𝛼+𝛽²|𝑧|^2(𝛼+𝛽)𝑧²𝑓;𝑧+(1−𝛼)𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝛽|𝑧|𝛼+𝛽1−^2(𝛼+𝛽)≤1, (34) where =𝑓𝑓;𝑧^′′(𝑧)𝑓^′(𝑧)^′−12𝑓^′′(𝑧)𝑓^′(𝑧)², (35) then the function 𝐹_𝛼 defined by (5) is analytic and univalent in 𝒰.

Remark 15. For special values of the parameters 𝛼 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].

For ℎ(𝑧)≡(1/(𝛼+𝛽))(1/𝑧−𝑓^′(𝑧)/𝑓(𝑧)) we have the following.

Corollary 16. Let 𝛼 and 𝛽 be complex numbers, ℜ𝛼>0, |𝛽|<ℜ(𝛼+𝛽), |𝛽|≤|𝛼|, and 𝑓∈𝒜. If for all 𝑧∈𝒰⧵{0}1−|𝑧|^2(𝛼+𝛽)𝑧𝑑𝛼+𝛽𝑧𝑑𝑧log²𝑓^′(𝑧)𝑓²(𝑧)+|𝑧|−𝛽1−^{2(𝛼+𝛽)2}𝛼+𝛽²|𝑧|^2(𝛼+𝛽)𝑑⋅𝑧𝑧𝑑𝑧log^1+𝛼𝑓^′(𝑧)𝑓^1+𝛼(𝑧)≤1, (36) then the function 𝐹_𝛼 defined by (5) is analytic and univalent in 𝒰.

Remark 17. Corollary 16 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].

For 𝑐=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. Let 𝛼 and 𝛽 be complex numbers, ℜ𝛼>0, |𝛽|<ℜ(𝛼+𝛽), |𝛽|≤|𝛼|, and 𝑓∈𝒜. If 𝑓 satisfies the inequalities 𝑧²𝑓^′(𝑧)𝑓²(𝑧)𝑧−1<1,₂𝑓^′(𝑧)𝑓²(𝑧)−1|𝑧|^2(𝛼+𝛽)+1−|𝑧|^2(𝛼+𝛽)×2𝑧𝛼+𝛽₂𝑓^′(𝑧)𝑓²(𝑧)+|𝑧|−1−𝛽1−^{2(𝛼+𝛽)2}𝛼+𝛽²|𝑧|^2(𝛼+𝛽)×𝑧²𝑓^′(𝑧)𝑓²(𝑧)+()𝑓(𝑧)−11−𝛼𝑧−1≤1, (37)(38) then the function 𝐹_𝛼 defined by (5) is analytic and univalent in 𝒰.

Remark 19. Corollary 18 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. Let 𝑛 be a natural number, 𝑛≥3. We consider the function 𝑓(𝑧)=𝑧1−𝑧^𝑛+1./𝑛 (39) We note that 𝑧²𝑓^′(𝑧)𝑓²(𝑧)−1=𝑧^𝑛+1,𝑓(𝑧)𝑧𝑧−1=^𝑛+1𝑛−𝑧^𝑛+1. (40) 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 𝑧^𝑛+1|𝑧|^𝑛+1|𝑧|+21−^𝑛+1𝑛+12𝑧^𝑛+1+𝑛−12|𝑧|+41−^𝑛+12(𝑛+1)²|𝑧|^𝑛+1𝑧^𝑛+1+()𝑧1−𝑛^𝑛+1𝑛−𝑧^𝑛+1≤|𝑧|^2(𝑛+1)+1−|𝑧|^𝑛+14|𝑧|𝑛+1^𝑛+1+8+𝑛−1(𝑛+1)²1−|𝑧|^𝑛+12=1(𝑛+1)²𝑛²−2𝑛+5|𝑧|^2(𝑛+1)−𝑛²−4𝑛+11|𝑧|^𝑛+1+𝑛²+7≤1, (41) because the greatest value of the function 𝑔(𝑥)=𝑛²−2𝑛+5𝑥²−𝑛²−4𝑛+11𝑥+𝑛²+7, (42) 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 𝐹_𝑛 defined by (5) is analytic and univalent in 𝒰.

Remark 20. Theorem 2 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.