Kudriasov Type Univalence Criteria for Some Integral Operators

Abstract

We consider some integral operators defined by analytic functions in the open unit disk and derive new univalence criteria for these operators, using Kudriasov condition for a function to be univalent.

1. Introduction

Let 𝒜 be the class of functions 𝑓 which are analytic in the open unit disk 𝑈={𝑧∈ℂ∶|𝑧|<1} of the form 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛, (1) normalized by 𝑓(0)=𝑓^′(0)−1=0.

We denote by 𝒮 the subclass of 𝒜 consisting of functions 𝑓∈𝒜, which are univalent in 𝑈.

We consider the integral operators 𝐻_𝑛(𝑧)=𝛽^𝑧₀𝑢^𝛽−1𝑓₁(𝑢)𝑢^𝛾1⋯𝑓_𝑛(𝑢)𝑢^𝛾𝑛𝑑𝑢^1/𝛽,𝑇_𝑛(𝑧)=𝛽^𝑧₀𝑢^𝛽−1𝑓^′₁(𝑢)^𝛾1⋯𝑓^′_𝑛(𝑢)^𝛾𝑛𝑑𝑢^1/𝛽, (2) for 𝛽, 𝛾_𝑗 being complex numbers, 𝛽≠0, 𝑗=1,𝑛, and the functions 𝑓_𝑗∈𝒜, 𝑗=1,𝑛.

Some univalence criteria for these integral operators were studied in [1]. Applying univalence conditions given by Kudriasov [2] and Pascu [3], we obtain new Kudriasov type univalence criteria for these two integral operators.

2. Preliminaries

Various generalizations of Becker’s univalence criteria for analytic functions given in [4] were obtained by many authors. For example, the result obtained by Pascu in [3] is also known as an improvement of Becker’s univalence criteria. This result or other similar generalizations of Becker’s univalence criteria have been used further to derive new univalence criteria for integral operators (see, e.g., some relatively recent works as [1, 5, 6]). In this paper we use Pascu improvement of Becker’s univalence criteria and also another univalence condition for a function to be univalent, given by Kudriasov in [2]. There are also some papers devoted to univalence criteria that use some Kudriasov type conditions (see, e.g., the work [1] containing a chapter dedicated to Kudriasov type univalence conditions and other papers as, e.g., [7–10]).

The following univalence criteria are given by Kudriasov for a regular function.

Lemma 1 (see [2]). Let 𝑓 be a regular function in 𝑈, 𝑓(𝑧)=𝑧+𝑎₂𝑧²+⋯. If 𝑓^′′(𝑧)𝑓^′(𝑧)≤𝐾,𝑧∈𝑈, (3) for all 𝑧∈𝑈, where 𝐾≅3.05, the function 𝑓 is univalent in 𝑈.

Remark 2. The constant 𝐾 is a solution of the equation 8[𝑥(𝑥−2)³]^1/2−3(4−𝑥)²=12. An approximation of this solution in MATLAB environment is 3.03902118847875. However, we call this constant in our further results like Kudriasov gave it, approximately equal to 3.05.

The improvement of Becker’s univalence condition is given by Pascu for integral operators as follows.

Lemma 3 (see [3]). Let 𝛼 be a complex number, Re𝛼>0, and the function 𝑓∈𝒜. If 1−|𝑧|^2Re𝛼Re𝛼𝑧𝑓^′′(𝑧)𝑓^′(𝑧)≤1, (4) for all 𝑧∈𝑈, then for every complex number 𝛽, Re𝛽≥Re𝛼, the function 𝐹_𝛽(𝑧)=𝛽^𝑧₀𝑢^𝛽−1𝑓^′(𝑢)𝑑𝑢^1/𝛽 (5) is regular and univalent in 𝑈.

3. Main Results

Theorem 4. Let 𝛼, 𝛾_𝑗 be complex numbers, Re𝛼>0, 𝑗=1,𝑛, the functions 𝑓_𝑗∈𝒜, 𝑓_𝑗=𝑧+𝑎_2𝑗𝑧²+⋯, 𝑗=1,𝑛, 𝑛∈ℕ−{0}, and 𝐾 the positive real number 𝐾≅3.05.
If 𝑓^′′_𝑗(𝑧)𝑓^′_𝑗(𝑧)≤𝐾,𝑧∈𝑈,𝑗=𝛾1,𝑛,₁+𝛾₂+⋯+𝛾_𝑛≤minRe𝛼4,14, (6)(7) then 𝑓_𝑗∈𝒮, 𝑗=1,𝑛, and for every complex number 𝛽, Re𝛽≥Re𝛼, the integral operator 𝐻_𝑛 is in the class 𝒮.

Proof. Let us consider the function ℎ_𝑛(𝑧)=^𝑧₀𝑓₁(𝑢)𝑢^𝛾1⋯𝑓_𝑛(𝑢)𝑢^𝛾𝑛𝑑𝑢. (8)
The function ℎ_𝑛 is regular in 𝑈 and ℎ_𝑛(0)=ℎ^′_𝑛(0)−1=0. We have 𝑧ℎ^′′_𝑛(𝑧)ℎ^′_𝑛(𝑧)=𝛾₁𝑧𝑓^′₁(𝑧)𝑓₁(𝑧)−1+⋯+𝛾_𝑛𝑧𝑓^′_𝑛(𝑧)𝑓_𝑛(𝑧)−1, (9) for all 𝑧∈𝑈.
From (9), we obtain further 𝑧ℎ^′′_𝑛(𝑧)ℎ^′_𝑛(𝑧)≤𝛾₁𝑧𝑓^′₁(𝑧)𝑓₁(𝑧)𝛾+1+⋯+_𝑛𝑧𝑓^′_𝑛(𝑧)𝑓_𝑛(𝑧)+1. (10)
By (6) and Lemma 1, we have 𝑓_𝑗∈𝒮, 𝑗=1,𝑛, and hence we obtain 𝑧𝑓^′_𝑗(𝑧)𝑓_𝑗(𝑧)≤1+|𝑧|1−|𝑧|,𝑧∈𝑈,𝑗=1,𝑛. (11)
From (10) and (11), we get 1−|𝑧|^2Re𝛼Re𝛼𝑧ℎ^′′_𝑛(𝑧)ℎ^′_𝑛(𝑧)≤1−|𝑧|^2Re𝛼2Re𝛼|𝑧|𝛾1−¹+⋯+𝛾_𝑛 (12) for all 𝑧∈𝑈.
Now we consider the following cases.
(1) 0<Re𝛼<1. The function 𝑠∶(0,1)→ℝ, 𝑠(𝑥)=1−𝑎^2𝑥, 𝑥=Re𝛼, 𝑎=|𝑧|, (0≤𝑎<1) is increasing and we obtain 1−|𝑧|^2Re𝛼≤1−|𝑧|²,𝑧∈𝑈. (13)
From (12) and (13), we obtain 1−|𝑧|^2Re𝛼Re𝛼𝑧ℎ^′′_𝑛(𝑧)ℎ^′_𝑛(𝑧)≤4𝛾Re𝛼¹+⋯+𝛾_𝑛 (14) for all 𝑧∈𝑈.
Using the hypothesis condition (7), from (14), we have 1−|𝑧|^2Re𝛼Re𝛼𝑧ℎ^′′_𝑛(𝑧)ℎ^′_𝑛(𝑧)≤1, (15) for all 𝑧∈𝑈.
(2) Re𝛼≥1. We notice that the function 𝑞∶[1,∞)⟶ℝ,𝑞(𝑥)=1−𝑎^2𝑥𝑥,𝑥=Re𝛼,𝑎=|𝑧|,(0≤𝑎<1) (16) is decreasing function, and we obtain 1−|𝑧|^2Re𝛼|𝑧|Re𝛼≤1−²,𝑧∈𝑈. (17)
Using the last inequality in (12), we have 1−|𝑧|^2Re𝛼Re𝛼𝑧ℎ^′′_𝑛(𝑧)ℎ^′_𝑛(𝑧)≤4𝛾₁+⋯+𝛾_𝑛 (18) for all 𝑧∈𝑈.
Now using the hypothesis condition (7), from (18), we get 1−|𝑧|^2Re𝛼Re𝛼𝑧ℎ^′′_𝑛(𝑧)ℎ^′_𝑛(𝑧)≤1 (19) for all 𝑧∈𝑈.
Hence, based on the conditions obtained in (15) and in (19), applying Lemma 3, we have that 𝐻_𝑛∈𝒮.

Theorem 5. Let 𝛼, 𝛾_𝑗 be complex numbers, 𝑗=1,𝑛, Re𝛼>0, the functions 𝑓_𝑗∈𝒜, 𝑓_𝑗(𝑧)=𝑧+𝑎_2𝑗𝑧²+⋯, 𝑗=1,𝑛, 𝑛∈ℕ−{0}, and 𝐾 the positive real number, 𝐾≅3.05.
If 𝑓^′′_𝑗(𝑧)𝑓^′_𝑗(𝑧)≤𝐾,𝑧∈𝑈,𝑗=𝛾1,𝑛,₁+𝛾₂+⋯+𝛾_𝑛≤(2Re𝛼+1)^{(2Re𝛼+1)/(2Re𝛼)},2𝐾 (20)(21) then 𝑓_𝑗∈𝒮, 𝑗=1,𝑛, and for every complex number 𝛽, Re𝛽≥Re𝛼, the integral operator 𝑇_𝑛 is in the class 𝒮.

Proof. By (20) and Lemma 1, we obtain that 𝑓_𝑗∈𝒮, 𝑗=1,𝑛.
We consider the function 𝑝_𝑛(𝑧)=^𝑧₀𝑓^′₁(𝑢)^𝛾1⋯𝑓^′_𝑛(𝑢)^𝛾𝑛𝑑𝑢. (22) The function 𝑝_𝑛 is regular in 𝑈 and 𝑝_𝑛(0)=𝑝^′_𝑛(0)−1=0.
We have 𝑧𝑝^′′_𝑛(𝑧)𝑝^′_𝑛(𝑧)=𝛾₁𝑧𝑓^′′₁(𝑧)𝑓^′₁(𝑧)+⋯+𝛾_𝑛𝑧𝑓^′′_𝑛(𝑧)𝑓^′_𝑛(𝑧), (23) for all 𝑧∈𝑈.
Further we obtain 1−|𝑧|^2Re𝛼Re𝛼𝑧𝑝^′′_𝑛(𝑧)𝑝^′_𝑛(𝑧)≤1−|𝑧|^2Re𝛼|𝑧|𝛾Re𝛼¹𝑓^′′₁(𝑧)𝑓^′₁(𝑧)+⋯+𝛾_𝑛𝑓^′′_𝑛(𝑧)𝑓^′_𝑛(𝑧). (24)
From (24) and from the Kudriasov condition within the hypothesis, (20), we have 1−|𝑧|^2Re𝛼Re𝛼𝑧𝑝^′′_𝑛(𝑧)𝑝^′_𝑛(𝑧)≤1−|𝑧|^2Re𝛼|𝑧×𝐾𝛾Re𝛼₁|+⋯+𝐾𝛾_𝑛, (25) for all 𝑧∈𝑈.
Let us consider the function 𝐺∶[0,1]→ℝ, 𝐺(𝑥)=((1−𝑥^2𝑎)/𝑎)𝑥, 𝑥=|𝑧|, 𝑎=Re𝛼. We have max_[]𝑥∈0,1𝐺(𝑥)=2(2𝑎+1)^{(2𝑎+1)/(2𝑎)}. (26)
By (25), (26), and (21) we obtain 1−|𝑧|^2Re𝛼Re𝛼𝑧𝑝^′′_𝑛(𝑧)𝑝^′_𝑛(𝑧)<1, (27) for all 𝑧∈𝑈.
Now from (27) and Lemma 3, it results that 𝑇_𝑛∈𝒮.

4. Corollaries

Corollary 1. Let 𝛾 be complex number, Re[𝑛(𝛾−1)+1]>0, the functions 𝑓_𝑗∈𝒜, 𝑗=1,𝑛, 𝑛∈ℕ−{0}, and 𝐾 the positive real number, 𝐾≅3.05.
If 𝑓^′′_𝑗(𝑧)𝑓^′_𝑗(𝑧)≤𝐾,𝑧∈𝑈,𝑗=𝑛𝑛1,𝑛,𝛾−1≤minRe𝛾−1+14,14, (28) then 𝑓_𝑗∈𝒮, 𝑗=1,𝑛, and the integral operator 𝐼_𝛼,𝑛 defined by 𝐼_𝛼,𝑛(𝑧)=𝑛(𝛾−1)+1^𝑧₀𝑓₁^𝛾−1(𝑢)⋯𝑓_𝑛^𝛾−1(𝑢)𝑑𝑢^{1/(𝑛(𝛾−1)+1)} (29) is in the class 𝒮.

Proof. From (29), we have 𝐼_𝛼,𝑛(𝑧)=𝑛𝛾−1+1^𝑧₀𝑢^{𝑛(𝛾−1)}𝑓₁(𝑢)𝑢^𝛾−1⋯𝑓_𝑛(𝑢)𝑢^𝛾−1𝑑𝑢^{1/(𝑛(𝛾−1)+1)}, (30) and for 𝛼=𝛽=𝑛(𝛾−1)+1, 𝛾₁=𝛾₂=⋯=𝛾_𝑛=𝛾−1, from Theorem 4, we obtain Corollary 1.

Corollary 2. Let 𝛼, 𝛾_𝑗 be complex numbers, 𝑗=1,𝑛, 0<Re𝛼≤1, the functions 𝑓_𝑗∈𝒜, 𝑓_𝑗(𝑧)=𝑧+𝑎_2𝑗𝑧²+⋯, 𝑗=1,𝑛, 𝑛∈ℕ−{0}, and 𝐾 the positive real number 𝐾≅3.05.
If 𝑓^′′_𝑗(𝑧)𝑓^′_𝑗(𝑧)≤𝐾,𝑧∈𝑈,𝑗=𝛾1,𝑛,₁+𝛾₂+⋯+𝛾_𝑛≤Re𝛼4, (31) then 𝑓_𝑗∈𝒮, 𝑗=1,𝑛 and the integral operator 𝐿_𝑛 defined by 𝐿_𝑛(𝑧)=^𝑧₀𝑓₁(𝑢)𝑢^𝛾1⋯𝑓_𝑛(𝑢)𝑢^𝛾𝑛𝑑𝑢 (32) belongs to the class 𝒮.

Corollary 3. Let 𝛼, 𝛾_𝑗 be complex numbers, 𝑗=1,𝑛, 0<Re𝛼≤1, the functions 𝑓_𝑗∈𝒜, 𝑓_𝑗(𝑧)=𝑧+𝑎_2𝑗𝑧²+⋯, 𝑗=1,𝑛, 𝑛∈ℕ−{0}, and 𝐾 the positive real number, 𝐾≅3.05.
If 𝑓^′′_𝑗(𝑧)𝑓^′_𝑗(𝑧)≤𝐾,𝑧∈𝑈,𝑗=𝛾1,𝑛,₁+𝛾₂+⋯+𝛾_𝑛≤(2Re𝛼+1)^{(2Re𝛼+1)/(2Re𝛼)},2𝐾 (33) then 𝑓_𝑗∈𝒮, 𝑗=1,𝑛, and the integral operator 𝐺_𝑛 defined by 𝐺_𝑛(𝑧)=^𝑧₀𝑓^′₁(𝑢)^𝛾1⋯𝑓^′_𝑛(𝑢)^𝛾𝑛𝑑𝑢 (34) is in the class 𝒮.