Certain Properties of Multivalent Functions Associated with the Dziok-Srivastava Operator

Abstract

By making use of the techniques of the differential subordination, we derive certain properties of 𝑝-valent functions associated with the Dziok-Srivastava operator.

1. Introduction

Let 𝐴(𝑝,𝑘) denote the class of functions of the form 𝑓(𝑧)=𝑧^𝑝+^∞_𝑛=𝑘𝑎_𝑛+𝑝𝑧^𝑛+𝑝𝑝,𝑘∈ℕ={1,2,3,…}, (1) which are analytic in the open unit disk  𝑈={𝑧∈ℂ∶|𝑧|<1}. We write 𝐴(𝑝,1)=𝐴(𝑝).

Suppose that 𝑓 and 𝑔 are analytic in 𝑈. We say that the function 𝑓 is subordinate to 𝑔 in 𝑈, or 𝑔 superordinate to𝑓 in 𝑈, and we write 𝑓≺𝑔 or 𝑓(𝑧)≺𝑔(𝑧)(𝑧∈𝑈), if there exists an analytic function 𝜔 in 𝑈 with 𝜔(0)=0 and |𝜔(𝑧)|<1, such that 𝑓(𝑧)=𝑔(𝜔(𝑧))(𝑧∈𝑈). If 𝑔 is univalent in 𝑈, then the following equivalence relationship holds true (see [1–3]): 𝑓(𝑧)≺𝑔(𝑧)⟺𝑓(0)=𝑔(0),𝑓(𝑈)⊂𝑔(𝑈). (2)

For functions 𝑓_𝑗∈𝐴(𝑝,𝑘) given by 𝑓_𝑗(𝑧)=𝑧^𝑝+^∞_𝑛=𝑘𝑎_{𝑛+𝑝,𝑗}𝑧^𝑛+𝑝𝑗=1,2;𝑝∈ℕ, (3) we define the Hadamard product (or convolution) of 𝑓₁ and 𝑓₂ by 𝑓₁∗𝑓₂(𝑧)=𝑧^𝑝+^∞_𝑛=𝑘𝑎_𝑛+𝑝,1𝑎_𝑛+𝑝,2𝑧^𝑛+𝑝=𝑓₂∗𝑓₁(𝑧). (4)

For complex parameters 𝑎₁,…,𝑎_𝑞 and 𝑏₁,…,𝑏_𝑠(𝑏𝑗∉ℤ⁻₀={0,−1,−2,…};𝑗=1,…,𝑠), the generalized hypergeometric function _𝑞𝐹_𝑠 is defined (see [4]) by the following infinite series: _𝑞𝐹_𝑠𝑎₁,…,𝑎_𝑖,…,𝑎_𝑞;𝑏₁,…,𝑏_𝑠;𝑧=^∞_𝑛=0𝑎_1𝑛⋯𝑎_𝑞𝑛𝑏_1𝑛⋯𝑏_𝑠𝑛𝑧^𝑛𝑛!𝑞≤𝑠+1;𝑞,𝑠∈ℕ₀=ℕ∪{0};𝑧∈𝑈, (5) where (𝜃)_𝑛 is the Pochhammer symbol defined, in terms of the Gamma function Γ, by (𝜃)_𝑛=Γ(𝜃+𝑛)Γ(𝜃)=(),𝜃()⋯(),().1,𝜐=0𝜃+1𝜃+𝑛−1𝜐∈ℕ (6) Corresponding a function ℎ_𝑝(𝑎₁,…,𝑎_𝑖,…,𝑎_𝑞;𝑏₁,…,𝑏_𝑠;𝑧) defined by ℎ_𝑝𝑎₁,…,𝑎_𝑖,…,𝑎_𝑞;𝑏₁,…,𝑏_𝑠;𝑧=𝑧^𝑝⋅_𝑞𝐹_𝑠𝑎₁,…,𝑎_𝑖,…,𝑎_𝑞;𝑏₁,…,𝑏_𝑠;𝑧(𝑧∈𝑈), (7) Dziok and Srivastava [5] considered a linear operator 𝐻_𝑝𝑎₁,…,𝑎_𝑞;𝑏₁,…,𝑏_𝑠∶𝐴𝑝,𝑘→𝐴𝑝,𝑘 (8) defined by the following Hadamard product: 𝐻_𝑝𝑎₁,…,𝑎_𝑞;𝑏₁,…,𝑏_𝑠𝑓(𝑧),=ℎ_𝑝𝑎₁,…,𝑎_𝑖,…,𝑎_𝑞;𝑏₁,…,𝑏_𝑠;𝑧∗𝑓(𝑧),𝑞≤𝑠+1;𝑞,𝑠∈ℕ₀;𝑧∈𝑈. (9) If 𝑓∈𝐴(𝑝,𝑘) is given by (1), then we have 𝐻_𝑝𝑎₁,…,𝑎_𝑞;𝑏₁,…,𝑏_𝑠𝑓(𝑧)=𝑧^𝑝+^∞_𝑛=𝑘Γ_𝑛𝑎_𝑛+𝑝𝑧^𝑛+𝑝(𝑧∈𝑈), (10) where Γ_𝑛=𝑎_1𝑛⋯𝑎_𝑞𝑛𝑏_1𝑛⋯𝑏_𝑠𝑛1,().𝑛!𝑛∈ℕ (11) To make the notation simple, we write 𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)=𝐻_𝑝𝑎₁,…,𝑎_𝑞;𝑏₁,…,𝑏_𝑠𝑓(𝑧). (12) It easily follows from (9) or (10) that 𝑧𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^′=𝑎₁𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)−𝑎+1₁−𝑝𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧),(𝑧∈𝑈). (13) It should be remarked that the linear operator 𝐻_{𝑝,𝑞,𝑠}(𝑎₁) is a generalization of many other linear operators considered earlier. In particular, for 𝑓∈𝐴(𝑝) we have the following observations:(i)𝐻_1,2,1(𝑎,𝑏;𝑐)𝑓(𝑧)=(𝐼_𝑐^𝑎,𝑏)𝑓(𝑧)(𝑎,𝑏∈ℂ;𝑐∉ℤ⁻₀), where the linear operator 𝐼_𝑐^𝑎,𝑏 was investigated by Hohlov [6];(ii)𝐻_𝑝,2,1(𝑛+𝑝,1;1)𝑓(𝑧)=𝐷^{𝑛+𝑝−1}𝑓(𝑧)(𝑛∈ℕ;𝑛>−𝑝), where the linear operator 𝐷^{𝑛+𝑝−1} was studied by Goel and Sohi [7]. In the case when 𝑝=1, 𝐷^𝑛𝑓(𝑧) is the Ruscheweyh derivative of 𝑓(𝑧) (see [8]);(iii)𝐻_𝑝,2,1(𝜇+𝑝,1;𝜇+𝑝+1)𝑓(𝑧)=𝐽_𝑝,𝛿(𝑓)(𝑧)=((𝑝+𝛿)/𝑧^𝛿)∫^𝑧₀𝑡^𝛿−1𝑓(𝑡)𝑑𝑡(𝛿>−𝑝), where 𝐽_𝑝,𝛿 is the generalized Bernardi-Libera-Livingston integral operator (see [9]);(iv)𝐻_𝑝,2,1(𝑝+1,1;𝑝+1−𝜆)𝑓(𝑧)=Ω_𝑧^(𝜆,𝑝)𝑓(𝑧)=(Γ(𝑝+1−𝜆)/Γ(𝑝+1))𝑧^𝜆𝐷^𝜆_𝑧𝑓(𝑧) (−∞≤𝜆<𝑝+1;𝑧∈𝑈), where 𝐷^𝜆_𝑧𝑓(𝑧) is the fractional integral of 𝑓 of order −𝜆 when −∞≤𝜆<0 and fractional derivative of 𝑓 of order 𝜆 when 0≤𝜆<𝑝+1. The extended fractional differintegral operator 𝐷_𝑧^(𝜆,𝑝) was introduced and studied by Patel and Mishra [10]. The fractional differential operator Ω_𝑧^(𝜆,𝑝) with 0≤𝜆<1 was investigated by Srivastava and Aouf [11]. The operator Ω_𝑧^(𝜆,1)=Ω^𝜆_𝑧 was introduced by Owa and Srivastava [12] (see also [13–15]).(v)𝐻_𝑝,2,1(𝑎,1;𝑐)𝑓(𝑧) = 𝐿_𝑝(𝑎,𝑐)𝑓(𝑧)(𝑎∈ℝ;𝑐∈ℝ⧵ℤ⁻₀), where the linear operator 𝐿_𝑝(𝑎,𝑐) was studied by Saitoh [16] which yields the operator 𝐿(𝑎,𝑐) introduced by Carlson and Shaffer [17] for 𝑝=1;(vi)𝐻_1,2,1(𝜇,1;𝜆+1)𝑓(𝑧) = 𝐼_𝜆,𝜇𝑓(𝑧)(𝜆>−1;𝜇>0), where 𝐼_𝜆,𝜇 is the Choi-Saigo-Srivastava operator [9] which is closely related to the Carlson-Shaffer [17] operator 𝐿(𝜇,𝜆+1)𝑓(𝑧);(vii)𝐻_𝑝,2,1(𝑝+1,1;𝑛+𝑝)𝑓(𝑧) = 𝐼_𝑛,𝑝𝑓(𝑧)(𝑛∈ℤ;𝑛>−𝑝), where the operator 𝐼_𝑛,𝑝 was considered by Liu and Noor [18];(viii)𝐻_𝑝,2,1(𝜆+𝑝,𝑐;𝑎)𝑓(𝑧) = 𝐼^𝜆_𝑝(𝑎,𝑐)𝑓(𝑧)(𝑎,𝑐∈ℝ⧵ℤ⁻₀;𝜆>−𝑝), where 𝐼^𝜆_𝑝(𝑎,𝑐) is the Cho-Kwon-Srivastava operator [19].

In recent years, many interesting subclasses of analytic functions, associated with the Dziok-Srivastava operator 𝐻_{𝑝,𝑞,𝑠}(𝑎₁) and its many special cases, were investigated by, for example, Dziok and Srivastava [5, 20], Gangadharan et al. [21], Liu and Noor [18], Liu [22], Liu and Srivastava [23], and others (see also [19, 24–26]). In the present paper, we shall use the method based upon the differential subordination to derive inclusion relationships and other interesting properties and characteristics of the Dziok-Srivastava operator 𝐻_{𝑝,𝑞,𝑠}(𝑎₁).

2. Main Results

Unless otherwise mentioned, we assume throughout the sequel that 𝑎_𝑖>0; 𝑎_𝑖∉ℤ⁻₀(𝑖=1,…,𝑞); 𝛼>0; 𝜇>0 and −1≤𝐵<𝐴≤1.

Let 𝑃[𝑘] denote the class of functions of the form 𝜑(𝑧)=1+𝑐_𝑘𝑧^𝑘+𝑐_𝑘+1𝑧^𝑘+1+⋯ (14) that are analytic in 𝑈, we write 𝑃[1]=𝑃. In our present investigation, we shall require the following lemmas.

Lemma 1 (see [2]). Let ℎ be analytic and convex (univalent) in 𝑈 with ℎ(0)=1 and 𝜑∈𝑃[𝑘]. If 𝜑(𝑧)+𝑧𝜑^′(𝑧)𝛾≺ℎ(𝑧), (15) then, for 𝛾≠0 and ℜ(𝛾)≥0, 𝜑(𝑧)(𝑧)=𝛾≺𝑞𝑘𝑧^{−𝛾/𝑘𝑧}₀𝑡^{𝛾/𝑘−1}ℎ(𝑡)𝑑𝑡≺ℎ(𝑧), (16) and 𝑞 is the best dominant.

Lemma 2 (see [1]). Let 𝐷 be a set in the complex plane ℂ and 𝑏 be a complex number satisfying ℜ(𝑏)>0. Suppose that the function Ψ∶ℂ²×𝑈→ℂ satisfies the condition Ψ(𝑖𝑥,𝑦)∉𝐷 for all real 𝑥,𝑦≤−|𝑏−𝑖𝑥|/2ℜ(𝑏) and for all 𝑧∈𝑈. If the functions 𝜑∈𝑃 and ℜ{Ψ(𝜑(𝑧),𝑧𝜑^′(𝑧);𝑧)}∈𝐷, then ℜ{𝜑(𝑧)}>0 in 𝑈.

Lemma 3 (see [27]). Let 𝜙 be analytic in 𝑈 with 𝜙(0)=1 and 𝜙(𝑧)≠0 for all 𝑧∈𝑈. If there exist two points 𝑧₁,𝑧₂∈𝑈 such that −𝜋2𝛿₁=arg𝜙𝑧₁<arg𝜙(𝑧)<arg𝜙𝑧₂=𝜋2𝛿₂ (17) for some 𝛿₁ and 𝛿₂(𝛿₁,𝛿₂>0) and for all 𝑧(|𝑧|<|𝑧₁|=|𝑧₂|), then 𝑧₁𝜙^′𝑧₁𝜙𝑧₁𝛿=−𝑖¹+𝛿₂2𝑚,𝑧₂𝜙^′𝑧₂𝜙𝑧₂𝛿=−𝑖¹+𝛿₂2𝑚, (18) where |𝑏|𝑚≥1−|𝑏|𝛿1+,𝑏=𝑖tan²−𝛿₁𝛿₂+𝛿₁. (19)

Theorem 4. Let 𝑚≥1, 𝛾>0. Let 𝑓∈𝐴(𝑘,𝑝), then ℜ𝐻_{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧)^(𝑗)𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)<𝑎₁+𝛾𝑎₁𝑧∈𝑈;0≤𝑗<𝑝, (20) implies ℜ𝐻_{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧)^(𝑗)𝑧^{𝑝−𝑗−1/2𝛾𝑚}>2^−1/𝑚𝑧∈𝑈;0≤𝑗<𝑝. (21) The bound 2^−1/𝑚 is the best possible.

Proof. It easily follows from (13) that 𝑧𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗+1)=𝑎₁𝐻_{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧)^(𝑗)−𝑎₁−𝑝+𝑗𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)𝑧∈𝑈;0≤𝑗<𝑝. (22) From (20) and (22), we have ℜ𝑧𝐻_{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧)^(𝑗+1)𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗).<𝛾+𝑝−𝑗𝑧∈𝑈;0≤𝑗<𝑝 (23) That is, −1𝑧𝐻2𝛾^{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧)^(𝑗+1)𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)≺𝑧−𝑝+𝑗().1−𝑧𝑧∈𝑈 (24) Let 𝜑(𝑧)=𝑝−𝑗!𝐻𝑝!^{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)𝑧^{𝑝−𝑗−1/2𝛾}(𝑧∈𝑈), (25) then (24) may be written as 𝑧log𝜑(𝑧)^′1≺𝑧log1−𝑧^′. (26) By using a well-known result (see [28]) to (26) we obtain that 𝜑(𝑧)≺1,1−𝑧 (27) or, equivalently, 𝑝−𝑗!𝐻𝑝!^{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)𝑧^{𝑝−𝑗−1/2𝛾𝑚}=11−𝜔(𝑧)^1/𝑚, (28) where 𝜔 is analytic in 𝑈, 𝜔(0)=0 and |𝜔(𝑧)|<1 for 𝑧∈𝑈. Since ℜ(𝑡^1/𝑚)≥(ℜ(𝑡))^1/𝑚 for ℜ(𝑡)>0 and 𝑚≥1, (28) yields ℜ𝑝−𝑗!𝐻𝑝!^{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)𝑧^{𝑝−𝑗−1/2𝛾𝑚}≥ℜ11−𝜔(𝑧)^1/𝑚≥2^−1/𝑚(𝑧∈𝑈). (29) To see that the bound 2^−1/𝑚 cannot be increased, we consider the function 𝑔(𝑧)=𝑧^𝑝+𝑝!𝑝−𝑗!^∞_𝑛=1−2𝛾_𝑛𝑛+𝑝−𝑗!𝑛!𝑛+𝑝!Γ_𝑛𝑧^𝑛+𝑝,(𝑧∈𝑈). (30) Since 𝑝−𝑗!𝐻𝑝!^{𝑝,𝑞,𝑠}𝑎₁𝑔(𝑧)^(𝑗)𝑧^𝑝−𝑗=(1−𝑧)^−2𝛾, (31) we easily have that 𝑔 satisfies (20) and ℜ𝑝−𝑗!𝐻𝑝!^{𝑝,𝑞,𝑠}𝑎₁𝑔(𝑧)^(𝑗)𝑧^{𝑝−𝑗−1/2𝛾𝑚}→2^−1/𝑚 (32) as ℜ(𝑧)=𝑧→1⁻. This completes the proof of Theorem 4.

Theorem 5. Let 𝛼≥0, 𝛾>1. If 𝑓∈𝐴(𝑝) satisfies the following inequality ℜ()𝐻1−𝛼^{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧)^(𝑗)𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)𝐻+𝛼^{𝑝,𝑞,𝑠}𝑎₁+2𝑓(𝑧)^(𝑗)𝐻_{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧)^(𝑗),<𝛾0≤𝑗<𝑝;𝑧∈𝑈 (33) then ℜ𝐻_{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧)^(𝑗)𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)<𝛽0≤𝑗<𝑝;𝑧∈𝑈, (34) where 𝛽∈(1,∞) is the positive root of the equation 2𝑎₁−𝛼+1𝑥²+3𝛼−2𝛾𝛼−2𝛾𝑥−𝛼=0. (35)

Proof. Let 𝜑(𝑧)=1𝐻𝛽−1𝛽−^{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧)^(𝑗)𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)(𝑧∈𝑈), (36) then 𝜑(𝑧) is analytic in 𝑈 and 𝜑(0)=1. Differentiating (36) and using (22), we obtain that ()𝐻1−𝛼^{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧)^(𝑗)𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)𝐻+𝛼^{𝑝,𝑞,𝑠}𝑎₁+2𝑓(𝑧)^(𝑗)𝐻_{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧)^(𝑗)𝛼=𝛽−𝛽−1𝑎₁−𝑎+1¹−𝛼+1𝛽−1𝑎₁𝜑(𝑧)−𝛼+1𝛽−1𝑎₁+1𝑧𝜑^′(𝑧)𝜑(𝑧)𝜑(𝑧)𝛽−𝛽−1=𝜓,𝑧𝜑^′(𝑧), (37) where 𝜓()𝛼𝑟,𝑠=𝛽−𝛽−1𝑎₁−𝑎+1¹−𝛼+1𝛽−1𝑎₁𝑟−𝛼+1𝛽−1𝑎₁𝑠+1𝑟.𝛽−𝛽−1 (38) Using (33) and (38), we have 𝜓𝜑(𝑧),𝑧𝜑^′(𝑧)∶𝑧∈𝑈⊂𝐷=𝑧∈ℂ∶ℜ(𝑧)<𝛾. (39) Now for all real 𝑥,𝑦≤−(1+𝑥²)/2, we have ℜ𝜓𝛼𝑖𝑥,𝑦=𝛽−𝛽−1𝑎₁−𝛼+1𝛽−1𝑎₁+1𝛽𝑦𝛽²+𝛽−1²𝑥²𝛼≥𝛽−𝛽−1𝑎₁++1𝛼𝛽𝛽−12𝑎₁+11+𝑥²𝛽²+𝛽−1²𝑥²𝛼≥𝛽−𝛽−1𝑎₁+𝛼+1𝛽−12𝛽𝑎₁𝛼+1=𝛽−𝛽−12𝛽−12𝛽𝑎₁+1=𝛾, (40) where 𝛽 is the positive root of (35).
Note that for 𝛼≥0, 𝛾>1, 𝑎₁>0 and ℎ(𝑥)=2𝑎₁−𝛼+1𝑥²+3𝛼−2𝛾𝛼−2𝛾𝑥−𝛼, (41) we have ℎ(0)=−𝛼≤0 and ℎ(1)=2𝑎₁(1−𝛾)−2𝛾<0. This shows 𝛽∈(0,+∞). Hence for each 𝑧∈𝑈, 𝜓(𝑖𝑥,𝑦)∉Ω. By Lemma 2, we get ℜ{𝜑(𝑧)}>0(𝑧∈𝑈), and this proves (34).

Theorem 6. Suppose that 0≤𝑗<𝑝; 𝛼>0 and 0<𝛿₁, 𝛿₂≤1. If 𝐹_𝛼 given by 𝐹_𝛼(𝑧)=1−𝛼−𝛼𝑎₁+𝛼𝑝𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)+𝛼𝑎₁𝐻_{𝑝,𝑞,𝑠}𝑎₁+1𝑓(𝑧) (42) satisfies −𝜋2𝛿₁𝐹<arg^𝛼(𝑗)(𝑧)𝑧^𝑝−𝑗<𝜋2𝛿₂(𝑧∈𝑈), (43) then −𝜋2𝜂₁𝐻<arg^{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)𝑧^𝑝−𝑗<𝜋2𝜂₂(𝑧∈𝑈), (44) where 𝜂₁ and 𝜂₂ are the solution of the equations: 𝛿₁=𝜂₁+2𝜋𝛼𝜂arctan¹+𝜂₂2|𝑏|1−𝛼+𝛼𝑝1−|𝑏|,𝛿1+₂=𝜂₂+2𝜋𝛼𝜂arctan¹+𝜂₂2|𝑏|1−𝛼+𝛼𝑝1−|𝑏|,1+ (45) where 𝑏 is given by (19).

Proof. Using (42) and the identity (22), it follows that 𝐹_𝛼^(𝑗)(𝑧)=1−𝛼+𝛼𝑗𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)+𝛼𝑧𝐻_{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗+1), (46) for 0≤𝑗<𝑝. Putting 𝜑(𝑧)=𝑝−𝑗!𝐻𝑝!^{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)𝑧^𝑝−𝑗(𝑧∈𝑈). (47) On differentiating (47) followed by a simple calculation, we get 𝐹_𝛼^(𝑗)(𝑧)𝑧^𝑝−𝑗=𝑝!1−𝛼+𝛼𝑝!×𝜑(𝑧)+𝛼𝑝−𝑗1−𝛼+𝛼𝑝𝑧𝜑^′(𝑧)(𝑧∈𝑈). (48) Let ℎ be the function which maps 𝑈 onto the angular domain {𝑤∈ℂ∶−(𝜋/2)𝛿₁<arg{𝑤}<(𝜋/2)𝛿₂} with ℎ(0)=1. By using (43) in (48), we get 𝜑(𝑧)+𝛼1−𝛼+𝛼𝑝𝑧𝜑^′(𝑧)≺ℎ(𝑧). (49) Further, an application of Lemma 1 yields ℜ{𝜑(𝑧)}>0 in 𝑈 and hence 𝜑(𝑧)≠0 for 𝑧∈𝑈.
Suppose there exist two points 𝑧₁,𝑧₂∈𝑈 such that the condition (28) is satisfied. Then by Lemma 3, we obtain (18) under the constraint (19). Therefore, we have arg1−𝛼+𝛼𝑝𝜑𝑧₁+𝛼𝑧𝜑^′𝑧₁=arg𝜑𝑧₁𝑧+arg1−𝛼+𝛼𝑝+𝛼¹𝜑^′𝑧₁𝜑𝑧₁𝜋=−2𝜂₁𝛼𝜂+arg1−𝛼+𝛼𝑝−𝑖¹+𝜂₂2𝑚𝜋=−2𝜂₁𝛼𝜂−arctan¹+𝜂₂2𝑚𝜋1−𝛼+𝛼𝑝≤−2𝜂₁𝛼𝜂−arctan¹+𝜂₂2|𝑏|1−𝛼+𝛼𝑝1−|𝑏|,𝜑𝑧1+arg1−𝛼+𝛼𝑝₂+𝛼𝑧𝜑^′𝑧₂𝜋≥−2𝜂₂𝛼𝜂−arctan¹+𝜂₂2|𝑏|1−𝛼+𝛼𝑝1−|𝑏|,1+ (50) which contradicts the assumption (43). This proves the assertion (44) of the Theorem 6.
For 𝛿₁=𝛿₂=𝛿, Theorem 6 reduces to the following corollary.

Corollary 7. Suppose that 0≤𝑗<𝑝 and 𝛼>0. If 𝐹_𝛼 defined by (42) satisfies 𝐹arg^𝛼(𝑗)(𝑧)𝑧^𝑝−𝑗<𝜋2𝛿(0<𝛿≤1;𝑧∈𝑈), (51) then 𝐻arg^{𝑝,𝑞,𝑠}𝑎₁𝑓(𝑧)^(𝑗)𝑧^𝑝−𝑗<𝜋2𝜂(𝑧∈𝑈), (52) where 𝜂(0<𝜂≤1) is the solution of the equation: 2𝛿=𝜂+𝜋arctan𝛼𝜂.1−𝛼+𝛼𝑝 (53)