Sufficient Conditions for 𝜆-Spirallike and 𝜆-Robertson Functions of Complex Order

Abstract

We obtain several sufficient conditions for 𝜆-spirallike and 𝜆-Robertson functions of complex order by making use of the well-known Jack Lemma.

1. Introduction and Definitions

Let 𝒜 denote the class of functions of the form 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛, (1) which are analytic in the open unit disk 𝒰={𝑧∶𝑧∈ℂ and |𝑧|<1}. A function 𝑓(𝑧)∈𝒜 is said to be the 𝜆-spirallike function of complex order 𝑏 and type 𝛼(0≤𝛼<1) in 𝒰, denoted by 𝒮^𝜆_𝛼(𝑏) if and only if 𝑒ℜ𝑒1+^𝑖𝜆𝑏cos𝜆𝑧𝑓^′(𝑧)𝑓(𝑧)−1>𝛼(𝑧∈𝒰), (2) for some real numbers 𝜆∈ℝ with |𝜆|<𝜋/2 and 𝑏∈ℂ^∗∶=ℂ⧵{0}. Furthermore, a function 𝑓(𝑧)∈𝒜 is also said to be the 𝜆-Robertson function of complex order 𝑏 and type 𝛼(0≤𝛼<1) in 𝒰 if and only if 𝑒ℜ𝑒1+^𝑖𝜆𝑏cos𝜆𝑧𝑓^′′(𝑧)𝑓^′(𝑧)>𝛼(𝑧∈𝒰), (3) for some real numbers 𝜆∈ℝ with |𝜆|<𝜋/2 and 𝑏∈ℂ^∗. We denote this class by 𝒞^𝜆_𝛼(𝑏).

Noting that the above function classes include several subclasses which have important role in the analytic and geometric function theory. From this reason, we want to state some of them. (i)𝒮^𝜆₀(𝑏)=∶𝒮^𝜆(𝑏)(𝑏∈ℂ^∗;|𝜆|<𝜋/2) is the class of the 𝜆-spirallike function of complex order 𝑏 introduced and studied by Al-Oboudi and Haidan [1].(ii)𝒞^𝜆₀(𝑏)=∶𝒞^𝜆(𝑏)(𝑏∈ℂ^∗;|𝜆|<𝜋/2) is the class of the 𝜆-Robertson function of complex order 𝑏 (see, [2]).(iii)𝒮⁰_𝛼(𝑏)=∶𝒮^∗_𝛼(𝑏)(𝑏∈ℂ^∗;0≤𝛼<1) is the class of starlike functions of complex order 𝑏 and type 𝛼 introduced and studied by the author [3], and 𝒮^∗_𝛼(1)=∶𝒮^∗_𝛼 is said to be the starlike functions class of order 𝛼(0≤𝛼<1) and was studied by Robertson [4].(iv)𝒞⁰_𝛼(𝑏)=∶𝒞_𝛼(𝑏)(𝑏∈ℂ^∗;0≤𝛼<1) is the class of convex functions of complex order 𝑏 and type 𝛼 introduced and studied by the author [3], and 𝒞_𝛼(1)=∶𝒞_𝛼 is said to be the convex functions class of order 𝛼(0≤𝛼<1) and was studied by Robertson [4].(v)𝒮^𝜆₀(1)=∶𝒮^𝜆(|𝜆|<𝜋/2) is known the 𝜆-spirallike univalent functions class and was defined by Spacek [5], 𝒮⁰₀(𝑏)=∶𝒮(𝑏)(𝑏∈ℂ^∗) is said to be the starlike functions class of complex order and was studied by Nasr and Aouf [6], and 𝒮^𝜆₀(1−𝛽)=∶𝒮^𝜆(𝛽)(0≤𝛽<1) is known the 𝜆-spirallike functions class of order 𝛽 and was studied by Libera [7].(vi)𝒞^𝜆₀(1)=∶𝒞^𝜆(|𝜆|<𝜋/2) is known the 𝜆-Robertson type functions class and was first studied by Robertson [8], 𝒞⁰₀(𝑏)=∶𝒞(𝑏) is called the convex functions class of complex order and was studied by Wiatrowski [9], Nasr and Aouf [10] and Aouf [11], and 𝒞^𝜆₀(1−𝛽)=∶𝒞^𝜆(𝛽)(0≤𝛽<1) is known the 𝜆-Robertson type functions class of order 𝛽 and was studied by Chichra [12].

In this paper, we obtain several sufficient conditions for the analytic functions 𝑓(𝑧) belonging to the classes 𝒮^𝜆_𝛼(𝑏),  𝒞^𝜆_𝛼(𝑏),  𝒮^∗_𝛼(𝑏),  𝒞_𝛼(𝑏),  𝒮(𝑏),  𝒞(𝑏),  𝒮^∗_𝛼, and  𝒞_𝛼 by making use of the well-known Jack Lemma [13].

2. Main Result

In order to derive our main result, we have to recall here the following Jack Lemma.

Lemma 1 (see [13]). Let 𝑤(𝑧) be analytic in 𝒰 such that 𝑤(0)=0. Then, if |𝑤(𝑧)| attains its maximum value on circle |𝑧|=𝑟<1 at a point 𝑧₀∈𝒰, one has 𝑧₀𝑤^′𝑧₀=𝑘𝑤𝑧₀, (4) where 𝑘>1 is a real number.

Now, with the help of Lemma 1, we can prove the following result.

Theorem 2. Let 𝑓(𝑧)∈𝒜, 𝜁,𝑏∈ℂ^∗  and |𝜆|<𝜋/2, and let 𝐹(𝑧) be defined by 𝐹(𝑧)=1−𝜇𝑓(𝑧)+𝜇𝑧𝑓^′(𝑧),0≤𝜇≤1. (5) If 𝐹(𝑧) satisfies any of the following inequalities: 𝑧𝑧𝐹^′(𝑧)^′/𝐹(𝑧)−𝑧𝐹^′(𝑧)/𝐹(𝑧)²1+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)/𝐹(𝑧)−11+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)/𝐹(𝑧)−1^𝜁<𝑏−1𝜁cos𝜆,(),2−𝛼𝑧∈𝒰;0≤𝛼<1𝑧𝑧𝐹^′(𝑧)^′/𝐹(𝑧)−𝑧𝐹^′(𝑧)/𝐹(𝑧)²1+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)(𝑧)<𝑏/𝐹−1𝜁(1−𝛼)cos𝜆,(),2−𝛼𝑧∈𝒰;0≤𝛼<1𝑧𝑧𝐹^′(𝑧)^′/𝐹(𝑧)−𝑧𝐹^′(𝑧)/𝐹(𝑧)²1+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)/𝐹(𝑧)−1^𝜁+1<𝑏𝜁(1−𝛼)cos𝜆(2−𝛼)²,(𝑧∈𝒰;0≤𝛼<1),𝑒1+^𝑖𝜆𝑏cos𝜆𝑧𝐹^′(𝑧)𝐹(𝑧)−1^𝜁−1×𝑧𝑧𝐹^′(𝑧)^′𝐹(𝑧)−𝑧𝐹^′(𝑧)𝐹(𝑧)²<𝑏𝜁(1−𝛼)cos𝜆,(𝑧∈𝒰;0≤𝛼<1),Re1+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)/𝐹(𝑧)−1^𝜁−1×𝑧𝑧𝐹^′(𝑧)^′/𝐹(𝑧)−𝑧𝐹^′(𝑧)/𝐹(𝑧)²×1+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)/𝐹(𝑧)−1^𝜁−1⁻¹<cos𝜆𝜁²Re𝑏𝜁𝑒^−𝑖𝜆,(𝑧∈𝒰), (6)(7)(8)(9)(10) then 𝑒Re1+^𝑖𝜆𝑏cos𝜆𝑧𝐹^′(𝑧)𝐹(𝑧)−1^𝜁>𝛼,(𝑧∈𝒰;0≤𝛼<1). (11) The powers are taken by their principal value.

Proof. Define a function 𝑤(𝑧) by 𝑒1+^𝑖𝜆𝑏cos𝜆𝑧𝐹^′(𝑧)𝐹(𝑧)−1^𝜁=1+(1−𝛼)𝑤(𝑧). (12)
Then, 𝑤(𝑧) is analytic in 𝒰 and 𝑤(0)=0. It follows from (12) that 𝑧𝑧𝐹^′(𝑧)^′/𝐹(𝑧)−𝑧𝐹^′(𝑧)/𝐹(𝑧)²1+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)(𝑧)+𝑒/𝐹−11+^𝑖𝜆𝑏cos𝜆𝑧𝐹^′(𝑧)𝐹(𝑧)−1^𝜁=()𝑤(𝑧−11−𝛼)1+𝑏𝑒^−𝑖𝜆cos𝜆𝜁𝑧𝑤^′(𝑧)𝑤(𝑧)1()𝑤(𝑧).1+1−𝛼 (13)
Thus, we have 𝔽₁(𝑧)=𝑧𝑧𝐹^′(𝑧)^′/𝐹(𝑧)−𝑧𝐹^′(𝑧)/𝐹(𝑧)²1+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)/𝐹(𝑧)−11+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)/𝐹(𝑧)−1^𝜁=−1𝑏𝑒^−𝑖𝜆cos𝜆𝜁𝑧𝑤^′(𝑧)𝑤(𝑧)1()𝑤(𝑧),1+1−𝛼𝔽₂(𝑧)=𝑧𝑧𝐹^′(𝑧)^′/𝐹(𝑧)−𝑧𝐹^′(𝑧)/𝐹(𝑧)²1+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)(𝑧)=/𝐹−1𝑏𝑒^−𝑖𝜆cos𝜆𝜁(1−𝛼)𝑧𝑤^′(𝑧)()𝑤(𝑧),1+1−𝛼𝔽₃(𝑧)=𝑧𝑧𝐹^′(𝑧)^′/𝐹(𝑧)−𝑧𝐹^′(𝑧)/𝐹(𝑧)²1+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)/𝐹(𝑧)−1^𝜁+1=𝑏𝑒^−𝑖𝜆cos𝜆𝜁(1−𝛼)𝑧𝑤^′(𝑧)[1+(1−𝛼)𝑤(𝑧)]²,𝔽₄(𝑧)=𝑒1+^𝑖𝜆𝑏cos𝜆𝑧𝐹^′(𝑧)𝐹(𝑧)−1^𝜁−1×𝑧𝑧𝐹^′(𝑧)^′𝐹(𝑧)−𝑧𝐹^′(𝑧)𝐹(𝑧)²=𝑏𝑒^−𝑖𝜆cos𝜆𝜁(1−𝛼)𝑧𝑤^′(𝑧),𝔽₅(𝑧)=1+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)/𝐹(𝑧)−1^𝜁−1×𝑧𝑧𝐹^′(𝑧)^′/𝐹(𝑧)−𝑧𝐹^′(𝑧)/𝐹(𝑧)²×1+𝑒^𝑖𝜆/𝑏cos𝜆𝑧𝐹^′(𝑧)/𝐹(𝑧)−1^𝜁−1⁻¹=𝑏𝑒^−𝑖𝜆cos𝜆𝜁𝑧𝑤^′(𝑧)𝑤(𝑧). (14)(15)(16)(17)(18)
We claim that |𝑤(𝑧)|<1 in 𝒰. For otherwise, by Lemma 1, there exists 𝑧₀∈𝒰 such that 𝑧₀𝑤^′(𝑧₀)=𝑘𝑤(𝑧₀), where |𝑤(𝑧₀)|=1 and 𝑘>1. Therefore, (14)–(18) yield 𝔽₁𝑧₀=𝑏𝑒^−𝑖𝜆cos𝜆𝜁𝑧₀𝑤^′𝑧₀𝑤𝑧₀11+(1−𝛼)𝑤𝑧₀=(𝑏cos𝜆)𝜁𝑘𝑤𝑧₀1+(1−𝛼)𝑤𝑧₀≥𝑏𝜁cos𝜆,𝔽2−𝛼₂𝑧₀=𝑏𝑒^−𝑖𝜆cos𝜆𝜁(1−𝛼)𝑧₀𝑤^′𝑧₀1+(1−𝛼)𝑤𝑧₀=(𝑏cos𝜆)𝜁𝑘(1−𝛼)𝑤𝑧₀1+(1−𝛼)𝑤𝑧₀≥𝑏𝜁(1−𝛼)cos𝜆,𝔽2−𝛼₃𝑧₀=𝑏𝑒^−𝑖𝜆cos𝜆𝜁(1−𝛼)𝑧₀𝑤^′𝑧₀1+(1−𝛼)𝑤(𝑧₀)²=(𝑏cos𝜆)𝜁𝑘(1−𝛼)𝑤𝑧₀1+(1−𝛼)𝑤𝑧₀²≥𝑏𝜁(1−𝛼)cos𝜆(2−𝛼)²,𝔽₄𝑧₀=𝑏𝑒^−𝑖𝜆cos𝜆𝜁(1−𝛼)𝑧₀𝑤^′𝑧₀𝑏=cos𝜆𝜁𝑘(1−𝛼)𝑤𝑧₀≥𝑏𝜁)𝔽(1−𝛼cos𝜆,Re₅𝑧₀=Re𝑏𝑒^−𝑖𝜆cos𝜆𝜁𝑧₀𝑤^′𝑧₀𝑤𝑧₀()𝑏=𝑘cos𝜆Re𝜁𝑒^−𝑖𝜆≥cos𝜆𝜁²Re𝑏𝜁𝑒^−𝑖𝜆, (19) which contradicts our assumptions (6)–(10), respectively. Therefore, |𝑤(𝑧)|<1 holds true for all 𝑧∈𝒰. We finally have 𝑒1+^𝑖𝜆𝑏cos𝜆𝑧𝐹^′(𝑧)𝐹(𝑧)−1^𝜁−1<1−𝛼, (20) thus, we have 𝑒Re1+^𝑖𝜆𝑏cos𝜆𝑧𝐹^′(𝑧)𝐹(𝑧)−1^𝜁>𝛼,(𝑧∈𝒰;0≤𝛼<1). (21)

Remark 3. Taking different choices of 𝛼,𝜆,  𝜇, and 𝜁 in Theorem 2, we obtain new sufficient conditions for functions to be in the classes 𝒮^𝜆_𝛼(𝑏), 𝒞^𝜆_𝛼(𝑏),𝒮^∗_𝛼(𝑏), 𝒞_𝛼(𝑏), 𝒮(𝑏), 𝒞(𝑏),  𝒮^∗_𝛼, and 𝒞_𝛼.

Acknowledgment

The author would like to thank the referee for his helpful comments and suggestions.