Classes of Meromorphic Functions Defined by the Hadamard Product

Abstract

The object of the present paper is to introduce new classes of meromorphic functions with varying argument of coefficients defined by means of the Hadamard product (or convolution). Several properties like the coefficients bounds, growth and distortion theorems, radii of starlikeness and convexity, and partial sums are investigated. Some consequences of the main results for well-known classes of meromorphic functions are also pointed out.

1. Introduction

Let ℳ denote the class of functions which are analytic in 𝒟=𝒟(1), where

𝒟(𝑟)={𝑧∈ℂ∶0<|𝑧|<𝑟}, (1.1) with a simple pole in the point 𝑧=0. By ℳ, we denote the class of functions 𝑓∈ℳ of the form

1𝑓(𝑧)=𝑧+^∞_𝑛=1𝑎_𝑛𝑧^𝑛(𝑧∈𝒟). (1.2) Also, by 𝒯^𝜀_𝜂(𝜂∈ℝ,𝜀∈{0,1}), we denote the class of functions 𝑓∈ℳ of the form (1.2) for which

arg(𝑎_𝑛)=𝜀𝜋−(𝑛+1)𝜂(𝑛∈ℕ∶={1,2,3,…}). (1.3) For 𝜂=0, we obtain the classes 𝒯⁰₀ and 𝒯¹₀ of functions with positive coefficients and negative coefficients, respectively.

Motivated by Silverman [1], we define the class

𝒯^𝜀∶=_𝜂∈ℝ𝒯^𝜀_𝜂. (1.4)

It is called the class of functions with varying argument of coefficients.

Let 𝛼∈⟨0,1),𝑟∈(0,1⟩. A function 𝑓∈ℳ is said to be meromorphically convex of order 𝛼 in 𝒟(𝑟) if

Re(1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧))<−𝛼(𝑧∈𝒟(𝑟)). (1.5)

A function 𝑓∈ℳ is said to be meromorphically starlike of order 𝛼 in 𝒟(𝑟) if

Re(𝑧𝑓^′(𝑧)𝑓(𝑧))<−𝛼(𝑧∈𝒟(𝑟)). (1.6) We denote by ℳ𝒮^𝑐(𝛼) the class of all functions 𝑓∈ℳ, which are meromorphically convex of order 𝛼 in 𝒟 and by ℳ𝒮^∗(𝛼), we denote the class of all functions 𝑓∈ℳ, which are meromorphically starlike of order 𝛼 in 𝒟. We also set

ℳ𝒮^𝑐=ℳ𝒮^𝑐(0),ℳ𝒮^∗=ℳ𝒮^∗(0). (1.7) It is easy to show that for a function 𝑓∈𝒯⁰_𝜂, the condition (1.6) is equivalent to the following:

|𝑧𝑓^′(𝑧)𝑓(𝑧)+1|<1−𝛼(𝑧∈𝒟(𝑟)). (1.8)

Let ℬ be a subclass of the class ℳ. We define the radius of starlikeness of order 𝛼 and the radius of convexity of order 𝛼 for the class ℬ by

𝑅^∗_𝛼(ℬ)=inf_𝑓∈ℬ𝑅(sup{𝑟∈(0,1]∶𝑓ismeromorphicallystarlikeoforder𝛼in𝒟(𝑟)});_𝑐𝛼(ℬ)=inf_𝑓∈ℬ(sup{𝑟∈(0,1]∶𝑓ismeromorphicallyconvexoforder𝛼in𝒟(𝑟)}), (1.9) respectively.

Let functions 𝑓,𝑔 be analytic in 𝒰∶=𝒟∪{0}. We say that the function 𝑓 is subordinate to the function 𝑔, and write 𝑓(𝑧)≺𝑔(𝑧) (or simply 𝑓≺𝑔) if there exists a function 𝜔 analytic in 𝒰, |𝜔(𝑧)|≤|𝑧|(𝑧∈𝒰), such that

𝑓(𝑧)=𝑔(𝜔(𝑧))(𝑧∈𝒰). (1.10) In particular, if 𝐹 is univalent in 𝒰, we have the following equivalence:

𝑓(𝑧)≺𝐹(𝑧)⇔𝑓(0)=𝐹(0),𝑓(𝒰)⊂𝐹(𝒰). (1.11)

For functions 𝑓,𝑔∈ℳ of the form

𝑓(𝑧)=^∞_𝑛=−1𝑎_𝑛𝑧^𝑛,𝑔(𝑧)=^∞_𝑛=−1𝑏_𝑛𝑧^𝑛, (1.12) by 𝑓∗𝑔 we denote the Hadamard product (or convolution) of 𝑓 and 𝑔, defined by

(𝑓∗𝑔)(𝑧)=^∞_𝑛=−1𝑎_𝑛𝑏_𝑛𝑧^𝑛(𝑧∈𝒟). (1.13)

Let 𝐴,𝐵 be real parameters, −1≤𝐴<𝐵≤1, and let 𝜑,𝜙 be given functions from the class ℳ.

By 𝒲(𝜙,𝜑;𝐴,𝐵), we denote the class of functions 𝑓∈ℳ such that (𝜑∗𝑓)(𝑧)≠0(𝑧∈𝒟) and

(𝜙∗𝑓)(𝑧)≺(𝜑∗𝑓)(𝑧)1+𝐴𝑧.1+𝐵𝑧 (1.14) Moreover, let us define

𝒯𝒲^𝜀(𝜙,𝜑;𝐴,𝐵)∶=𝒯^𝜀∩𝒲(𝜙,𝜑;𝐴,𝐵),𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵)∶=𝒯^𝜀_𝜂∩𝒲(𝜙,𝜑;𝐴,𝐵), (1.15) where the functions 𝜑,𝜙 have the form

1𝜙(𝑧)=𝑧+(−1)^∞𝜀+1_𝑛=1𝛽_𝑛𝑧^𝑛1,𝜑(𝑧)=𝑧+(−1)^𝜀∞_𝑛=1𝛼_𝑛𝑧^𝑛(𝑧∈𝒟), (1.16) and the sequences {𝛼_𝑛},{𝛽_𝑛} are nonnegative real, with

𝛼_𝑛+𝛽_𝑛>0(𝑛∈ℕ). (1.17) Moreover, let us put

𝑑_𝑛∶=(1+𝐵)𝛽_𝑛+(1+𝐴)𝛼_𝑛(𝑛∈ℕ). (1.18)

It is easy to show that

𝑓(𝑧)∈𝒯𝒲(𝑧𝜙^′(𝑧),𝑧𝜑^′(𝑧);𝐴,𝐵)⇔−𝑧𝑓^′(𝑧)∈𝒯𝒲(𝜙,𝜑;𝐴,𝐵),𝑓(𝑧)∈𝒯𝒲⁰_𝜂(𝑧𝜙^′(𝑧),𝑧𝜑^′(𝑧);𝐴,𝐵)⇔−𝑧𝑓^′(𝑧)∈𝒯𝒲¹_𝜂(𝜙,𝜑;𝐴,𝐵). (1.19)

The object of the present paper is to investigate the coefficient estimates, distortion properties and the radii of starlikeness and convexity, and partial sums for the classes of meromorphic functions with varying argument of coefficients. Some remarks depicting consequences of the main results are also mentioned.

2. Coefficients Estimates

First we mention a sufficient condition for functions to belong to the class 𝒲(𝜙,𝜑;𝐴,𝐵).

Theorem 2.1. Let {𝑑_𝑛} be defined by (1.18), −1≤𝐴<𝐵≤1. If a function 𝑓 of the form (1.2) satisfies the condition ^∞_𝑛=1𝑑_𝑛|𝑎_𝑛|≤𝐵−𝐴, (2.1) then 𝑓 belongs to the class 𝒲(𝜙,𝜑;𝐴,𝐵).

Proof. A function 𝑓 of the form (1.2) belongs to the class 𝒲^𝜀(𝜙,𝜑;𝐴,𝐵) if and only if there exists a function 𝜔,|𝜔(𝑧)|≤|𝑧|(𝑧∈𝒟), such that (𝜙∗𝑓)(𝑧)=(𝜑∗𝑓)(𝑧)1+𝐴𝜔(𝑧)1+𝐵𝜔(𝑧)(𝑧∈𝒟), (2.2) or equivalently |𝑧(𝜙∗𝑓)(𝑧)−𝑧(𝜑∗𝑓)(𝑧)𝐵𝑧(𝜙∗𝑓)(𝑧)−𝐴𝑧(𝜑∗𝑓)(𝑧)|<1(𝑧∈𝒟). (2.3) Thus, it is sufficient to prove that |𝑧(𝜙∗𝑓)(𝑧)−𝑧(𝜑∗𝑓)(𝑧)|−|𝐵𝑧(𝜙∗𝑓)(𝑧)−𝐴𝑧(𝜑∗𝑓)(𝑧)|<0(𝑧∈𝒟). (2.4) Indeed, letting |𝑧|=𝑟(0<𝑟<1), we have |𝑧(𝜑∗𝑓)(𝑧)−𝑧(𝜙∗𝑓)(𝑧)|−|𝐵𝑧(𝜙∗𝑓)(𝑧)−𝐴𝑧(𝜑∗𝑓)(𝑧)|=|^∞_𝑛=1(𝛽_𝑛+𝛼_𝑛)𝑎_𝑛𝑧^𝑛+1|−|(𝐵−𝐴)−^∞_𝑛=1(𝐵𝛽_𝑛+𝐴𝛼_𝑛)𝑎_𝑛𝑧^𝑛+1|≤^∞_𝑛=1(𝛽_𝑛+𝛼_𝑛)|𝑎_𝑛|𝑟^𝑛+1−(𝐵−𝐴)+^∞_𝑛=1(𝐵𝛽_𝑛+𝐴𝛼_𝑛)|𝑎_𝑛|𝑟^𝑛+1≤^∞_𝑛=1𝑑_𝑛|𝑎_𝑛|𝑟^𝑛+1−(𝐵−𝐴)<0, (2.5) whence 𝑓∈𝒲(𝜙,𝜑;𝐴,𝐵).

Theorem 2.2. Let 𝑓 be a function of the form (1.2), with (1.3). Then 𝑓 belongs to the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵) if and only if the condition (2.1) holds true.

Proof. In view of Theorem 2.1, we need only to show that each function 𝑓 from the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵) satisfies the coefficient inequality (2.1). Let 𝑓∈𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵). Then by (2.3) and (1.2), we have |∑^∞_𝑛=1(𝛽_𝑛+𝛼_𝑛)𝑎_𝑛𝑧^𝑛+1𝐵−𝐴−∑^∞_𝑛=1(−1)^𝜀(𝐵𝛽_𝑛+𝐴𝛼_𝑛)𝑎_𝑛𝑧^𝑛+1|<1(𝑧∈𝒟). (2.6) Therefore, putting 𝑧=𝑟𝑒^𝑖𝜂(0≤𝑟<1), and applying (1.3), we obtain ∑^∞_𝑛=1(𝛽_𝑛+𝛼_𝑛)|𝑎_𝑛|𝑟^𝑛+1𝐵−𝐴−∑^∞_𝑛=1(𝐵𝛽_𝑛+𝐴𝛼_𝑛)|𝑎_𝑛|𝑟^𝑛+1<1. (2.7) It is clear, that the denominator of the left hand said cannot vanish for 𝑟∈⟨0,1). Moreover, it is positive for 𝑟=0, and in consequence for 𝑟∈⟨0,1). Thus, by (2.7), we have ^∞_𝑛=1[(1+𝐵)𝛽_𝑛+(1+𝐴)𝛼_𝑛]|𝑎_𝑛|𝑟^𝑛+1<𝐵−𝐴, (2.8) which, upon letting 𝑟→1⁻, readily yields the assertion (2.1).

From Theorem 2.2, we obtain coefficients estimates for the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵).

Corollary 2.3. If a function 𝑓 of the form (1.2) belongs to the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵),  then |𝑎_𝑛|≤𝐵−𝐴𝑑_𝑛(𝑛∈ℕ), (2.9) where {𝑑_𝑛} is defined by (1.18). The result is sharp. The functions 𝑓_𝑛,𝜂 of the form 𝑓_𝑛,𝜂1(𝑧)=𝑧+𝐵−𝐴𝑒^{𝑖{(𝑛+1)𝜂−𝜀𝜋}}𝑑_𝑛𝑧^𝑛(𝑧∈𝒟;𝑛∈ℕ) (2.10) are the extremal functions.

3. Distortion Theorems

From Theorem 2.2, we have the following lemma.

Lemma 3.1. Let a function 𝑓 of the form (1.2) belong to the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵). If the sequence {𝑑_𝑛}defined by (1.18) satisfies the inequality 𝑑₁≤𝑑_𝑛(𝑛∈ℕ), (3.1) then ^∞_𝑛=1𝑎_𝑛≤𝐵−𝐴𝑑₁. (3.2) Moreover, if 𝑛𝑑₁≤𝑑_𝑛(𝑛∈ℕ), (3.3) then ^∞_𝑛=1𝑛𝑎_𝑛≤𝐵−𝐴𝑑₁. (3.4)

Theorem 3.2. Let a function 𝑓 belong to the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵). If the sequence {𝑑_𝑛} defined by (1.18) satisfies (3.1), then 1𝑟−𝐵−𝐴𝑑₁1𝑟≤|𝑓(𝑧)|≤𝑟+𝐵−𝐴𝑑₁𝑟(|𝑧|=𝑟<1). (3.5) Moreover, if (3.3) holds, then 1𝑟²−𝐵−𝐴𝑑₁≤|𝑓^′1(𝑧)|≤𝑟²+𝐵−𝐴𝑑₁(|𝑧|=𝑟<1). (3.6) The result is sharp, with the extremal function 𝑓_1,𝜂 of the form (2.10).

Proof. Let a function 𝑓 of the form (1.2) belong to the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵),|𝑧|=𝑟<1. Since 1|𝑓(𝑧)|=|𝑧+^∞_𝑛=1𝑎_𝑛𝑧^𝑛1|≤𝑟+^∞_𝑛=1|𝑎_𝑛|𝑟^𝑛=1𝑟+𝑟^∞_𝑛=1|𝑎_𝑛|𝑟^𝑛−1≤1𝑟+𝑟^∞_𝑛=1|𝑎_𝑛|,|𝑓(𝑧)|=|𝑧+^∞_𝑛=1𝑎_𝑛𝑧^𝑛1|≥𝑟−^∞_𝑛=1|𝑎_𝑛|𝑟^𝑛=1𝑟−𝑟^∞_𝑛=1|𝑎_𝑛|𝑟^𝑛−1≥1𝑟−𝑟^∞_𝑛=1|𝑎_𝑛|, (3.7) then by Lemma 3.1 we have (3.5). Analogously we prove (3.6).

4. The Radii of Convexity and Starlikeness

Theorem 4.1. The radius of starlikeness of order 𝛼 for the class 𝒯𝒲⁰_𝜂(𝜙,𝜑;𝐴,𝐵) is given by 𝑅^∗_𝛼(𝒯𝒲⁰_𝜂(𝜙,𝜑;𝐴,𝐵))=inf_𝑛∈ℕ((1−𝛼)𝑑_𝑛)(𝑛−𝛼)(𝐵−𝐴)^1/(𝑛+1), (4.1) where 𝑑_𝑛 is defined by (1.18).

Proof. The function 𝑓∈𝒯⁰_𝜂 of the form (1.2) is meromorphically starlike of order 𝛼 in the disk 𝒟(𝑟),0<𝑟≤1, if and only if it satisfies the condition (1.8). Since |𝑧𝑓^′(𝑧)∑𝑓(𝑧)+1|=|^∞𝑛=1(𝑛+1)𝑎_𝑛𝑧^𝑛+11+∑^∞_𝑛=1𝑎_𝑛𝑧^𝑛+1∑|≤^∞𝑛=1(𝑛+1)|𝑎_𝑛||𝑧|^𝑛+11−∑^∞_𝑛=1|𝑎_𝑛||𝑧|^𝑛+1, (4.2) putting |𝑧|=𝑟, the condition (1.8) is true if ^∞_𝑛=1𝑛−𝛼1−𝛼|𝑎_𝑛|𝑟^𝑛+1≤1. (4.3) By Theorem 2.2, we have ^∞_𝑛=1𝑑_𝑛𝐵−𝐴|𝑎_𝑛|≤1. (4.4) Thus, the condition (4.3) is true if 𝑛−𝛼𝑟1−𝛼^𝑛+1≤𝑑_𝑛𝐵−𝐴(𝑛∈ℕ), (4.5) that is, if 𝑟≤((1−𝛼)𝑑_𝑛)(𝑛−𝛼)(𝐵−𝐴)^1/(𝑛+1)(𝑛∈ℕ). (4.6) It follows that each function 𝑓∈𝒯𝒲⁰_𝜂(𝜙,𝜑;𝐴,𝐵) is meromorphically starlike of order 𝛼 in the disk 𝒟(𝑟), where 𝑟=𝑅^∗(𝒯𝒲⁰_𝜂(𝜙,𝜑;𝐴,𝐵)) is defined by (4.1). Moreover, the radius of starlikeness of the functions 𝑓_𝑛,𝜂 defined by (2.10) is given by 𝑟^∗_𝑛=min{1,((1−𝛼)𝑑_𝑛)(𝑛−𝛼)(𝐵−𝐴)^1/(𝑛+1)}(𝑛∈ℕ). (4.7) Thus we have (4.1).

Theorem 4.2. The radius of convexity of order 𝛼 for the class 𝒯𝒲¹_𝜂(𝜙,𝜑;𝐴,𝐵) is given by 𝑅^𝑐_𝛼(𝒯𝒲¹_𝜂(𝜙,𝜑;𝐴,𝐵))=inf_𝑛∈ℕ((1−𝛼)𝑑_𝑛)𝑛(𝑛−𝛼)(𝐵−𝐴)^1/(𝑛+1), (4.8) where 𝑑_𝑛 is defined by (1.18).

Proof. The proof is analogous to that of Theorem 4.1, and we omit the details.

5. Partial Sums

Let 𝑓∈ℳ be a function of the form (1.2). Motivated by Silverman [2] and Silvia [3] (see also [4]), we define the partial sums 𝑓_𝑚 defined by

𝑓_𝑚1(𝑧)=𝑧+^𝑚_𝑛=1𝑎_𝑛𝑧^𝑛(𝑚∈ℕ). (5.1) In this section, we consider partial sums of functions from the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵) and obtain sharp lower bounds for the real part of ratios of 𝑓 to 𝑓_𝑚 and 𝑓^′ to 𝑓^′_𝑚.

Theorem 5.1. Let 𝑚∈ℕ and let the sequence {𝑑_𝑛}, defined by (1.18), satisfy the inequalities 𝐵−𝐴≤𝑑_𝑛≤𝑑_𝑛+1(𝑚∈ℕ). (5.2) If a function 𝑓 belongs to the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵), then Re{𝑓(𝑧)𝑓_𝑚(𝑧)}≥1−𝐵−𝐴𝑑_𝑚+1𝑓(𝑧∈𝒟),Re{^𝑚(𝑧)𝑑𝑓(𝑧)}≥^𝑚+1𝐵−𝐴+𝑑_𝑚+1(𝑧∈𝒟). (5.3)(5.4) The bounds are sharp, with the extremal function 𝑓_𝑚+1,𝜂 of the form (2.10).

Proof. Let a function 𝑓 of the form (1.2) belong to the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵). Then by (5.2) and Theorem 2.2, we have ^𝑚_𝑛=1|𝑎_𝑛𝑑|+^𝑚+1𝐵−𝐴^∞_𝑛=𝑚+1|𝑎_𝑛|≤^∞_𝑛=1𝑑_𝑛𝐵−𝐴|𝑎_𝑛|≤1. (5.5) If we put 𝑑𝑔(𝑧)=^𝑚+1{𝐵−𝐴𝑓(𝑧)𝑓_𝑚(𝑧)−(1−𝐵−𝐴𝑑_𝑚+1)}=1+(𝑑_𝑚+1/(𝐵−𝐴))∑^∞_𝑛=𝑚+1𝑎_𝑛𝑧^𝑛+11+∑^𝑚_𝑛=1𝑎_𝑛𝑧^𝑛+1(𝑧∈𝒟), (5.6) then it suffices to show that Re𝑔(𝑧)≥0(𝑧∈𝒟), (5.7) or |𝑔(𝑧)−1𝑔(𝑧)+1|≤1(𝑧∈𝒟). (5.8) Applying (5.5), we find that |𝑔(𝑧)−1𝑔(𝑧)+1|≤(𝑑_𝑚+1/(𝐵−𝐴))∑^∞_𝑛=𝑚+1|𝑎_𝑛|2−2∑^𝑛_𝑛=1|𝑎_𝑛|−(𝑑_𝑚+1/(𝐵−𝐴))∑^∞_𝑛=𝑚+1|𝑎_𝑛|≤1(𝑧∈𝒟), (5.9) which readily yields the assertion (5.3). In order to see that 𝑓=𝑓_𝑚+1,𝜂 gives the result sharp, we observe that for 𝑧=𝑟𝑒^𝑖𝜂 we have 𝑓(𝑧)𝑓_𝑚(𝑧)=1−(𝐵−𝐴)𝑟^𝑚+2𝑑_𝑚+1^𝑟→1−−−−−−→1−𝐵−𝐴𝑑_𝑚+1. (5.10) Similarly, if we take ℎ(𝑧)=(𝐵−𝐴+𝑑_𝑚+1𝑓){^𝑚(𝑧)−𝑑𝑓(𝑧)𝐵−𝐴+^𝑚+1𝑑_𝑚+1}(𝑧∈𝒟), (5.11) and make use of (5.5), we can deduce that |ℎ(𝑧)−1ℎ(𝑧)+1|≤(1+(𝑑_𝑚+1/(𝐵−𝐴)))∑^∞_𝑛=𝑚+1|𝑎_𝑛|2−2∑^𝑚_𝑛=1|𝑎_𝑛|−(1−(𝑑_𝑚+1/(𝐵−𝐴)))∑^∞_𝑛=𝑚+1|𝑎_𝑛|≤1(𝑧∈𝒟), (5.12) which leads us immediately to the assertion (5.4). The bound in (5.4) is sharp for each 𝑚∈𝑁, with the extremal function 𝑓=𝑓_𝑚+1,𝜂, given by (2.10).

Theorem 5.2. Let 𝑚∈𝑁 and let the sequence {𝑑_𝑛}, defined by (1.18), satisfy the inequalities (5.2). If a function 𝑓 belongs to the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵), then 𝑓Re{^′(𝑧)𝑓_𝑚(𝑧)}≥1−(𝐵−𝐴)(𝑚+1)𝑑_𝑚+1𝑓(𝑧∈𝒟),Re{^′𝑚(𝑧)𝑑𝑓(𝑧)}≥^𝑚+1(𝐵−𝐴)(𝑚+1)+𝑑_𝑚+1(𝑧∈𝒟). (5.13) The bounds are sharp, with the extremal function 𝑓_𝑚+1,𝜂 of the form (2.10).

Proof. The proof is analogous to that of Theorem 5.1, and we omit the details.

Remark 5.3. We observe that the obtained results are true if we replace the class 𝒯𝒲^𝜀_𝜂(𝜙,𝜑;𝐴,𝐵) by 𝒯𝒲^𝜀(𝜙,𝜑;𝐴,𝐵).

6. Concluding Remarks

We conclude this paper by observing that, in view of the subordination relation (1.18), by choosing the functions 𝜙 and 𝜑, we can define new classes of functions. In particular, the class

𝒲(𝜑;𝐴,𝐵)∶=𝒲(−𝑧𝜑^′(𝑧),𝜑(𝑧);𝐴,𝐵) (6.1) contains functions 𝑓∈ℳ, such that

−𝑧(𝜑∗𝑓)^′(𝑧)≺(𝜑∗𝑓)(𝑧)1+𝐴𝑧.1+𝐵𝑧 (6.2) A function 𝑓∈ℳ belongs to the class

𝒲(𝜑;𝛼)∶=𝒲(𝜑;2𝛼−1,𝛼)(0≤𝛼<1) (6.3) if it satisfies the condition

Re{−𝑧(𝜑∗𝑓)^′(𝑧)(𝜑∗𝑓)(𝑧)}>𝛼(𝑧∈𝒟). (6.4) The class 𝒲(𝜑;𝛼) is related to the class of starlike function of order 𝛼. In particular, we have the following relationships:

𝒮^∗1(𝛼)=𝒲(𝒮𝑧(1−𝑧);𝛼),_𝑐(𝛼)=𝒲(1−2𝑧𝑧(1−𝑧)²;𝛼). (6.5)

Let 𝜆 be a convex parameter. A function 𝑓∈ℳ belongs to the class

𝒲_𝜆(𝜑;𝐴,𝐵)∶=𝒲(𝜆𝑧𝜑(𝑧)+(𝜆−1)𝑧²𝜑^′1(𝑧),𝑧;𝐴,𝐵) (6.6) if it satisfies the condition

𝜆𝑧(𝜑∗𝑓)(𝑧)+(𝜆−1)𝑧²(𝜑∗𝑓)^′(𝑧)≺1+𝐴𝑧.1+𝐵𝑧 (6.7) The classes 𝒲(𝜑;𝐴,𝐵),𝒲(𝜑;𝛼), and 𝒲_𝜆(𝜑;𝐴,𝐵) generalize well-known important classes, which were investigated in earlier works; see for example [5–10].

If we apply the results presented in this paper to the classes discussed above, we can obtain several additional results. Some of these results were obtained in earlier works; see for example [5–10].