New subclass of generalized close-to-convex function related with quasi-subordination

1.   Introduction

Let $\mathcal{A}$ denote the class of functions of the normalized form

which are analytic in the unit open disk

and all coefficients are complex numbers. Let $\Phi$ denote the set of analytic function with positive real part on $\mathbb{D}$ with

and $\phi(z)$ maps $\mathbb{D}$ onto a region starlike with respect to 1 and symmetric with respect to the $x$-axis. And, the function $\phi(z)$ has a series expansion of the form

where all coefficients $A_k(k\geq 1)$ are real number and $A_1 > 0$. Also let $\mathcal{U}$ denote the class of Schwartz functions, which is analytic in $\mathbb{D}$ satisfying

In 1970, Robertson ^[1] introduced the concept of quasi-subordination. For two analytic functions $f_1(z)$ and $f_2(z)$, the function $f_1(z)$ is quasi-subordinate to $f_2(z)$ in $\mathbb{D}$, denoted by

if there exists a Schwarz function $u(z)\in\mathcal{U}$ and an analytic function $h(z)$ with

such that

Observe that when

then

and it is said that $f_1(z)$ is subordinate to $f_2(z)$ and written

in $\mathbb{D}$. Also notice that if

then

and it is said that $f_1(z)$ is majorized by $f_2(z)$ and written

in $\mathbb{D}$. Hence it is obvious that quasi-subordination is a generalization of subordination as well as majorization. For works related to early study of the quasi-subordination concept, see ^[2,3,4].

In order to further explore the concept of quasi-subordination, some researchers have extended the construction of function classes and obtained some geometric properties of function classes. In 2012, Mohd and Darus generalized Ma-Minda starlike and convex classes in ^[5] and defined the generalized starlike class $\mathcal{S}^*_q(\phi)$ and the generalized convex class $\mathcal{C}_q(\phi)$ by using quasi-subordination, as below

And, they defined the following function class (also see ^[6])

In 2015, El-Ashwah et al. ^[7] introduced the generalized starlike class $\mathcal{S}^*_q(\mu; \phi)$ of complex order and the generalized convex class $\mathcal{C}_q(\mu; \phi)$ of complex order as follows,

In 2020, Ramachandran et al. ^[8] defined the class $\mathcal{M}^*_q(\alpha, \beta, \lambda; \psi)$ by using quasi-subordination. The function $f(z)\in \mathcal{A}$ is in the class $\mathcal{M}^*_q(\alpha, \beta, \lambda; \phi)$ if

where $\phi(z)\in\Phi$ and $0\leq\alpha, \beta, \lambda\leq1$. Many authors have studied various function subclasses defined by quasi-subordination. For example, Vays et al. ^[9], Altinkaya et al. ^[10], Goyal et al. ^[11] and Choi et al. ^[12] studied bi-univalent functions using quasi-subordination. Shah et al. ^[13] and Aoen et al. ^[14] introduced meromorphic functions using quasi-subordination. Karthikeyan et al. ^[15] studied Bazilević function using quasi-subordination. Shah et al. ^[16] studied non-Bazilević function using quasi-subordination. And, there are some function subclasses of linear and nonlinear operators (such as, hohlov operator ^[17], difference operator ^[18] and derivative operator ^[19]) using quasi-subordination.

Recently, some researchers have begun to generalize close-to-convex function classes by using quasi-subordination relationship. In 2019, Gurmeet Singh et al. ^[20] introduced the subclass of bi-close-to-convex function defined by quasi-subordination. In 2023, Aoen et al. ^[21] introduced the class of generalized close-to-convex function with complex order written as $\mathcal{K}_q(\gamma; \phi, \psi)$. This class were defined as below

In order to denote a new function class, we need to introduce the following function subclasses.

Definition 1.1. Let

Also let $\phi(z)\in\Phi$. A function $f(z)\in\mathcal{A}$ given by $(1.1)$ is said to be in the class $\mathcal{M}_{q}(\alpha, \mu; \phi)$ if the following condition is satisfied

Example 1.2. Let

The function

defined by the following

belongs to the class $\mathcal{M}_{q}(\alpha, \mu; \phi)$.

Remark 1.3. There are some suitable choices of $\alpha, \mu$ which would provide some classical subclasses of analytic functions.

(1) By taking $\mu = 1$ in Definition 1.1, we have

which is introduced by Mohd et al. ^[5].

(2) By taking $\alpha = 0$ in Definition 1.1, we have

which is introduced by El-Ashwah et al.^[7]. Specially, for $\mu = 1$ we have

which is introduced and studied by Mohd et al. ^[5].

(3) By taking $\alpha = 1$ in Definition 1.1, we have

which is introduced by El-Ashwah et al. ^[7]. Specially, for $\mu = 1$ we have

which is introduced and studied by Mohd et al. ^[5].

Now we define a generalization class of close-to-convex function by using quasi-subordination relationship.

Definition 1.4. Let

Also let

A function $f(z)\in\mathcal{A}$ given by $(1.1)$ is said to be in the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ if the following condition is satisfied

Example 1.5. Let

The function

defined by the following

belongs to the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$.

Remark 1.6. There are some suitable choices of $\alpha, \beta, \mu, \gamma$ which would provide the following subclasses of the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$.

(1) By taking $\beta = 0$ in Definition 1.4, the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ reduces to the new subclass $\mathcal{K}_{q}(\alpha, \mu, \gamma; \phi, \psi)$ which is the class of generalized close-to-convex function satisfied by

Specially, for $\alpha = 1, \mu = 1$ in the class $\mathcal{K}_{q}(\alpha, \mu, \gamma; \phi, \psi)$, we have

for $\alpha = 0, \mu = 1$ in the class $\mathcal{K}_{q}(\alpha, \mu, \gamma; \phi, \psi)$, we have

which is introduced and studied by Aoen et al. ^[21]. Also, for $\gamma = 1$ in the class $\mathcal{K}_{q}(\gamma; \phi, \psi)$, we have the class $\mathcal{K}_{q}(\phi, \psi)$ which is introduced and studied by Aoen et al. ^[21].

(2) By taking $\beta = 1$ in Definition 1.4, the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ reduces to the new subclass $\mathcal{C}^*_{q}(\alpha, \mu, \gamma; \phi, \psi)$ which is the class of generalized quasi-convex function satisfied by

Specially, for $\alpha = 1, \mu = 1$ in the class $\mathcal{C}^*_{q}(\alpha, \mu, \gamma; \phi, \psi)$, we have

for $\alpha = 0, \mu = 1$ in the class $K_{q}(\alpha, \mu, \gamma; \phi, \psi)$, we have

Studying the theory of analytic functions has been an area of concern for many researchers. The study of coefficients estimate is a more special and important field in complex analysis. For example, the bound for the second coefficient $a_2$ of normalized univalent functions readily yields the growth and distortion bounds for univalent functions. The coefficient functional $|a_3-\mu a_2^2|$ (that is, Fekete-Szegö problem) also naturally arises in the investigation of univalency of analytic functions. There are now many results of this type in the literature, each of them dealing with coefficient estimate for various classes of functions. In particular, some authors start to study the coefficient estimates for various classes using quasi-subordination. For example, Arikan et al. ^[22] and Marut et al. ^[23] studied the Fekete-Szegö problem for some function subclasses using quasi-subordination. Aoen et al. ^[24] and Ahman et al. ^[25] obtained the results on coefficient estimates for various subclasses using quasi-subordination. The purpose of this paper is to study some properties of the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ and some of its subclasses, such as the integral expression, the first two coefficient estimate problems and Fekete-Szegö problem. Our results are new in this direction and they give birth to many corollaries.

In order to derive our main results, we have to recall here the following lemmas.

Lemma 1.7. Let $f(z)\in \mathcal{C}_{q}(\phi)$, then

where

Proof. Since

then there exist two analytic functions $h(z), u(z)$ with

such that

By substitution, the Eq (1.3) can be reduced to a first-order differential equation. According to the method of solving the first-order differential equations, we can obtain the general solution of the equation. That is,

Integrating both sides of Eq (1.4), we get (1.2). Thus, the proof of Lemma 1.7 is complete.□

Lemma 1.8. ^[21] Let $f(z)\in\mathcal{S}_q^*(\phi)$, then

where

Lemma 1.9. ^[26] Let

be an analytic function in $\mathbb{D}$ with $|\varphi(z)|\leq1$, then

Lemma 1.10. ^[27] Let

be an analytic function in $\mathbb{D}$ with $|t(z)| < 1$, then

where $\mu\in\mathbb{C}.$ The result is sharp for the functions

2.   Integral expressions

In this section, we discuss the integral expressions for the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ and some of its subclasses by using methods for solving differential equations.

Theorem 2.1. Let

the function $f(z)\in\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ be given by (1.1). Then,

(ⅰ) If $\beta\neq0$, then

(ⅱ) If $\beta = 0$, then

where

Proof. Since $f(z)\in\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$, then there exist two analytic functions $h(z), u(z)$ with

such that

Then, we have

Let

then we have

Then, the above equation is a first-order nonhomogeneous linear differential equation. According to the method of solving first-order linear differential equations, we can obtain the general solution of the equation. That is,

Integrating both sides of Eq (2.2), we get (2.1). Thus, the proof of Theorem 2.1 is complete. □

By taking $\beta = 1$ in Theorem 2.1, we obtain the following result.

Corollary 2.2. Let the function $f(z)\in \mathcal{C}^*_{q}(\alpha, \mu, \gamma; \phi, \psi)$ be given by (1.1). Then

where

According to Lemmas 1.7 and 1.8 and Corollary 2.2, we can obtain the following two results.

Corollary 2.3. Let the function $f(z)\in \mathcal{C}^*_{q}(\gamma; \phi, \psi)$ be given by (1.1). Then

where

Corollary 2.4. Let the function $f(z)\in \mathcal{L}_{q}(\gamma; \phi, \psi)$ be given by (1.1). Then

where

3.   Coefficient estimates problem

In this section, we obtain the first two coefficient estimate and Fekete-Szegö problem for the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ and some subclasses of this class by using algebraic operations, fundamental inequalities of analytic functions.

In addition to special statements, suppose the Taylor series expression for the following functions, as follows

In order to derive our main results, we have to discuss the first two coefficient estimates and Fekete-Szegö problem for the class $\mathcal{M}_{q}(\alpha, \mu; \phi)$.

Theorem 3.1. Let $\alpha\in[0, 1], \mu\in\mathbb{C}\backslash\{0\},$ the function $f(z)\in\mathcal{M}_{q}(\alpha, \mu; \phi)$ be given by (1.1). Then

and for any $\eta\in\mathbb{C}$,

where

Proof. If $f(z)\in\mathcal{M}_{q}(\alpha, \mu; \phi)$, according to Definition 1.1, there exist analytic functions $\varphi(z)$ and $u(z)$, with

such that

By substituting the Taylor series expression for the function $f(z)$ to the left of the above expression, we have

Thus we get the following expression

And by substituting the power series expression of the functions $\varphi(z), \phi(z), u(z)$ to the right of (3.4), we can get the following expression

By substituting (3.5) and (3.6) into (3.4) and comparing the coefficients of the same power terms on both sides, we can get

Further,

Applying Lemmas 1.9 and 1.10 to $(3.7)$, we obtain

Since $\varphi(z)$ is analytic and bounded in $\mathbb{D}$, using ^[28], for some

Replacing the value of $c_1$ as defined above, we get

If $c_0 = 0$, then applying Lemmas 1.9 and 1.10 to $(3.9)$, we obtain

If $c_0\neq0$, let

which is a polynomial in $c_0$ and have analytic in $|c_0|\leq1$.

According to Maximum modulus principle, we get

Thus

Applying Lemma 1.10 to (3.11), we can conclude (3.3). For $\eta = 0$ in (3.3), we have (3.2).

Let

or

then the results of (3.1)–(3.3) are sharp. Thus, the proof of Theorem 3.1 is complete. □

Remark 3.2. (1) For $\mu = 1$ in Theorem 3.1, we can obtain the result which is Theorem 2.10 in ^[5].

(2) For $\mu = 1, \alpha = 0$ and $\mu = 1, \alpha = 1$ in Theorem 3.1, we can obtain the results which are Theorems 2.1 and 2.4 in ^[5], respectively.

(3) For $\alpha = 0$ and $\alpha = 1$ in Theorem 3.1, we improve the results which are Theorems 2.1 and 2.7 in ^[7], respectively.

Theorem 3.3. Let

the function $f(z)\in\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$ be given by (1.1). Then

and for any $\tau\in\mathbb{C}$

where

Proof. If $f(z)\in\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$, then there exist analytic functions $h(z)$ and $v(z)$, with

such that

By substituting the Taylor series expression for the functions $f(z), g(z)$ to the left of the above expression, we have

Thus we get the following expression

Similar to the proof of Theorem 3.1, by substituting the power series expression of the functions $h(z), \psi(z), v(z)$ to the right of (3.15), we can get the following expression

By substituting (3.16) and (3.17) into (3.15) and comparing the coefficients of the same power terms on both sides, we can get

and

Further,

Applying Lemmas 1.9 and 1.10 to (3.18) and (3.19), we obtain

and

According to Theorem 3.1, it follows that

and, for any complex number $\tau$, we have

where

Similar to the proof of Theorem 3.1, we can also get the following inequality

where

By substituting (3.22) into (3.20), we get (3.12). And by substituting (3.22)–(3.24) into (3.21), we can conclude (3.14). For $\tau = 0$ in (3.14), we have (3.13).

The results of (3.12) and (3.13) are sharp for $\beta\neq0$ if

or

and the results of (3.14) and (3.15) are sharp for $\beta = 0$ if

or

Thus, the proof of Theorem 3.3 is complete. □

By taking special values of parameters $\alpha, \beta, \mu$ in Theorem 3.3, we can obtain coefficient estimates for functions belonging to some subclasses of the class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$.

Corollary 3.4. Let the function $f(z)\in\mathcal{K}_{q}(\alpha, \mu, \gamma; \phi, \psi)$. Then

and for any $\tau\in\mathbb{C}$,

Remark 3.5. For $\alpha = \beta = 0, \mu = 1$ in Theorem 3.3 or $\alpha = 0, \mu = 1$ in Corollary 3.4, we obtain the result which is Corollary 3 in ^[21].

Corollary 3.6. Let the function $f(z)\in\mathcal{H}_{q}(\gamma; \phi, \psi)$. Then

and for any $\tau\in\mathbb{C}$

Especially, let

we can obtain the following result.

Remark 3.7. Let the function

Then

and for any $\tau\in\mathbb{C}$

The sharpness of the estimates is demonstrated by the functions

or

and the graph of the functions $f_1(z)$ and $f_1(z)$ are shown as follows,

In Figures 1 and 2, the three-dimensional coordinate system, coupled with color, is used to represent complex functions. Specifically, the $x$-axis corresponds to the real part of the variable $z$, the $y$-axis to the imaginary part of $z$, the $z$-axis indicates the real part of the function, and the color signifies the imaginary part of the function.

Corollary 3.8. Let the function $f(z)\in\mathcal{C}^*_{q}(\alpha, \mu, \gamma; \phi, \psi)$. Then

and for any $\tau\in\mathbb{C}$,

Corollary 3.9. Let the function $f(z)\in\mathcal{C}^*_{q}(\gamma; \phi, \psi)$. Then

and for any $\tau\in\mathbb{C}$,

Corollary 3.10. Let the function $f(z)\in\mathcal{L}_{q}(\gamma; \phi, \psi)$. Then

and for any $\tau\in\mathbb{C}$,

4.   Conclusions

In this paper, we introduce the new function class $\mathcal{C}_{q}(\alpha, \beta, \mu, \gamma; \phi, \psi)$, which is a expanded close-to-convex functions defined by quasi-subordination. We mainly study the integral expression, the first two coefficient estimates and Fekete-Szegö problem for this class and some of its subclasses. In the future, we can consider to study other forms of coefficient estimation, such as Milin coefficient eatimate, Zal-cman functional estimate, high order Hankel Determinant estimate for these classes using the concepts dealt with in the paper.

Author contributions

Aoen: conceptualization, methodology, software, investigation, writing—original draft preparation, writing—review and editing, project administration, funding acquisition, visualization; Shuhai Li: conceptualization, methodology, formal analysis, resources; Tula: validation, data curation; Shuwen Li and Hang Gao: supervision. All authors have read and agreed to the published version of the manuscript.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The present investigation was supported by the Natural Science Foundation of Inner Mongolia Autonomous Region of China under Grant (No. 2025MS01034, 2024MS01014, 2020MS01010), the National Natural Science Foundation of China (Grant No.11561001) and the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant (No. NJYT-18-A14).

Conflicts of interest

The authors declare no conflicts of interest.