On new subclasses of bi-starlike functions with bounded boundary rotation

1.   Introduction

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

which are analytic in the open unit disk $\mathbb{U} = \{z: z \in \mathbb{C}\quad {\rm and}\quad |z| < 1\}.$ Further, denote by $\mathcal{S}$ the class of all functions in $\mathcal{A}$ which are univalent in $\mathbb{U}$ and normalized by the condition $f(0) = 0 = f'(0)-1$.

One of the important and well examined subclasses of $\mathcal{S}$ is the class $\mathcal{S}^*(\alpha)$ of starlike functions of order $\alpha, (0\leq \alpha < 1)$, defined by the condition

and the class $\mathcal{K}(\alpha)\subset\mathcal S$ of convex functions of order $\alpha, (0\leq \alpha < 1)$, is defined by the condition

The class $\mathcal{B}_\lambda(\alpha)$ of $\lambda$-pseudo-starlike functions of order $\alpha, (0\leq \alpha < 1)$ was introduced and investigated by Babalola ^[1]. A function $f$, $f\in\mathcal A$ is in the class $\mathcal{B}_\lambda(\alpha)$ if it satisfies

In ^[1] it was showed that all pseudo-starlike functions are Bazilevič functions of type $\left(1-1/\lambda\right)$ and of order $\alpha^{1/\lambda}$ and univalent in $\mathbb{U}$.

In ^[13] Padmanabhan and Parvatham defined the classes of functions $\mathcal{P}_{m}(\beta)$ as follows:

Definition 1.1. ^[13] Let $\mathcal{P}_{m}(\beta)$, with $m\geq2$ and $0\leq\beta < 1$, denote the class of univalent analytic functions $P$, normalized with $P(0) = 1$, and satisfying

where $z = re^{i\theta}\in\mathbb{U}$.

For $\beta = 0$, we denote $\mathcal{P}_m: = \mathcal{P}_m(0)$, hence the class $\mathcal{P}_m$ represents the class of functions $p$ analytic in $\mathbb{U}$, normalized with $p(0) = 1$, and having the representation

where $\mu$ is a real-valued function with bounded variation, which satisfies

Details referring the above integral representation could be found in [13,Lemma 1]. Remark that $\mathcal{P}: = \mathcal{P}_2$ is the well-known class of Carathéodory functions, i.e. the normalized functions with positive real part in $\mathbb{U}$.

Lemma 1.1.([6,Lemma 2.1]) Let the function $\Phi(z) = 1+\sum\limits_{n = 1}^\infty{h_n}{z^n}$, $z\in\mathbb{U}$, be such that $\Phi\in\mathcal{P}_m(\beta)$. Then,

Supposing that the functions $p, q\in\mathcal{P}_m(\beta)$, with

from Lemma 1.1 it follows that

It is well known that every univalent function $f\in \mathcal{S}$ of the form (1.1), has an inverse $f^{-1}(w)$ defined in $\left(|w| < r_0(f); r_0(f)\geq\frac{1}{4}\right),$ where

A function $f\in \mathcal{S}$ is said to be bi-univalent in $\mathbb{U}$ if there exists a function $g\in \mathcal{S}$ such that $g(z)$ is an univalent extension of $f^{-1}$ to $\mathbb{U}$. Let $\Sigma$ denote the class of bi-univalent functions in $\mathbb{U}$. The functions $\frac{z}{1-z}$, $-\log(1-z)$, $\frac{1}{2}\log\left(\frac{1+z}{1-z}\right)$ are in the class $\Sigma$ ^[14]. However, the familiar Koebe function is not bi-univalent. Lewin ^[8] investigated the class of bi-univalent functions $\Sigma$ and obtained a bound $|a_2|\leqq 1.51$. Further Brannan and Clunie ^[3], Brannan and Taha ^[4] also worked on certain subclasses of the bi-univalent function class $\Sigma$ and obtained estimates for their initial coefficients. Various classes of bi-univalent functions were introduced and studied in recent times, the study of bi-univalent functions gained momentum mainly due to the work of Srivastava et al.^[14]. Motivated by this, many researchers ^{[2,5,11,14,15,16,17,18,19,20]} recently investigated several interesting subclasses of the class $\Sigma$ and found non-sharp estimates on the first two Taylor-Maclaurin coefficients.

Motivated by the aforementioned work on bi-univalent functions and recent works in ^[7,10], in this paper we define two new subclasses $\mathcal{B}_{\Sigma}^\lambda(m, \mu),$ $\lambda$-bi-pseudo-starlike functions and $\mathcal{L}_{\Sigma}^\eta(m, \beta)$ of $\Sigma$ and determine the bounds for the initial Taylor-Maclaurin coefficients of $|a_2|$ and $|a_3|$ for $f\in\mathcal{B}_{\Sigma}^\lambda(m, \mu)$ and $f\in\mathcal{L}_{\Sigma}^\eta(m, \beta)$.

Definition 1.2. Assume that $f\in \Sigma$, $\lambda\geq 1$ and $(f'(z))^\lambda$ is analytic in $\mathbb U$ with $(f'(0))^\lambda = 1$. Furthermore, assume that $g(z)$ is an univalent extension of $f^{-1}$ to $\mathbb{U}$, and $(g'(z))^\lambda$ is analytic in $\mathbb U$ with $(g'(0))^\lambda = 1$. Then $f(z)$ is said to be in the class $\mathcal{B}_{\Sigma}^\lambda(m, \mu)$ of $\lambda$-bi-pseudo-starlike functions if the following conditions are satisfied:

and

where $0\leq \mu\leq 1$.

Remark 1.1. For $\lambda = 1,$ a function $f\in \Sigma$ is in the class $\mathcal{B}_{\Sigma}^1(m, \mu)\equiv\mathcal{M}_{\Sigma}(m, \mu)$ if the following conditions are satisfied:

where $z, w\in\mathbb{ U}$ and the function $g$ is described in (1.4).

Remark 1.2. For $\lambda = 1;\mu = 1,$ a function $f\in \Sigma$ is in the class $\mathcal{B}_{\Sigma}^1(m, 1)\equiv\mathcal{S}^*_{\Sigma}(m)$ if the following conditions are satisfied:

where $z, w\in\mathbb{ U}$ and the function $g$ is described in (1.4).

Remark 1.3. For $\lambda = 2; \mu = 1,$ a function $f\in \Sigma$ is in the class $\mathcal{B}_{\Sigma}^2(m, 1)\equiv\mathcal{G}_{\Sigma}(m)$ if the following conditions are satisfied:

where $z, w\in\mathbb{ U}$ and the function $g$ is described in (1.4).

Remark 1.4. For $\mu = 0,$ a function $f\in \Sigma$ is in the class $\mathcal{B}_{\Sigma}^\lambda(m, 0)\equiv\mathcal{R}^\lambda_{\Sigma}(m)$ if the following conditions are satisfied:

where $z, w\in\mathbb{ U}$ and the function $g$ is described in (1.4).

Remark 1.5. For $\lambda = 1; \mu = 0,$ a function $f\in \Sigma$ is in the class $\mathcal{B}_{\Sigma}^1(m, 0)\equiv\mathcal{N}_{\Sigma}(m)$ if the following conditions are satisfied:

where $z, w\in\mathbb{ U}$ and the function $g$ is described in (1.4).

2.   Coefficient estimates for $f\in\mathcal{B}_{\Sigma}^\lambda(m, \mu)$

Theorem 2.1. Let $f(z)$ given by (1.1) be in the class $\mathcal{B}_{\Sigma}^\lambda(m, \mu)$, then

and

where

Proof. It is known that $g$ has the form

Since $f\in \mathcal{B}_{\Sigma}^\lambda(m, \mu)$, there exists two analytic functions

and

then

On the other hand, we have

Using (2.3), (2.4), (2.7) and (2.8) and comparing the like coefficients of $z$ and $z^2$, we get

From (2.9) and (2.11), we find that

from Lemma 1.1 it follows that

Adding (2.10) and (2.12), we have

Hence by Lemma 1.1

Subtracting (2.10) from (2.12), we obtain

By using (2.9) and (2.10) and by simple computation, we get

Again by using (2.9) and (2.12)

From (2.12) we have

Furthermore by

where

This completes the proof of Theorem 2.1.

Remark 2.1. Specializing $\lambda, \mu$ suitably as mentioned in Remarks 1.1 to 1.5 we can state the initial Taylor coefficients $|a_2|, \; |a_3|$ and the inequality $|a_3-\delta a_2^2|$ for the function classes defined in Remarks 1.1 to 1.5.

3.   Coefficient estimates for $f\in\mathcal{L}_{\Sigma}^\eta(m, \beta)$

In ^[12], Obradovic et al. gave some criteria for univalence expressing by $\Re(f'(z)) > 0,$ for the linear combinations

Based on the above definition recently, in ^[9], Lashin introduced and studied the new subclass of bi-univalent functions. We define the following new bi-univalent function class:

Definition 3.1. A function $f(z)\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{L}_{\Sigma}^\eta(m, \beta)$ if it satisfies the following conditions :

and

where $\eta \geq 1, z, w \in \mathbb{U}$ and the function $g$ is given by (1.4).

Theorem 3.1. Let $f(z)$ be given by (1.1) be in the class $\mathcal{L}_{\Sigma}^\eta(m, \beta)$, $\eta \geq 1$. Then

and

where

Proof. It follows from (3.1) and (3.2) that

and

From (3.5) and (3.6), we have

and

Now, equating the coefficients, we get

and

From (3.7) and (3.9), we get

it follows that

Now by adding (3.8) and (3.10), we obtain

which, by virtue of Lemma 1.1, implies that

Hence

Subtracting (3.10) from (3.8), we obtain

By using (3.7) and (3.8) and by simple computation, we get

Again by using (3.7) in (3.10)

From (3.10) we have

Furthermore by

where

This completes the proof of Theorem 3.1.

Corollary 3.2. Let $f(z)$ be given by (1.1) be in the class $\mathcal{L}_{\Sigma}^\eta(m, \beta)$, $\eta = 1$. Then

and

where

4.   Conclusion

In this paper, we introduce two new classes $\mathcal{B}_{\Sigma}^\lambda(m, \mu)$ of $\lambda$-pseudo bi-starlike functions and $\mathcal{L}_{\Sigma}^\eta(m, \beta)$ and obtain the estimates of $|a_2|$, $|a_3|$ and the upper bounds of the Fekete-Szegö inequality, where $a_2$ and $a_3$ belong to $f\in\mathcal{B}_{\Sigma}^\lambda(m, \mu)$ and $f\in\mathcal{L}_{\Sigma}^\eta(m, \beta),$ respectively. In addition, we observe that, if we choose some suitable parameters $\lambda$, $\mu$, $\eta$ and $m$ in the results involved, we can get some corresponding bounds.

Acknowledgments

This work was supported by the Natural Science Foundation of the People's Republic of China (Grant No. 11561001), the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region (Grant No. NJYT-18-A14), the Natural Science Foundation of Inner Mongolia of the People's Republic of China (Grant No. 2018MS01026), the Higher School Science Research Foundation of Inner Mongolia of the People's Republic of China (Grant No. NJZY18217) and the Natural Science Foundation of Chifeng of Inner Mongolia. Also, the authors would like to thank the referees for their valuable comments and suggestions, which was essential to improve the quality of this paper.

Conflict of interest

The authors declare no conflicts of interest.