Certain new subclasses of bi-univalent function associated with bounded boundary rotation involving sǎlǎgean derivative

1.   Introduction

Indicate $\mathcal{A}$ as the class of all functions $h:\mathbb{E}\rightarrow \mathbb{C}$ defined by

which are analytic in open unit disk $\mathbb{E}: = \left\{u \in \mathbb{C} : |u| < 1\right\}.$ Let $\mathcal{S}$ be the subclass of $\mathcal{A},$ which is univalent in $\mathbb{E}.$ Fix $0\leq\delta < 1.$ The well known subclasses $\mathcal{S}^{\ast}(\delta),$ $\mathcal{C}(\delta)$ and $\mathcal{R}(\delta)$ of class $\mathcal{S}$ are the class of starlike, convex, and the class of functions whose derivatives have positive real part of order $\delta,$ respectively. The analytic descriptions of the above classes are given by

and

Indicate $\mathcal{V}_{\vartheta}$ as the class of functions $h$ given in (1.1), which maps the open unit disk $\mathbb{E}$ conformally onto an image domain $h(\mathbb{E})$ of boundary rotation at most $\vartheta\pi.$ The functions belonging to the class $\mathcal{V}_{\vartheta}$ are known as functions of bounded boundary rotation. Pinchuk ^[15] introduced class $\mathcal{V}_{\vartheta}.$ Any function $h\in\mathcal{V}_{\vartheta}$ is expressed as

Assume $\mathcal{R}_{\vartheta}$ as the class of functions $h$ given in (1.1) which map open unit $\mathbb{E}$ conformally onto an image domain $h(\mathbb{E})$ of boundary radius rotation at most $\vartheta\pi.$ The functions belonging to the class $\mathcal{R}_{\vartheta}$ are known as functions of bounded radius rotation. If a function $h\in\mathcal{R} _{\vartheta},$ then it can be expressed as

Let $\mathcal{P}_{\vartheta}$ be the class of functions $t$ with $t(0) = 1$ in $\mathbb{E}$ and having an integral representation

where $\theta(\mu)$ is a function of bounded variation and satisfying

Assume $\mathcal{S}_{\vartheta}$ be the subclass of $\mathcal{V}_{\vartheta}$ whose members are univalent in $\mathbb{E}.$ Paatero ^[13] proved that $\mathcal{V}_{\vartheta}$ coincides with $\mathcal{S}_{\vartheta}$ whenever $2\leq\vartheta\leq4.$ i.e., If $2\leq\vartheta\leq4,$ $h\in\mathcal{V}_{\vartheta}$ contains only univalent functions in $\mathbb{E}.$ If $\vartheta > 4,$ then functions in the class $\mathcal{V}_{\vartheta}$ is fail to univalent conditions.

Noonan ^[11] gave the concept of order of a function for both $\mathcal{V}_{\vartheta}$ and $\mathcal{R}_{\vartheta}$ in 1971 and Padmanabhan and Parvatham ^[14] introduced the concept of order of a function for $\mathcal{P}_{\vartheta}$ in 1975. Let $\mathcal{P} _{\vartheta}(\delta)$ be the class of function $t$ in $\mathbb{E}$ normalized by the conditions $t(0) = 1$ and

It is well known that ^[5] every function $h\in\mathcal{S}$ has an inverse $h^{-1},$ defined by

and

Hence, the inverse function $h^{-1}$ is given by

If both $h$ and $h^{-1}$ are univalent in $\mathbb{E},$ then $h$ is said to be bi-univalent in $\mathbb{E}.$ Let us indicate $\Sigma$ as the class of bi-univalent functions in $\mathbb{E}.$ Lewin ^[7] introduced the class $\Sigma$ and it was proved that $|h_{2}| < 1.51.$ The coefficient problem for each of the following Taylor-Maclaurin coefficients:

is an open problem. Subsequently Brannan and Clunie ^[3] conjectured that $|h_{2}|\leq\sqrt{2}$ and Netanyahu ^[10] showed that for $h\in\Sigma,$ $\max|h_{2}| = \frac{4}{3}.$ Several authors ^[6,9,20] introduced and investigated various subclasses of the class $\Sigma$ and obtained estimates for their coefficients $|h_{2}|$ and $|h_{3}|$ for the functions in these subclasses. Brannan and Taha ^[4] introduced the subclasses of bi-univalent functions $\mathcal{S}_{\Sigma}^{\ast}(\delta)$ and $\mathcal{K}_{\Sigma}(\delta)$, called bi-starlike functions of order $\delta$ and $\mathcal{K}_{\Sigma}(\delta)$ bi-convex functions of order $\delta,$ respectively.

In geometric function theory and its related field, the study of operators plays an important role. Several authors ^[1,12,17,18] introduced and investigated various subclasses of the class $\Sigma$ using different operators. For $h\in\mathcal{A},$ Sălăgean ^[16] introduced the differential operator $\mathcal{D}^{\eta},$ which is defined by

then

where $\eta\in\mathbb{N}_{0} = \mathbb{N}\cup\{0\}.$

Lemma 1. ^[2] If a function $t\in\mathcal{P}_{\vartheta}(\delta)$ is given in the form

then for each $m\geq1,$

This result is sharp.

By applying the Sǎlǎgean operator, three new subclasses of bi-univalent functions associated with bounded boundary rotations in open unit disk $\mathbb{E}$ are introduced and investigated. For these new classes, the initial coefficient estimates and the Fekete-Szegö inequality are obtained. Some of our findings improved the earlier existing results available in the literature and few of the bounds presented here generalize the result of Sharma ^[19].

2.   Main results

Definition 1. A function $h\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{L}_{\Sigma}^{\eta, a}(\vartheta, \delta)$ if the following conditions

and

hold where $0\leq a\leq1,$ $2\leq\vartheta\leq4,$ $0\leq\delta < 1$ and the function $\gamma(\omega)$ is as defined by (1.2).

Remark 1. If $a = 1$ in Definition 1, we have $\mathcal{L}_{\Sigma}^{\eta, a}(\vartheta, \delta)\equiv \mathcal{L}_{\Sigma}^{\eta, 1}(\vartheta, \delta) \equiv \mathcal{H}_{\Sigma}^{\eta}(\vartheta, \delta).$ That is, a function $h\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{H} _{\Sigma}^{\eta}(\vartheta, \delta)$ if the following conditions

and

hold where $2\leq\vartheta\leq4,$ $0\leq\delta < 1$ and the function $\gamma(\omega)$ is as defined by (1.2).

Remark 2. If $a = 0$ in Definition 1, we have $\mathcal{L}_{\Sigma}^{\eta, a}(\vartheta, \delta)\equiv \mathcal{L}_{\Sigma}^{\eta, 0}(\vartheta, \delta) \equiv \mathcal{L}_{\Sigma}^{\eta}(\vartheta, \delta).$ That is a function $h\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{L} _{\Sigma}^{\eta}(\vartheta, \delta)$ if the following conditions

and

hold where $2\leq\vartheta\leq4,$ $0\leq\delta < 1$ and the function $\gamma(\omega)$ is as defined by (1.2).

Remark 3. ^[19] If $\eta = 0$ in Definition 1, we have $\mathcal{L} _{\Sigma}^{\eta, a}(\vartheta, \delta)\equiv \mathcal{L}_{\Sigma}^{0, a}(\vartheta, \delta)\equiv \mathcal{L}_{\Sigma}^{a}(\vartheta, \delta).$ A function $h\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{L}_{\Sigma}^{\eta}(\vartheta, \delta)$ if the following conditions

and

hold where $0\leq a\leq1,$ $2\leq\vartheta\leq4,$ $0\leq\delta < 1$ and the function $\gamma(\omega)$ is as defined by (1.2).

Theorem 1. Let $h\in\mathcal{L}_{\Sigma}^{\eta, a}(\vartheta, \delta)$ be given in the form (1.1). Then

and

For any $\aleph\in\mathbb{R},$

Proof. As $h\in\mathcal{L}_{\Sigma}^{\eta, a}(\vartheta, \delta)$, from Definition 1,

and

where $t(u)$ and $s(\omega)$ are analytic functions belonging to the class $\mathcal{P}_{\vartheta}(\delta)$ given by

and

Comparing the coefficients by using (2.4)–(2.7), we have

and

Adding (2.9) and (2.11), we have

Now by using Lemma 1, we have

gives the bound of $|h_{2}|$ given in (2.1). Now by using Lemma 1, in (2.9), we have

gives the bound of $|h_{3}|$ given in (2.2). Now fix $\aleph\in\mathbb{R}$ and by using (2.9) and (2.12), we have

Now by using Lemma 1, we have

gives the bound of $|h_{3}-\aleph h_{2}^{2}|$ given in (2.3) finishing Theorem 1. □

Definition 2. A function $h\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{B}_{\Sigma}^{\eta, b}(\vartheta, \delta)$ if the conditions

and

hold where $b\geq0,$ $2\leq\vartheta\leq4,$ $0\leq\delta < 1$ and the function $\gamma(\omega)$ is as defined by (1.2).

Remark 4. If $b = 0$ in Definition 2, we have $\mathcal{B}_{\Sigma}^{\eta, b}(\vartheta, \delta)\equiv \mathcal{B}_{\Sigma}^{\eta, 0}(\vartheta, \delta) \equiv \mathcal{B}_{\Sigma}^{\eta}(\vartheta, \delta).$ That is, a function $h\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{B} _{\Sigma}^{\eta}(\vartheta, \delta)$ if

and

where $2\leq\vartheta\leq4,$ $0\leq\delta < 1$ and the function $\gamma(\omega)$ is as defined by (1.2).

Remark 5. ^[19] If $\eta = 0$ in Definition 2, we have $\mathcal{B} _{\Sigma}^{\eta, b}(\vartheta, \delta)\equiv \mathcal{B}_{\Sigma}^{0, b}(\vartheta, \delta)\equiv \mathcal{S}_{\Sigma}^{\ast}(b, \vartheta, \delta).$ That is, a function $h\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{S}_{\Sigma}^{\ast}(b, \vartheta, \delta)$ if

and

where $b\geq0,$ $2\leq\vartheta\leq4,$ $0\leq\delta < 1$ and the function $\gamma(\omega)$ is as defined by (1.2).

Theorem 2. If $h\in\mathcal{B}_{\Sigma}^{\eta, b}(\vartheta, \delta)$ is of the form (1.1), then

and

For any $\aleph\in\mathbb{R},$ then

where

Proof. As $h\in\mathcal{B}_{\Sigma}^{\eta, b}(\vartheta, \delta)$, we have

and

where $t(u)$ and $s(\omega)$ are analytic functions belonging to the class $\mathcal{P}_{\vartheta}(\delta)$ given by (2.6) and (2.7). Comparing the coefficients using (2.6), (2.7), (2.16), and (2.17), we have

and

Adding (2.19) and (2.21), we have

Now, using Lemma 1, in (2.22), we have

where $b\geq0$ and $\eta\in\mathbb{N}_{0} = \mathbb{N}\cup\{0\}$ and (2.23) gives the bound of $|h_{2}|$ given in (2.13). Again from (2.19) and (2.21), we have

Now, using (2.22) in (2.24), we have

Now, using Lemma 1, in (2.25), we have

Equation (2.26) gives the bound of $|h_{3}|$ given in (2.14). Now fix $\aleph\in\mathbb{R}$ and by using (2.22) and (2.25), we have

Now, using Lemma 1, we have

gives the bound of $|h_{3}-\aleph h_{2}^{2}|$ given in (2.15) finishing Theorem 2. □

Definition 3. A function $h\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{N}_{\Sigma}^{\eta, d}(\vartheta, \delta)$ if the conditions

and

are satisfied where $0\leq d\leq1,$ $2\leq\vartheta\leq4,$ $0\leq\delta < 1$ and the function $\gamma(\omega)$ is as defined by (1.2).

Remark 6. If $d = 0$ in Definition 3, we have $\mathcal{N}_{\Sigma}^{\eta, d}(\vartheta, \delta)\equiv \mathcal{N}_{\Sigma}^{\eta, 0}(\vartheta, \delta) \equiv \mathcal{B}_{\Sigma}^{\eta}(\vartheta, \delta).$ A function $h\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{B} _{\Sigma}^{\eta}(\vartheta, \delta)$ if

and

where $2\leq\vartheta\leq4,$ $0\leq\delta < 1$ and the function $\gamma(\omega)$ is as defined by (1.2).

Remark 7. If $d = 1$ in Definition 3, we have $\mathcal{N}_{\Sigma}^{\eta, d}(\vartheta, \delta)\equiv \mathcal{N}_{\Sigma}^{\eta, 1}(\vartheta, \delta) \equiv \mathcal{N}_{\Sigma}^{\eta}(\vartheta, \delta).$ A function $h\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{N} _{\Sigma}^{\eta}(\vartheta, \delta)$ if

and

where $0\leq d\leq1,$ $2\leq\vartheta\leq4,$ $0\leq\delta < 1$ and the function $\gamma(\omega)$ is as defined by (1.2).

Remark 8. ^[19] If $\eta = 0$ in Definition 3, we have $\mathcal{N} _{\Sigma}^{\eta, d}(\vartheta, \delta)\equiv \mathcal{N}_{\Sigma}^{0, d}(\vartheta, \delta)\equiv \mathcal{M}_{\Sigma}^{d}(\vartheta, \delta).$ A function $h\in\Sigma$ given by (1.1) is said to be in the class $\mathcal{M}_{\Sigma}^{\eta, d}(\vartheta, \delta)$ if

and

where $0\leq d\leq1,$ $2\leq\vartheta\leq4,$ $0\leq\delta < 1$ and the function $\gamma(\omega)$ is as defined by (1.2).

Theorem 3. If $h\in\mathcal{N}_{\Sigma}^{\eta, d}(\vartheta, \delta)$ is given in the form (1.1), then

and

For any $\aleph\in\mathbb{R},$ then

where

Proof. As $h\in\mathcal{N}_{\Sigma}^{\eta, d}(\vartheta, \delta)$, we have,

and

where $t(u)$ and $s(\omega)$ are analytic functions belonging to the class $\mathcal{P}_{\vartheta}(\delta)$ given by (2.6) and (2.7). Comparing the coefficients using (2.6), (2.7), (2.31) and (2.32), we have

and

Adding (2.34) and (2.36), we have

Now, using Lemma 1, in (2.37), we have

Equation (2.38) gives the bound of $|h_{2}|$ given in (2.28). Again from (2.34) and (2.36), we have

Now, using (2.37) in (2.39), we have

Now, by using Lemma 1, in (2.40), we have

Equation (2.41) gives the bound of $|h_{3}|$ given in (2.29). Now fix $\aleph\in\mathbb{R}$ and by using (2.37) and (2.40), we have

Now, using Lemma 1, in (2.42), we have

Equation (2.43) gives the bound of $|h_{3}-\aleph h_{2}^{2}|$ given in (2.30), which completes the proof of Theorem 3. □

3.   Corollaries and remarks

For the choices of $a = 1$, $a = 0$ and $\eta = 0$ in Theorem 1, we get the following Corollaries namely Corollary 1, Corollary 2 and Corollary 3, respectively.

Corollary 1. If $h\in\mathcal{H}_{\Sigma}^{\eta}(\vartheta, \delta)$ is given in the form (1.1), then

and

For any $\aleph\in\mathbb{R},$ then

Corollary 2. If $h\in\mathcal{L}_{\Sigma}^{\eta}(\vartheta, \delta)$ is of the form (1.1), then

and

For any $\aleph\in\mathbb{R},$ then

Corollary 3. If $h\in\mathcal{L}_{\Sigma}^{a}(\vartheta, \delta)$ is given in the form (1.1), then

and

For any $\aleph\in\mathbb{R},$ then

Remark 9. $\vartheta = 2$ in Corollary 3, verifies the results obtained in ^[6].

For the selection of $b = 0$, $\eta = 0$ in Theorem 2, we get the Corollaries Corollary 4, Corollary 5, respectively.

Corollary 4. If $h\in\mathcal{B}_{\Sigma}^{\eta}(\vartheta, \delta)$ is represented in the form (1.1), then

and

For any $\aleph\in\mathbb{R},$ then

where

Corollary 5. If $h\in\mathcal{S}_{\Sigma}^{\ast}(b, \vartheta, \delta)$ is given in the form (1.1), then

and

For any $\aleph\in\mathbb{R},$ then

where

If $\eta = 0$ and $b = 0$ in Theorem 2, we get Corollary 6, which verifies the results obtained in ^[8,19].

Corollary 6. If $h\in\mathcal{S}_{\Sigma}^{\ast}(\vartheta, \delta)$ given in the form (1.1), then

and

For any $\aleph\in\mathbb{R},$ then

Remark 10. $\vartheta = 2$ in Corollary 6, verifies the results obtained in ^[4].

For the choices $d = 1$, $\eta = 0$ in Theorem 3, we get corollaries Corollary 7 and Corollary 8, respectively.

Corollary 7. If $h\in\mathcal{N}_{\Sigma}^{\eta}(\vartheta, \delta)$ is of the form (1.1), then

and

For any $\aleph\in\mathbb{R},$ then

where

Corollary 8. If $h\in\mathcal{M}_{\Sigma}^{d}(\vartheta, \delta)$ is given in the form (1.1), then

and

For any $\aleph\in\mathbb{R},$ then

where

Remark 11. Corollary 8, verifies the results obtained in ^[19].

If $\eta = 0$ and $d = 1$ in Theorem 3, we get the following corollary.

Corollary 9. If $h\in\mathcal{M}_{\Sigma}(\vartheta, \delta)$ given in the form (1.1), then

and

For any $\aleph\in\mathbb{R},$ then

Remark 12. Corollary 9, verifies the results obtained in ^[8]. If $\vartheta = 2$ in Corollary 9, verifies the results obtained in ^[4].

4.   Conclusions

By an application of the Sǎlǎgean operator, three new subclasses of bi-univalent functions associated with bounded boundary rotation in open unit disk $\mathbb{E}$ are considered in this article. We first established initial coefficient bounds as well as the Fekete-Szegö estimates for the classes $\mathcal{L}_{\Sigma}^{\eta, a}(\vartheta, \delta),$ $\mathcal{B}_{\Sigma }^{\eta, b}(\vartheta, \delta),$ and $\mathcal{N}_{\Sigma }^{\eta, d}(\vartheta, \delta).$ Interesting remarks for the major results, including improvements of the earlier bounds, are also quoted. More corollaries and remarks could be reported for the selection of parameters, and those details have been omitted.

Author contributions

All authors of this article have been contributed equally. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The authors would like to thank the referees for their comments and suggestions on the original manuscript.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (No. NRF-2022R1A2C2004874). This work was supported by the Korea Institute of Energy Technology Evaluation and Planning(KETEP) and the Ministry of Trade, Industry & Energy(MOTIE) of the Republic of Korea (No. 20214000000280).

Conflict of interest

The authors declare no conflict of interest.