1 Introduction and lemmas

Let \(\mathcal{A}\) be the class of functions of the form

$$ f(z)=z+\sum_{k=2}^{\infty}a_{k}z^{k}, $$
(1.1)

which are analytic in the open unit disk \(\mathbb{U}=\{z\in\mathbb{C}: |z|<1\}\). We also denote by \(\mathcal{S}\) the subclass of the normalized analytic function class \(\mathcal{A}\) consisting of all functions in which are also univalent in \(\mathbb{U}\).

Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk \(\mathbb{U}\). In fact, the Koebe one-quarter theorem [3] ensures that the image of \(\mathbb{U}\) under every univalent function \(f\in\mathcal{S}\) contains a disk of radius \(1/4\). Thus every function \(f\in\mathcal{S}\) has an inverse \(f^{-1}\), which is defined by

$$ f^{-1}\bigl(f(z)\bigr)=z\quad(z\in\mathbb{U}), $$

and

$$ f \bigl(f^{-1}(w) \bigr)=w \quad \biggl( \vert w \vert < r_{0}(f); r_{0}(f)\geq \frac{1}{4} \biggr). $$

A function \(f\in\mathcal{A}\) is said to be bi-univalent in the open unit disk \(\mathbb{U}\) if both the function and its inverse are univalent in \(\mathbb{U}\). Let σ denote the class of analytic and bi-univalent functions in \(\mathbb{U}\) given by the Taylor–Maclaurin series expansion as in (1.1). For a brief history and interesting examples of functions in the class σ, see [19]. In fact, the aforecited work of Srivastava et al. [19] essentially revived the investigation of various subclasses of the bi-univalent function class σ in recent years; it was followed by works of, e.g., Frasin and Aouf [6], Srivastava et al. [18, 20], Xu et al. [21, 22] and others (see, for example, [1, 2, 4, 7, 11, 14]).

In this paper, the concept of bi-univalency is extended to the class of meromorphic functions defined on \(\Delta=\{z:z\in\mathbb{C}, 1<|z|<\infty\}\). The class of functions

$$ g(z)=z+\sum_{k=0}^{\infty} \frac{b_{k}}{z^{k}} $$
(1.2)

that are meromorphic and univalent in Δ is denoted by Σ, and every univalent function g has an inverse \(g^{-1}\) satisfying the series expansion

$$ g^{-1}(w)=w+\sum_{k=0}^{\infty} \frac{B_{k}}{z^{k}}, $$
(1.3)

where \(0< M<|w|<\infty\). Analogous to the bi-univalent analytic functions, a function \(g\in\Sigma\) is said to be meromorphic and bi-univalent if both g and \(g^{-1}\) are meromorphic and univalent in Δ given by (1.2). The class of all meromorphic and bi-univalent functions is denoted by \(\Sigma_{\sigma}\). A simple calculation shows that

$$ h(w)=g^{-1}(w)=w-b_{0}-\frac{b_{1}}{w}- \frac{b_{2}+b_{0}b_{1}}{w^{2}}-\frac{b_{3}+2b_{0}b_{2}+b_{0}^{2}b_{1}+b_{1}^{2}}{w^{3}}+\cdots. $$
(1.4)

Estimates on the coefficients of the inverses of meromorphic univalent functions were widely investigated in the literature. For example, Schiffer [15] showed that if g, defined by (1.2), is in Σ with \(b_{0}=0\), then \(|b_{2}|\leq2/3\). In 1971, Duren [5] obtained the inequality \(|b_{n}|\leq 2/(n+1)\) for \(g\in\Sigma\) with \(b_{k}=0\), \(1\leq k< n/2\). For \(g^{-1}\) being the inverse of g, Springer [17] showed that

$$ \vert B_{3} \vert \leq1 \quad \text{and} \quad \biggl\vert B_{3}+\frac{1}{2}B_{1}^{2} \biggr\vert \leq\frac{1}{2}, $$

and conjectured that

$$ \vert B_{2n-1} \vert \leq \frac{(2n-2)!}{n!(n-1)!} \quad (n=1,2,\dots). $$

In 1977, Kubota [9] proved that the Springer conjecture is true for \(n=3,4,5\), and subsequently Schober [16] obtained sharp bounds for the coefficients \(B_{2n-1}\) (\(1\leq n \leq7\)). Recently, Kapoor and Mishra [8] found coefficient estimates for inverses of meromorphic starlike functions of positive order α in Δ.

In the present investigation, certain subclasses of meromorphic bi-univalent functions are introduced and estimates for the coefficients \(|b_{0}|\) and \(|b_{1}|\) of functions in the newly introduced subclasses are obtained. These coefficient results are obtained by associating with the functions having positive real part. An analytic function p of the form \(p(z)=1+c_{1}z+c_{2}z^{2}+\cdots\) is called a function with positive real part in \(\mathbb{U}\) if \(\Re p(z)>0\) for all \(z\in\mathbb{U}\). The class of all functions with positive real part is denoted by \(\mathcal{P}\). We need the following lemmas [13] to prove our main results.

Lemma 1.1

If \(\phi(z)\in\mathcal{P}\), the class of functions analytic in \(\mathbb{U}\) with positive real part, is given by

$$ \phi(z)=1+c_{1}z+c_{2}z^{2}+c_{3}z^{3}+ \cdots\quad(z\in\mathbb{U}), $$

then \(|c_{n}|\leq2\) for each \(n\in\mathbb{N}\).

In 1972, the following univalence criterion was proved by Ozaki and Nunokawa [12].

Lemma 1.2

If for \(f\in\mathcal{A}\)

$$ \biggl\vert \frac{z^{2}f'(z)}{[f(z)]^{2}}-1 \biggr\vert < 1\quad(z\in\mathbb{U}), $$

then f is univalent in \(\mathbb{U}\) and hence \(f\in\mathcal{S}\).

Also, let \(\mathcal{T}(\mu)\) denote the class of functions \(f\in\mathcal{A}\) such that

$$ \biggl\vert \frac{z^{2}f'(z)}{[f(z)]^{2}}-1 \biggr\vert < \mu\quad(z\in\mathbb{U}), $$

where μ is a real number with \(0<\mu\leq1\) and \(\mathcal{T}(1)=\mathcal{T}\). It is clear that \(\mathcal{T}(\mu)\subset\mathcal{T}\subset\mathcal{S}\).

Further (see Kuroki et al. [10]), for \(f\in\mathcal{T}(\mu)\) we have that:

$$ \Re \biggl(\frac{z^{2}f'(z)}{[f(z)]^{2}} \biggr)>1-\mu\quad(z\in\mathbb{U}). $$

2 Main results

Definition 2.1

A function \(g\in\Sigma_{\sigma}\) given by (1.2) is said to be in the class \(\mathcal{T}_{\Sigma_{\sigma}}^{\alpha}\) if the following conditions are satisfied:

$$ \begin{aligned} &\biggl\vert \arg \biggl(\frac{z^{2}g'(z)}{[g(z)]^{2}} \biggr) \biggr\vert < \frac{\alpha\pi}{2}\quad(z\in\Delta; 0< \alpha\leq1) \quad \text{and} \quad \\ &\biggl\vert \arg \biggl(\frac{w^{2}h'(w)}{[h(w)]^{2}} \biggr) \biggr\vert < \frac{\alpha\pi}{2}\quad(w\in\Delta; 0< \alpha\leq1) \end{aligned} $$
(2.1)

where the function h is an extension of \(g^{-1}\) to Δ defined by (1.4).

Definition 2.2

A function \(g\in\Sigma_{\sigma}\) given by (1.2) is said to be in the class \(\mathcal{T}_{\Sigma_{\sigma}}(\mu)\) if the following conditions are satisfied:

$$ \begin{aligned} &\Re \biggl(\frac{z^{2}g'(z)}{[g(z)]^{2}} \biggr)>1-\mu\quad(z\in\Delta; 0< \mu\leq1) \quad \text{and} \\ &\Re \biggl(\frac{w^{2}h'(w)}{[h(w)]^{2}} \biggr)>1-\mu\quad(w\in\Delta; 0< \mu\leq1) \end{aligned} $$
(2.2)

where the function h is an extension of \(g^{-1}\) to Δ defined by (1.4).

Theorem 2.1

Let the function g, given by the series expansion (1.2), be in the function class \(\mathcal{T}_{\Sigma_{\sigma}}^{\alpha}\), \(0<\alpha\leq1\). Then

$$\begin{aligned} &\vert b_{0} \vert \leq\sqrt{\frac{2}{3}} \alpha , \end{aligned}$$
(2.3)
$$\begin{aligned} &\vert b_{1} \vert \leq \textstyle\begin{cases} \frac{2}{3}\alpha, & 0< \alpha\leq\frac{\sqrt{2}}{2}, \\ \frac{2\sqrt{2}}{3}\alpha^{2}, & \frac{\sqrt{2}}{2}\leq\alpha\leq1. \end{cases}\displaystyle \end{aligned}$$
(2.4)

Proof

It follows from (2.1) that

$$ \frac{z^{2}g'(z)}{[g(z)]^{2}}=\bigl[s(z)\bigr]^{\alpha} \quad \text{and} \quad \frac{w^{2}h'(w)}{[h(w)]^{2}}=\bigl[t(w)\bigr]^{\alpha}\quad (z\in\Delta) , $$
(2.5)

respectively, where \(s(z)\) and \(t(z)\) are functions with positive real part in Δ and have the forms

$$ s(z)=1+\frac{s_{1}}{z}+\frac{s_{2}}{z^{2}}+\cdots $$
(2.6)

and

$$ t(w)=1+\frac{t_{1}}{w}+\frac{t_{2}}{w^{2}}+\cdots, $$
(2.7)

respectively. Now, upon equating the coefficients in (2.5), we get

$$\begin{aligned} &{-}2b_{0}=\alpha s_{1} , \end{aligned}$$
(2.8)
$$\begin{aligned} &{-}3\bigl(b_{1}-b_{0}^{2}\bigr)= \alpha s_{2}+\frac{\alpha(\alpha-1)}{2}s_{1}^{2} , \end{aligned}$$
(2.9)
$$\begin{aligned} &2b_{0}=\alpha t_{1} , \end{aligned}$$
(2.10)
$$\begin{aligned} &3\bigl(b_{1}+b_{0}^{2}\bigr)= \alpha t_{2}+\frac{\alpha(\alpha-1)}{2}t_{1}^{2} , \end{aligned}$$
(2.11)

and, from (2.8) and (2.10), we find that

$$\begin{aligned} &s_{1}=-t_{1} , \end{aligned}$$
(2.12)
$$\begin{aligned} &8b_{0}^{2}=\alpha^{2} \bigl(s_{1}^{2}+t_{1}^{2}\bigr). \end{aligned}$$
(2.13)

Also from (2.9) and (2.11) we obtain

$$ 6b_{0}^{2}=\alpha(s_{2}+t_{2})+ \frac{\alpha(\alpha-1)}{2}\bigl(s_{1}^{2}+t_{1}^{2} \bigr). $$
(2.14)

Since \(\mathfrak{R}(s(z))>0\) and \(\mathfrak{R}(t(z))>0\) in Δ, the functions \(s(1/z),t(1/z)\in\mathcal{P}\) and hence the coefficients \(s_{k}\) and \(t_{k}\) for each k satisfy the inequality in Lemma 1.1. Applying the triangle inequality, and then Lemma 1.1, in (2.13) and (2.14) gives us the desired estimates on \(|b_{0}|\), as asserted in (2.3).

Next, in order to find the bound on the coefficient \(|b_{1}|\), we subtract (2.11) from (2.9), and we thus get

$$ -6b_{1}=\alpha(s_{2}-t_{2}). $$
(2.15)

Hence

$$ |b_{1|}\leq\frac{2}{3}\alpha. $$
(2.16)

On the other hand, using (2.9) and (2.11) yields

$$\begin{aligned} &9\bigl(b_{1}-b_{0}^{2} \bigr)^{2}+9\bigl(b_{1}+b_{0}^{2} \bigr)^{2} \\ &\quad =\alpha^{2}\bigl(s_{2}^{2}+t_{2}^{2} \bigr)+\frac{\alpha^{2}(\alpha-1)^{2}}{4}\bigl(s_{1}^{4}+t_{1}^{4} \bigr)+\alpha^{2}(\alpha-1) \bigl(s_{1}^{2}s_{2}+t_{1}^{2}t_{2} \bigr). \end{aligned}$$
(2.17)

By using (2.13), we have from the above equality

$$\begin{aligned} 18b_{1}^{2}={}&\alpha^{2} \bigl(s_{2}^{2}+t_{2}^{2}\bigr)+ \frac{\alpha^{2}(\alpha-1)^{2}}{4}\bigl(s_{1}^{4}+t_{1}^{4} \bigr)+\alpha^{2}(\alpha-1) \bigl(s_{1}^{2}s_{2}+t_{1}^{2}t_{2} \bigr) \\ &{}-\frac{9\alpha^{4}(s_{1}^{2}+t_{1}^{2})^{2}}{32}. \end{aligned}$$
(2.18)

From Lemma 1.1 we obtain

$$ \vert b_{1} \vert ^{2}\leq\frac{13}{9} \alpha^{4}, $$

and therefore,

$$ \vert b_{1} \vert \leq\frac{\sqrt{13}}{3} \alpha^{2}. $$
(2.19)

Also, by using (2.14), we have from equality (2.17) that

$$\begin{aligned} 18b_{1}^{2} =&\alpha^{2}\bigl(s_{2}^{2}+t_{2}^{2} \bigr)+\frac{\alpha^{2}(\alpha-1)^{2}}{4}\bigl(s_{1}^{4}+t_{1}^{4} \bigr)+\alpha^{2}(\alpha-1) \bigl(s_{1}^{2}s_{2}+t_{1}^{2}t_{2} \bigr)\\ &{}-18 \biggl[\frac{\alpha(s_{2}+t_{2})}{6}+\frac{\alpha(\alpha-1)(s_{1}^{2}+s_{2}^{2})}{12} \biggr]^{2}. \end{aligned}$$

From Lemma 1.1 we obtain

$$ \vert b_{1} \vert \leq\frac{2\sqrt{2}}{3} \alpha^{2}. $$
(2.20)

Comparing (2.16), (2.19) and (2.20), we get the desired estimate on the coefficient \(|b_{1}|\), as asserted in (2.4). □

Theorem 2.2

Let the function g, given by the series expansion (1.2), be in the function class \(\mathcal{T}_{\Sigma_{\sigma}}(\mu)\), \(0<\mu\leq1\). Then

$$\begin{aligned} &\vert b_{0} \vert \leq\sqrt{\frac{2\mu}{3}} , \end{aligned}$$
(2.21)
$$\begin{aligned} & \vert b_{1} \vert \leq\frac{2\sqrt{2}}{3}\mu . \end{aligned}$$
(2.22)

Proof

It follows from (2.2) that

$$\begin{aligned} &\frac{z^{2}g'(z)}{[g(z)]^{2}}=(1-\mu)+\mu s(z)\quad(z\in\Delta), \end{aligned}$$
(2.23)
$$\begin{aligned} &\frac{w^{2}h'(w)}{[h(w)]^{2}}=(1-\mu)+\mu t(z)\quad(z\in\Delta), \end{aligned}$$
(2.24)

respectively, where \(s(z)\) and \(t(w)\) are functions with positive real part in Δ and have the forms (2.6) and (2.7), respectively. Now, upon equating the coefficients in (2.23) and (2.24), we get

$$\begin{aligned} &{-}2b_{0}=\mu s_{1} , \end{aligned}$$
(2.25)
$$\begin{aligned} &{-}3\bigl(b_{1}-b_{0}^{2}\bigr)=\mu s_{2} , \end{aligned}$$
(2.26)
$$\begin{aligned} &2b_{0}=\mu t_{1} , \end{aligned}$$
(2.27)
$$\begin{aligned} &3\bigl(b_{1}+b_{0}^{2}\bigr)=\mu t_{2}. \end{aligned}$$
(2.28)

From (2.25) and (2.27) we obtain

$$\begin{aligned} &s_{1}=-t_{1} , \end{aligned}$$
(2.29)
$$\begin{aligned} &8b_{0}^{2}=\mu^{2} \bigl(s_{1}^{2}+t_{1}^{2}\bigr). \end{aligned}$$
(2.30)

Also from (2.26) and (2.28) we obtain

$$ 6b_{0}^{2}=\mu(s_{2}+t_{2}). $$
(2.31)

Since \(\mathfrak{R}(s(z))>0\) and \(\mathfrak{R}(t(z))>0\) in Δ, the functions \(s(1/z),t(1/z)\in\mathcal{P}\) and hence the coefficients \(s_{k}\) and \(t_{k}\) for each k satisfy the inequality in Lemma 1.1. Therefore we find from (2.30) and (2.31) that

$$ \vert b_{0} \vert \leq\mu \quad \text{and} \quad \vert b_{0} \vert \leq\sqrt{\frac{2\mu }{3}} , $$
(2.32)

respectively. So we get the desired estimate on the coefficient \(|b_{0}|\), as asserted in (2.21).

Next, in order to find the bound on the coefficient \(|b_{1}|\), we subtract (2.28) from (2.26), and obtain

$$ 6b_{1}=\mu(t_{2}-s_{2}). $$
(2.33)

Hence

$$ \vert b_{1} \vert \leq\frac{2}{3}\mu. $$
(2.34)

On the other hand, using (2.26) and (2.28) yields

$$ -9\bigl(b_{1}-b_{0}^{2}\bigr) \bigl(b_{1}+b_{0}^{2}\bigr)=\mu^{2}s_{2}t_{2} , $$
(2.35)

or equivalently

$$ 9b_{1}^{2}=9b_{0}^{4}- \mu^{2}s_{2}t_{2}. $$
(2.36)

Upon substituting the value of \(b_{0}^{2}\) from (2.30) and (2.31) into (2.36), respectively, it follows that

$$ \vert b_{1} \vert ^{2}\leq b_{0}^{4}- \frac{4}{9}\mu^{2} $$

and

$$ \vert b_{1} \vert \leq\frac{2\sqrt{2}}{3}\mu. $$
(2.37)

Comparing (2.34) and (2.37), we get the estimate desired on the coefficient \(|b_{1}|\), as given in (2.22). □

3 Conclusion

Lemma 3.1

If \(b_{0}=0\) for the function \(g\in\Sigma\), the series expansion (1.2) becomes

$$ g^{-1}(w)=w-\frac{b_{1}}{w}-\frac{b_{2}}{w^{2}}-\frac{b_{1}^{2}+b_{3}}{w^{3}}+ \cdots . $$

This series expansion was obtained by Schober [16].

Example 3.2

The function \(g(z)=z+1/z\) is clearly a univalent meromorphic function. Direct calculation shows that

$$ g^{-1}(w)=\frac{w+\sqrt{w^{2}-4}}{2}. $$

This function has the series expansion given by

$$ g^{-1}(w)=w-\frac{1}{w}-\frac{1}{w^{3}}-\frac{2}{w^{5}}- \frac{5}{w^{7}}-\frac{14}{w^{9}}-\cdots . $$

Corollary 3.1

If g, given by (1.2), is in the class \(\mathcal{T}_{\Sigma_{\sigma}}(\mu)\), \(0<\mu\leq1\), and \(b_{0}=0\) then

$$ \vert b_{1} \vert \leq\frac{2}{3}\mu. $$

Proof

Assume that the function \(g(z)=z+\sum_{n=1}^{\infty}\frac{b_{n}}{z^{n}}\in\mathcal{T}_{\Sigma_{\sigma}}(\mu)\) where \(0<\mu\leq1\). Since \(b_{0}=0, \ s_{1}=t_{1}=0\), the result can be verified by a direct calculation of (2.36). □

Corollary 3.2

Let \(g\in\mathcal{T}_{\Sigma_{\sigma}}^{\alpha}\), where \(0<\alpha\leq1\). Then

$$ \vert b_{1} \vert \leq\frac{2}{3}\alpha. $$

Proof

Since the function \(g(z)=z+\sum_{n=1}^{\infty}\frac{b_{n}}{z^{n}}\in\mathcal{T}_{\Sigma_{\sigma}}^{\alpha}\) where \(0<\alpha\leq1\) and \(b_{0}=0\), it follows that \(s_{1}=t_{1}=0\). The result can now be seen by a direct calculation of (2.17). □