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# Notes on analytic functions with a bounded positive real part

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 370 (2013)

## Abstract

For real *α* and *β* such that 0\le \alpha <1<\beta, we denote by \mathcal{S}(\alpha ,\beta ) the class of normalized analytic functions *f* such that \alpha <Re\{z{f}^{\prime}(z)/f(z)\}<\beta in . We find some properties, including inclusion properties, Fekete-Szegö problem and coefficient problems of inverse functions.

**MSC:**30C45, 30C55.

## Dedication

Dedicated to Professor Hari M Srivastava

## 1 Introduction

Let ℋ denote the class of analytic functions in the unit disc \mathbb{U}=\{z:|z|<1\} on the complex plane ℂ. Let denote the subclass of ℋ consisting of functions normalized by f(0)=0 and {f}^{\prime}(0)=1. Let denote the subclass of consisting of univalent functions. Denote by {\mathcal{S}}^{\ast} and , the class of starlike functions and convex functions, respectively. It is well-known that \mathcal{K}\subset {\mathcal{S}}^{\ast}\subset \mathcal{S}.

We say that *f* is subordinate to *F* in , written as f\prec F if and only if f(z)=F(w(z)) for some Schwarz function with w(0)=0 and |w(z)|<1, z\in \mathbb{U}. If F(z) is univalent in , then the subordination f\prec F is equivalent to f(0)=F(0) and f(\mathbb{U})\subset F(\mathbb{U}).

We denote by {\mathcal{S}}^{\ast}(A,B), the class of Janowski starlike functions, namely, the functions satisfying the subordination equation: z{f}^{\prime}(z)/f(z)\prec (1+Az)/(1+Bz). Note that {\mathcal{S}}^{\ast}(1,-1)={\mathcal{S}}^{\ast}.

Now, we shall introduce the class of analytic functions used in the sequel.

**Definition 1** Let *α* and *β* be real numbers such that 0\le \alpha <1<\beta. The function f\in \mathcal{A} belongs to the class \mathcal{S}(\alpha ,\beta ) if *f* satisfies the following inequality:

We remark that for given *α*, *β* (0\le \alpha <1<\beta), f\in \mathcal{S}(\alpha ,\beta ) if and only if *f* satisfies the following two subordination equations:

since the functions (1+(1-2\alpha )z)/(1-z) and (1+(1-2\beta )z)/(1-z) map onto the right half plane, having real part greater than *α*, and the left half plane, having real part smaller than *β*, respectively. The above class \mathcal{S}(\alpha ,\beta ) is introduced by Kuroki and Owa [1]. They investigated coefficient estimates for f\in \mathcal{S}(\alpha ,\beta ) and found the necessary and sufficient condition for f\in \mathcal{S}(\alpha ,\beta ) using the following subordination.

**Lemma 1** (Kuroki and Owa [1])

*Let* f\in \mathcal{A} *and* 0\le \alpha <1<\beta. *Then* f\in \mathcal{S}(\alpha ,\beta ) *if and only if*

Lemma 1 means that the function *p* defined by

maps the unit disk onto the strip domain *w* with \alpha <Re(w)<\beta. We note that the function f\in \mathcal{A}, given by

is in the class \mathcal{S}(\alpha ,\beta ).

## 2 Inclusion properties

**Theorem 1** *For given* 0\le \alpha <1<\beta, *let* *A* *and* *B* *be real numbers such that*

*Then* {\mathcal{S}}^{\ast}(A,B)\subset \mathcal{S}(\alpha ,\beta ).

*Proof* At first, we note that

For f\in {\mathcal{S}}^{\ast}(A,B), we know that the following inequality holds:

Therefore, it suffices to show that *α* and *β* satisfy the following inequalities:

Using inequality (4), we can derive that

Also,

By the above inequalities (5) and (6), we can easily obtain the inequalities (3), so the proof of Theorem 1 is completed. □

**Lemma 2** (Miller and Mocanu [2])

*Let* *Ξ* *be a set in the complex plane* ℂ *and let* *b* *be a complex number such that* Re(b)>0. *Suppose that a function* \psi :{\mathbb{C}}^{2}\times \mathbb{U}\to \mathbb{C} *satisfies the condition*

*for all real* *ρ*, \sigma \le -{|b-i\rho |}^{2}/(2Re(b)) *and all* z\in \mathbb{U}. *If the function* p(z) *defined by* p(z)=b+{b}_{1}z+{b}_{2}{z}^{2}+\cdots *is analytic in* *and if*

*then* Re(p(z))>0 *in* .

**Theorem 2** *Let* f\in \mathcal{A}, 1/2\le \alpha <1 *and* Re\{z{f}^{\prime}(z)/f(z)\}>\alpha *in* . *Then*

*Proof* Write \gamma (\alpha ):=\gamma and note that \frac{1}{2}\le \gamma <1 for \frac{1}{2}\le \alpha <1. Let *p* be defined by

Then *p* is analytic in , p(0)=1 and

where

Also,

Now for all real *ρ*, \sigma \le -\frac{1}{2}(1+{\rho}^{2}),

Now, we let

Then

hence {g}^{\prime}(\rho )=0 occurs at only \rho =0 and *g* satisfies

and

Since 1/2\le \gamma <1, we have

hence we get

This shows that Re\{\psi (i\rho ,\sigma )\}\notin {\Omega}_{\alpha}. By Lemma 2, we get Re(p(z))>0 in , and this shows that inequality (7) holds and the proof of Theorem 2 is completed. □

**Theorem 3** *Let* f\in \mathcal{A}, 1<\beta <3/2 *and* Re\{z{f}^{\prime}(z)/f(z)\}<\beta *in* . *Then*

*Proof* Note that \delta :=\delta (\beta )=\frac{1}{3-2\beta}>1 for \beta >1. Let *p* be defined by

Then *p* is analytic in , p(0)=1 and

where *ψ* is given in (8). Also

Now, for all real *ρ*, \sigma \le -\frac{1}{2}(1+{\rho}^{2}),

where g(\rho ) is given (9). Since

for all \delta >1, we have

This shows that Re\{\psi (i\rho ,\sigma )\}\notin {\Omega}_{\beta}. By Lemma 2, we get Re(p(z))>0 in , and this is equivalent to

and the proof of Theorem 3 is completed. □

Combining the above Theorems 2 and 3, we can obtain the following result:

**Theorem 4** *Let* f\in \mathcal{A}, 1/2\le \alpha <1<\beta <3/2 *and* \alpha <Re\{z{f}^{\prime}(z)/f(z)\}<\beta *in* . *Then*

*where* \gamma (\alpha ) *and* \delta (\beta ) *is given* (7) *and* (10).

## 3 Some coefficient problems

In this section, we investigate coefficient problems for functions in the class \mathcal{S}(\alpha ,\beta ). In [1], Kuroki and Owa investigated the coefficient of the function *p* given by (2); the function *p* can be written as

where

We denote by {\mathcal{S}}_{\sigma}(\alpha ,\beta ) the class of bi-univalent functions *f* such that f\in \mathcal{S}(\alpha ,\beta ) and the inverse function {f}^{-1}\in \mathcal{S}(\alpha ,\beta ). Srivastava *et al.* investigated the estimates on the initial coefficient for certain subclasses of analytic and bi-univalent functions in [3, 4]. Ali *et al.* have studied similar problems in [5].

In theorem, we shall solve the Fekete-Szegö problem for f\in S(\alpha ,\beta ). We need the following lemma:

**Lemma 3** (Keogh and Merkers [6])

*Let* p(z)=1+{c}_{1}z+{c}_{2}{z}^{2}+\cdots *be a function with positive real part in* . *Then*, *for any complex number* *ν*,

The following result holds for the coefficient of f\in S(\alpha ,\beta ).

**Theorem 5** *Let* 0\le \alpha <1<\beta *and let the function* *f* *given by* f(z)=z+{\sum}_{n=2}^{\mathrm{\infty}}{a}_{n}{z}^{n} *be in the class* \mathcal{S}(\alpha ,\beta ). *Then*, *for a complex number* *μ*,

*Proof* Let us consider a function *q* given by q(z)=z{f}^{\prime}(z)/f(z). Then, since f\in \mathcal{S}(\alpha ,\beta ), we have q(z)\prec p(z), where

Let

Then *h* is analytic and has positive real part in the open unit disk . We also have

We find from equation (12) that

and

which imply that

where

Applying Lemma 3, we can obtain

And substituting

and

in (13), we can obtain the result as asserted. □

Using the above Theorem 5, we can get the following result.

**Corollary 1** *Let the function* *f*, *given by* f(z)=z+{\sum}_{n=2}^{\mathrm{\infty}}{a}_{n}{z}^{n}, *be in the class* \mathcal{S}(\alpha ,\beta ). *Also let the function* {f}^{-1}, *defined by*

*be the inverse of* *f*. *If*

*then*

*and*

*Proof* Relations (16) and (17) give

Thus, we can get the estimate for |{b}_{2}| by

immediately. An application of Theorem 5 (with \mu =2) gives the estimates for |{b}_{3}|, hence the proof of Corollary 1 is completed. □

Next, we shall estimate on some initial coefficient for the bi-univalent functions f\in {\mathcal{S}}_{\sigma}(\alpha ,\beta ).

**Theorem 6** *Let* *f* *be given by* f(z)=z+{\sum}_{n=2}^{\mathrm{\infty}}{a}_{n}{z}^{n} *be in the class* {\mathcal{S}}_{\sigma}(\alpha ,\beta ). *Then*

*and*

*where* {B}_{1} *and* {B}_{2} *are given by* (14) *and* (15).

*Proof* If f\in {\mathcal{S}}_{\sigma}(\alpha ,\beta ), then f\in \mathcal{S}(\alpha ,\beta ) and g\in \mathcal{S}(\alpha ,\beta ), where g={f}^{-1}. Hence

where p(z) is given by (2). Let

and

Then *h* and *k* are analytic and have positive real part in . Also, we have

By suitably comparing coefficients, we get

and

where {B}_{1} and {B}_{2} are given by (14) and (15), respectively. Now, considering (20) and (22), we get

Also, from (21), (22), (23) and (24), we find that

Therefore, we have

This gives the bound on |{a}_{2}| as asserted in (18). Now, further computations from (21), (23), (24) and (25) lead to

This equation, together with the well-known estimates:

lead us to inequality (19). Therefore, the proof of Theorem 6 is completed. □

## References

Kuroki K, Owa S: Notes on new class for certain analytic functions.

*RIMS Kokyuroku*2011, 1772: 21–25.Miller SS, Mocanu PT Series of Monographs and Textbooks in Pure and Applied Mathematics 225. In

*Differential Subordinations: Theory and Applications*. Dekker, New York; 2000.Srivastava HM, Mishra AK, Gochhayat P: Certain subclasses of analytic and bi-univalent functions.

*Appl. Math. Lett.*2010, 23: 1188–1192. 10.1016/j.aml.2010.05.009Xu QH, Gui YC, Srivastava HM: Coefficient estimates for a certain subclass of analytic and bi-univalent functions.

*Appl. Math. Lett.*2012, 25: 990–994. 10.1016/j.aml.2011.11.013Ali RM, Lee SK, Ravichandran V, Supramaniam S: Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions.

*Appl. Math. Lett.*2012, 25: 344–351. 10.1016/j.aml.2011.09.012Keogh F, Merkers E: A coefficient inequality for certain classes of analytic functions.

*Proc. Am. Math. Soc.*1969, 20: 8–12. 10.1090/S0002-9939-1969-0232926-9

## Acknowledgements

The research was supported by Kyungsung University Research Grants in 2013.

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### Authors’ contributions

The corresponding author, OSK carried out the subclasses of analytic functions studies and conceived of the study. YJS participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.

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Sim, Y.J., Kwon, O.S. Notes on analytic functions with a bounded positive real part.
*J Inequal Appl* **2013**, 370 (2013). https://doi.org/10.1186/1029-242X-2013-370

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DOI: https://doi.org/10.1186/1029-242X-2013-370