Abstract

We consider the Fekete-Szegö inequalities for classes which were defined by Murugusundaramoorthy et al. (2013). These inequalities will result in bounds of the third coefficient which are better than these obtained by Murugusundaramoorthy et al. (2013). Moreover, we discuss two other classes of bi-univalent functions. The estimates of initial coefficients in these classes are obtained.

1. Introduction

Let denote the class of all functions of the form

analytic in the unit disk , and let denote the class of these functions in which are univalent. It is known that if then there exists the inverse function . Because of the normalization , is defined in some neighbourhood of the origin. In some cases, can be defined in the whole . Clearly, is also univalent. For this reason, the class is defined as follows.

A function is called bi-univalent in if both and are univalent in . The set of all bi-univalent functions is usually denoted by (or, following Lewin, by ).

It is easy to check that a bi-univalent function given by (1) has the inverse with the Taylor series of the form

The research into was started by Lewin ([1], 1967). It focused on problems connected with coefficients. Many papers concerning bi-univalent functions have been published recently. We owe the revival of these topics to Srivastava et al. ([2], 2010). The investigations in this direction have also been carried out, among others, by Ali et al. [3], Frasin and Aouf [4], and Xu et al. [5]. Hamidi and Jahangiri (e.g., [6]) have revealed the importance of the Faber polynomials in general studies on the coefficients of bi-univalent functions.

In fact, little is known about exact bounds of the initial coefficients of . For the most general families of functions given by (1) we know that for bi-univalent functions (Lewin, [1]), for bi-starlike functions (Kędzierawski, [7]), and for bi-convex functions (Brannan and Taha, [8]). Only the last estimate is sharp; equality holds only for and its rotations.

In the papers [1–15], the authors present some estimates for and , while is taken from various subclasses of . In 2013 Murugusundaramoorthy et al. (see [16]) obtained some coefficient bounds in two classes: and . For a function and its inverse function , let and be defined as below: where , , and .

The definitions of and are the following.

Definition 1. A function is said to be in the class if the functions and , defined by (3), corresponding to and , satisfy

Definition 2. A function is said to be in the class if the functions and , defined by (3), corresponding to and , satisfy

In particular, for , the classes and become the class of strongly bi-starlike functions of order and the class of bi-starlike functions of order , respectively. If additionally or , these two classes reduce to the class of bi-starlike functions.

Conditions (4) and (5) in the above definitions can be rewritten as follows:

respectively, where and are functions in and have the form

Throughout the paper, stands for the set of all analytic functions such that and for .

In [11] the authors proved the following theorems.

Theorem 3. If , , and , then

Theorem 4. If , , and , then

The above results can be improved. In order to do this, we consider the Fekete-Szegö inequalities for the discussed classes. This type of problems has been considered by many authors. The results concerning this problem are given, for example, in [17–20]. Moreover, it seems to be interesting to discuss two other classes defined in a similar way to and . The results presented in the paper are not sharp, but, unfortunately, no method which gives sharp results with regard to these problems is known.

In the proofs of the main theorems we need two lemmas.

Lemma 5. If then for all positive integers .

Lemma 6. Let and . If and then

The proof of Lemma 6 is easy. It is enough to observe that and to discuss three cases with respect to .

From Lemma 6 we immediately obtain the following.

Lemma 7. Let and . If and then

2. Results for and

Now, we will formulate two theorems concerning the Fekete-Szegö inequalities for and .

Theorem 8. If , , , and , then

Theorem 9. If , , , and , then

Proof of Theorem 8. Let given by (1) be in and let , , and . From Definition 1 and from (6) we know that where and are functions in which have the form (8).
Comparing the coefficients in each equality in (16), it follows that
From (17) and (19), there is . Summing and subtracting (18) and (20), we have two equalities Applying (17) and (19) we dispose of and in (21). Hence where is nonnegative. From Lemmas 5 and 7 we conclude

The proof of Theorem 9 is similar to that of Theorem 8 and can be omitted. From Theorems 8 and 9 we get the following corollaries.

Corollary 10. If , , and , then

Corollary 11. If , , and , then

The result in Corollary 10 improves the corresponding result in Theorem 3. Similarly, for the bound in Corollary 11 is better that the one obtained in Theorem 4.

If we get the bounds for and which are better than these obtained in [3, 11]. It is worth mentioning that recently Hamidi and Jahangiri ([6]) and Srivastava et al. ([13]) have provided an improvement of the result from Corollaries 10 and 11.

If additionally , we obtain that for the class of bi-starlike functions (see [3, 6, 13]).

3. Results for and

To begin with, we can observe that the operators which were used by Murugusundaramoorthy et al. in the definitions of and can be written as the weighted harmonic mean of two expressions: and 1; that is, where .

Let us define two new classes. In definitions of and we consider the weighted harmonic mean of and ; namely, where , , and . In fact, in the above functions the range of can be extended to the set .

Now, we can define the classes and .

Definition 12. A function is said to be in the class if the functions and , defined by (28), corresponding to and , satisfy

Definition 13. A function is said to be in the class if the functions and , defined by (28), corresponding to and , satisfy

The idea of considering the weighted mean of and first appeared in the paper by Miller et al. (see [16]). They did their research into the class of so-called -convex functions defined as the arithmetic weighted mean of the expressions mentioned above.

Now we are ready to establish the main theorems of this section.

Theorem 14. If , , , and , then (1)(2)

Theorem 15. If , , , and , then (1)(2)

Proof of Theorem 14. Assume that , , and . From Definition 12 it follows that if then where and are functions in and have the form (8).
Hence, comparing the coefficients in each equality in (35), we can write
From (36) and (38), it yields that . Putting (36) into (37) and (38) into (39), we obtain
Now, summing and subtracting (40) we have two equalities
Substituting in (41) and taken from (36) and (38), we get
By Lemma 5, the first part of our assertion follows.
From (41) and (42),
Applying Lemmas 5 and 7 completes the second part of the assertion.

Theorem 14 gives the following corollaries.

Corollary 16. If , , and , then

For we obtain the bounds for . For , the class reduces to the class of strongly bi-convex functions of order . Hence we have the following.

Corollary 17. If and , then(1)(2)

This result improves the result given in [3].

Proof of Theorem 15. Let with and . From Definition 13 we obtain where .
Hence,
Applying the same method as in the proof of Theorem 15 we get
Observe that the estimate for obtained from (53), that is, , is not always better than the estimate which follows directly from (49), that is, . The comparison of these two bounds completes the first part of the proof.