Estimates of Initial Coefficients for Bi-Univalent Functions

and Applied Analysis 3 Comparing the coefficients in each equality in (16), it follows that (1 − λ) a 2 = αp 1 , (17) 2 (1 − λ) a 3 − (1 − λ2) a 2 2 = 1 2 α (α − 1) p 1 2 + αp 2 , (18) − (1 − λ) a 2 = αq 1 , (19) −2 (1 − λ) a 3 + (λ2 − 4λ + 3) a 2 2 = 1 2 α (α − 1) q 1 2 + αq 2 . (20) From (17) and (19), there is p 1 = −q 1 . Summing and subtracting (18) and (20), we have two equalities 2(1 − λ)2a 2 2 = 1 2 α (α − 1) (p 1 2 + q 1 2) + α (p 2 + q 2 ) , (21) a 3 = a 2 2 + α 4 (1 − λ) (p 2 − q 2 ) . (22) Applying (17) and (19) we dispose of p 1 and q 1 in (21). Hence a 3 − μa 2 2 = p 2 [h (α) (1 − μ) + α 4 (1 − λ) ] + q 2 [h (α) (1 − μ) − α 4 (1 − λ) ] , (23) where h(α) = α2/(1 − λ)2(1 + α) is nonnegative. From Lemmas 5 and 7 we conclude 󵄨󵄨󵄨󵄨a3 − μa2 2 󵄨󵄨󵄨󵄨 ≤ {{{{{ {{{{{ { 4h (α) 󵄨󵄨󵄨1 − μ 󵄨󵄨󵄨 for h (α) 󵄨󵄨󵄨1 − μ 󵄨󵄨󵄨 ≥ α 4 (1 − λ) , α 1 − λ for h (α) 󵄨󵄨󵄨1 − μ 󵄨󵄨󵄨 ≤ α 4 (1 − λ) . (24) 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. Iff ∈ S Σ (α, λ), 0 < α ≤ 1, and 0 ≤ λ < 1, then 󵄨󵄨󵄨a3 󵄨󵄨󵄨 ≤ {{ {{ { 4α2 (1 − λ)2 (1 + α) for 4α ≥ (1 + α) (1 − λ) α 1 − λ for 4α ≤ (1 + α) (1 − λ) . (25) Corollary 11. If f ∈ M Σ (β, λ), 0 ≤ β < 1, and 0 ≤ λ < 1, then 󵄨󵄨󵄨a3 󵄨󵄨󵄨 ≤ 2 (1 − β) (1 − λ)2 . (26) The result in Corollary 10 improves the corresponding result in Theorem 3. Similarly, for 0 ≤ β ≤ 1/2 the bound in Corollary 11 is better that the one obtained inTheorem 4. If λ = 0 we get the bounds for Sα Σ and S∗ Σ (β) 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 α = 1, we obtain that |a 3 | ≤ 2 for the class S∗ Σ of bi-starlike functions (see [3, 6, 13]). 3. Results for K Σ (α, λ) and N Σ (β, λ) To begin with, we can observe that the operators which were used byMurugusundaramoorthy et al. in the definitions of S Σ (α, λ) and M Σ (β, λ) can be written as the weighted harmonic mean of two expressions: zf󸀠(z)/f(z) and 1; that is, F (z) = [ 1 − λ zf(z)/f(z) + λ 1 ] −1


Introduction
Let A denote the class of all functions of the form analytic in the unit disk D ≡ { ∈ C : || < 1}, and let S denote the class of these functions in A which are univalent.It is known that if  ∈ S then there exists the inverse function  −1 .Because of the normalization (0) = 0,  −1 is defined in some neighbourhood of the origin.In some cases,  −1 can be defined in the whole D. Clearly,  −1 is also univalent.For this reason, the class Σ is defined as follows.
A function  ∈ A is called bi-univalent in D if both  and  −1 are univalent in D. The set of all bi-univalent functions is usually denoted by Σ (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 Σ 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 particular, for  = 0, the classes S Σ (, ) and M Σ (, ) become the class S  Σ of strongly bi-starlike functions of order  and the class S * Σ () of bi-starlike functions of order , respectively.If additionally  = 1 or  = 0, these two classes reduce to the class S * Σ of bi-starlike functions.Conditions (4) and (5) in the above definitions can be rewritten as follows: respectively, where  and  are functions in P and have the form Throughout the paper, P stands for the set of all analytic functions ℎ such that ℎ(0) = 1 and Rℎ() > 0 for  ∈ D.
In [11] the authors proved the following theorems.
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][18][19][20].Moreover, it seems to be interesting to discuss two other classes defined in a similar way to S Σ (, ) and M Σ (, ).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.
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.
Proof of Theorem 8. Let  given by (1) be in S Σ (, ) and let 0 <  ≤ 1, 0 ≤  < 1, and  ∈ R. From Definition 1 and from (6) we know that where  and  are functions in P which have the form (8).
If  = 0 we get the bounds for S  Σ and S * Σ () 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.

Results for K Σ (𝛼, 𝜆) and N Σ (𝛽, 𝜆)
To begin with, we can observe that the operators which were used by Murugusundaramoorthy et al. in the definitions of S Σ (, ) and M Σ (, ) can be written as the weighted harmonic mean of two expressions:   ()/() and 1; that is, where  =  −1 .
Let us define two new classes.In definitions of  and  we consider the weighted harmonic mean of 1+  ()/  () and   ()/(); namely, where 0 ≤  ≤ 1,  ∈ D, and  =  −1 .In fact, in the above functions the range of  can be extended to the set [0, ∞).Now, we can define the classes K Σ (, ) and N Σ (, ).
The idea of considering the weighted mean of 1 +   ()/  () 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 gives the following corollaries.
Proof of Theorem 15.Let  ∈ N Σ (, ) with 0 ≤  < 1 and  ≥ 0. From Definition 13 we obtain where ,  ∈ P. Hence, which results in the second part of the assertion.
From Theorem 15 we get the following corollaries.For  = 0 we obtain the bounds for S * Σ ().For  = 1, the set N Σ (, ) is the class K Σ () of bi-convex functions of order .Hence we have the following.
This bound is better than the one proved in [3].