On Certain Classes of Convex Functions

Received 25 February 2013; Accepted 7 May 2013 Academic Editor: Heinrich Begehr Copyright © 2013 Y. J. Sim and O. S. Kwon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For real numbers α and β such that 0 ≤ α < 1 < β, we denote by K(α, β) the class of normalized analytic functions which satisfy the following two sided-inequality: α < R{1 + (zf 󸀠󸀠


Introduction
Let A denote the class of analytic functions in the unit disc which is normalized by Also let S denote the subclass of A which is composed of functions which are univalent in U. And, as usual, we denote by K the class of functions in A which are convex in U.
We say that  is subordinate to  in U, written as  ≺  ( ∈ U), if and only if () = (()) for some Schwarz function () such that If  is univalent in U, then the subordination  ≺  is equivalent to Definition 1.Let  and  be real numbers such that 0 ≤  < 1 < .The function  ∈ A belongs to the class K(, ) if  satisfies the following inequality: It is clear that K(, ) ⊂ K.And we remark that, for given real numbers  and  (0 ≤  < 1 < ),  ∈ K(, ) if and only if  satisfies each of the following two subordination relationships: Now, we define an analytic function  : U → C by The above function  was introduced by Kuroki and Owa [1], and they proved that  maps U onto a convex domain conformally.Using this fact and the definition of subordination, we can obtain the following lemma, directly.
International Journal of Mathematics and Mathematical Sciences Lemma 2. Let () ∈ A and 0 ≤  < 1 < .Then  ∈ K(, ) if and only if And we note that the function , defined by (7), has the form where For given real numbers  and  such that 0 ≤  < 1 < , we denote by K  (, ) the class of biunivalent functions consisting of the functions in A such that where  −1 is the inverse function of .
In our present investigation, we first find some relationships for functions in bounded positive class K(, ).And we solve several coefficient problems including Fekete-Szegö problems for functions in the class.Furthermore, we estimate the bounds of initial coefficients of inverse functions and biunivalent functions.For the coefficient bounds of functions in special subclasses of S, the readers may be referred to the works [2][3][4].

Relations Involving Bounds on the Real Parts
In this section, we will find some relations involving the functions in K(, ).And the following lemma will be needed in finding the relations.Then Proof.First of all, we put  = 1/(2 − ) and note that 1/2 ≤  < 1 for 0 ≤  < 1.Let Differentiating (17), we can obtain where Using (15), we have Now for all real ,  ∈ R with Define a function  : R → R by Then  is a continuous even function and Hence   (0) = 0 and  is increasing on (0, ∞), since 1/2 ≤  < 1. Hence  satisfies that for all  ∈ R. Therefore, by combining (21) and (24), we can get And this shows that R{(, )} ∉ Ω  for all ,  ∈ R with  ≤ −(1 +  2 )/2.By Lemma 3, we get R{()} > 0 for all  ∈ U, and this shows that the inequality (16) holds and the proof of Theorem 4 is completed.
And let As in the proof of Theorem 4, we can get by ( 26).And for all real ,  with where () is given by Since  > 1,  satisfies the inequality for all  ∈ R. Therefore, And this shows that R{(, )} ∉ Ω  for all ,  ∈ R with  ≤ −(1 +  2 )/2.By Lemma 3, we get R{()} > 0 for all  ∈ U, and this shows that the inequality (27) holds and the proof of Theorem 5 is completed.

Coefficient Problems Involving Functions in K(𝛼,𝛽)
In the present section, we will solve some coefficient problems involving functions in the class K(, ).And our first result on the coefficient estimates involves the function class K(, ) and the following lemma will be needed.
where | 1 | is given by Proof.Let us define Then, the subordination (9) can be written as follows: Note that the function () defined by ( 41) is convex in U and has the form where If we let then by Lemma 7, we see that the subordination (42) implies that where Now, equality (40) implies that Then, the coefficients of  −1 in both sides lead to where | 1 | is given by ( 47).This completes the proof of Theorem 8.
And now, we will solve the Fekete-Szegö problem for  ∈ K(, ), and we will need the following lemma.
Proof.Let us consider a function () given by () = 1 +   ()/  ().Then, since  ∈ K(, ), we have () ≺ ()( ∈ U), where where   is given by (11).Let Then ℎ is analytic and has positive real part in the open unit disk U. We also have We find from (58) that which imply that where Applying Lemma 9, we can obtain And substituting in (62), we can obtain the result as asserted.The estimate is sharp for the function  : U → C defined by where the function  is given by (7).Hence the proof of Theorem 10 is completed.
Using Theorem 10, we can get the following result.
Finally, we will estimate some initial coefficients for the bi-univalent functions  ∈ K  (, ).
be analytic and univalent in U and suppose that () maps U onto a convex domain.If () = ∑ ∞ =1     is analytic in U and satisfies the following subordination: Theorem 8. Let  and  be real numbers such that 0 ≤  < 1 < .If the functions Thus, we can get the estimate for | 2 | by ∞ =2     , be in the class K  (, ).Then