A Certain Subclass of Analytic Functions with Bounded Positive Real Part

  , T For real numbers  and  such that 0 1      , we denote by   the class of normalized analytic functions which satisfy     Re f z     z   , T   , where denotes the open unit disk. We find some relationships involving functions in the class      . And we estimate the bounds of coefficients and solve Fekete-Szegö problem for functions in this class. Furthermore, we investigate the bounds of initial coefficients of inverse functions or bi-univalent functions.


 
Let A denote the class of analytic functions in the unit disk which is normalized by and .Also let denote the subclass of  A which is composed of functions which are univalent in . We say that f is subordinate to F in , written as and .

   
We remark that, for given real numbers  and    if and only if f satisfies each of the following two subordination relationships: . Now, we define an analytic function by The above function was introduced by Kuroki and Owa [1] and they proved maps onto a convex domain , conformally.Using this fact and the definition of subordination, we can obtain the following Lemma, directly.

 
in .And we note that the function , defined by (1), has the form , where  and  such that For given real numbers 0 1 , where f  is the inverse function of f .
In our present investigation, we first find some relationships for functions in bounded positive class   , 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 bi-univalent functions.For the coefficient bounds of functions in special subclasses of , the readers may be referred to the works [2][3][4].

Relations Involving Bounds on the Real Parts
In this section, we shall find some relations involving the functions in   .And the following Lemma will be needed in finding the relations.Lemma 2.1 (see Miller and Mocanu [5]) Let  be a set in the complex plane and let b be a complex number such that .Suppose that a function satisfies the condition Then Proof.We put Then is analytic in and .And  where :R e : .p z zp z z w w Now, let .And we    shall find the maximum value of , where and are real numbers.Then We note that F  is continuous on and is even.
. By Lemma 2.1, we get  p z in and this shows that the inequality (5) holds and the proof of Theorem 2.2 is completed. Then Proof.We put And, we put where and are real numbers.As in the proof of Theorem 2.2, we can get by (6).And Now we define a function by We note that for  .Hence     F  is continuous on and is even.Since and . By Lemma 2.1, we get and this shows that the inequality (7)


In the present section, we will solve some coefficient problems involving functions in the class   .And our first result on the coefficient estimates involves the  and the following Lemma will be needed.
Lemma 3.1.(see Rogosinski [6]) Let  be analytic and univalent in and suppose that is analytic in and satisfies the following subordination: Let  and  be real numbers such that 0 1 where 1 B is given by and Then, the subordination (2) can be written as follows: Note that the function defined by (10) is convex in and has the form then by Lemma 3.1, we see that the subordination (11) implies that which is the sum of 2 n terms.Hence,  which leads to the inequality (8).If is odd, terms in the bracket.Hence, we get which leads to the inequality (8).Therefore, the proof of Theorem 3.
Now, the following result holds for the coefficient of where with is given by (3).Let Then is analytic and has positive real part in the open unit disk .We also have We find from the equations ( 12) and (13) that 1; 1 2 .

B h h
And substituting and   Finally, we shall estimate on some initial coefficients for the bi-univalent functions  .   Theorem 3.5.For given  and  such that f be given by 0 1 and , where is given by (1).Let where lead us to the inequality (20).Therefore, the proof of Theorem 3.5 is completed.

 f F Definition 1 . 1 .
Let  and  be real numbers .The function f A  belongs to the class  ,  T   if f satisfies the following inequality: 16)    in (14), we can obtain the result as asserted.Using Theorem 3.4, we can get the following result.

2 immediately.
Furthermore, an application of Theorem 3.4 (with   ) gives the estimates for 3 proof of Corollary 3.1 is completed.
holds and the proof of Theorem 2.3 is completed.By combining Theorem 2.2 and 2.3, we can get the following Theorem.