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International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 294378, 6 pages
On Certain Classes of Convex Functions
Department of Mathematics, Kyungsung University, Busan 608-736, Republic of Korea
Received 25 February 2013; Accepted 7 May 2013
Academic Editor: Heinrich Begehr
Copyright © 2013 Young Jae Sim and Oh Sang 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 and such that , we denote by the class of normalized analytic functions which satisfy the following two sided-inequality: where denotes the open unit disk. We find some relationships involving functions in the class . And we estimate the bounds of coefficients and solve the Fekete-Szegö problem for functions in this class. Furthermore, we investigate the bounds of initial coefficients of inverse functions or biunivalent functions.
Let denote the class of analytic functions in the unit disc which is normalized by Also let denote the subclass of which is composed of functions which are univalent in . And, as usual, we denote by the class of functions in which are convex in .
We say that is subordinate to in , written as , if and only if for some Schwarz function such that If is univalent in , then the subordination is equivalent to
Definition 1. Let and be real numbers such that . The function belongs to the class if satisfies the following inequality:
It is clear that . And we remark that, for given real numbers and , if and only if satisfies each of the following two subordination relationships:
Now, we define an analytic function by The above function was introduced by Kuroki and Owa , and they proved that maps onto a convex domain conformally. Using this fact and the definition of subordination, we can obtain the following lemma, directly.
Lemma 2. Let and . Then if and only if
And we note that the function , defined by (7), has the form where
For given real numbers and such that , we denote by the class of biunivalent functions consisting of the functions in such that where is the inverse function of .
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 [2–4].
2. Relations Involving Bounds on the Real Parts
In this section, we will find some relations involving the functions in . And the following lemma will be needed in finding the relations.
Lemma 3 (see Miller and Mocanu [5, Theorem ]). Let be a set in the complex plane and let be a complex number such that . Suppose that a function satisfies the condition for all real and all . If the function defined by is analytic in and if then in .
Theorem 4. Let , and Then
Proof. First of all, we put and note that for . Let Differentiating (17), we can obtain where Using (15), we have Now for all real with , Define a function by Then is a continuous even function and Hence and is increasing on , since . Hence satisfies that for all . Therefore, by combining (21) and (24), we can get And this shows that for all with . By Lemma 3, we get for all , and this shows that the inequality (16) holds and the proof of Theorem 4 is completed.
Theorem 5. Let , and Then
Proof. We put and note that for . And let As in the proof of Theorem 4, we can get by (26). And for all real , with , where is given by Since satisfies the inequality for all . Therefore, And this shows that for all with . By Lemma 3, we get for all , and this shows that the inequality (27) holds and the proof of Theorem 5 is completed.
Theorem 6. Let , and Then
3. Coefficient Problems Involving Functions in
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 function class and the following lemma will be needed.
Lemma 7 (see Rogosinski [6, Theorem 10]). Let be analytic and univalent in and suppose that maps onto a convex domain. If is analytic in and satisfies the following subordination: then
Theorem 8. Let and be real numbers such that . If the functions , then where 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 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 in both sides lead to A simple calculation combined with the inequality (46) yields that where is given by (47) and . Hence, we have . To prove the assertion of the theorem, we need to show that We now use the mathematical induction for the proof of the theorem. For the case , it is clear. We assume that the inequality (51) holds for . Then, some calculation gives us that which implies that the inequality (51) is true for . Hence, by the mathematical induction, we prove that where is given by (47). This completes the proof of Theorem 8.
And now, we will solve the Fekete-Szegö problem for , and we will need the following lemma.
Lemma 9 (see Keogh and Merkes ). Let be a function with positive real part in . Then, for any complex number ,
Now, the following result holds for the coefficient of .
Theorem 10. Let and let the function given by be in the class . Then, for a complex number , The result is sharp.
Proof. Let us consider a function given by . Then, since , we have , where where is given by (11). Let Then is analytic and has positive real part in the open unit disk . 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 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.
Corollary 11. Let and let the function , given by , be in the class . Also let the function , defined by be the inverse of . If then
Proof. The relations (66) and (67) give Thus, we can get the estimate for by immediately. Furthermore, an application of Theorem 10 (with ) gives the estimates for ; hence, the proof of Corollary 11 is completed.
Finally, we will estimate some initial coefficients for the bi-univalent functions .
Proof. If , then and , where . Hence where is given by (7). Let Then and are analytic and have positive real part in . Also, we have By suitably comparing coefficients, we get where and are given by (63) and (64), respectively. Now, considering (76) and (78), we get Also, from (77), (78), (79), and (80), we find that Therefore, we have This gives the bound on as asserted in (71). Now, further computations from (77), (79), (80), and (81) lead to This equation, together with the well-known estimates leads us to the inequality (72). Therefore, the proof of Theorem 12 is completed.
The research was supported by Kyungsung University Research Grants in 2013.
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