Coefficient Estimate Problem for a New Subclass of Biunivalent Functions
We introduce a unified subclass of the function class of biunivalent functions defined in the open unit disc. Furthermore, we find estimates on the coefficients and for functions in this subclass. In addition, many relevant connections with known or new results are pointed out.
Let denote the class of functions of the form which are analytic in the open unit disc . Further, by , we will denote the class of all functions in which are univalent in .
Some of the important and well-investigated subclasses of the univalent function class include, for example, the class of starlike functions of order in and the class of convex functions of order in .
It is well known that every function has an inverse , defined by where
A function is said to be biunivalent in if both and are univalent in . Let denote the class of biunivalent functions in given by (1).
In 1967, Lewin  investigated the biunivalent function class and showed that ; on the other hand Brannan and Clunie  (see also [3–5]) and Netanyahu  made an attempt to introduce various subclasses of biunivalent function class and obtained nonsharp coefficient estimates on the first two coefficients and of (1). But the coefficient problem for each of the following Taylor-Maclaurin coefficients for ; is still an open problem. In this line, following Brannan and Taha , recently, many researchers have introduced and investigated several interesting subclasses of biunivalent function class and they have found nonsharp estimates on the first two Taylor-Maclaurin coefficients and ; for details, one can refer to the works of [7–13].
Now, we define of function satisfying the following conditions: for some , where is the extension of to . Similarly, we say that a function belongs to the class if satisfies the following inequalities: for some , where is the extension of to . The classes and were introduced by Prema and Keerthi ; furthermore, for these classes, they have found the following estimates on the first two Taylor-Maclaurin coefficients in (1).
Theorem 1. If , , and , then
Theorem 2. If , , and , then
Definition 3. Let , and let the functions be so constrained that We say that if the following conditions are satisfied: where and the function is the extension of to .
The objective of the present paper is to introduce a new subclass and to obtain the estimates on the coefficients and for the functions in theaforementioned class, employing the techniques used earlier by Xu et al. [12, 13].
2. Main Result
In this section, we find the estimates on the coefficients and for the functions in the class .
Theorem 4. Let be of the form (1). If , then
Proof. Since , from (9), we have,
satisfy the conditions of Definition 3. Now, equating the coefficients in (12), we get
From (14) and (16), we get
From (15) and (17), we obtain
Since and , we immediately have
This gives the bound on as asserted in (10).
Next, in order to find the bound on , by subtracting (17) from (15), we get It follows from (19) and (21) that Since and , we readily get as asserted in (11). This completes the proof of Theorem 4.
By setting , where and , in Theorem 4, we get the following corollary.
Corollary 5. Let be of the form (1) and in the class . Then,
If we choose and in Corollary 5, we have the following corollary.
Corollary 6. Let be of the form (1) and in the class , and . Then,
If we set , , where and in Corollary 5, we readily have the following corollary.
Corollary 8. Let be of the form (1) and in the class , and . Then
Remark 10. For , the bounds obtained in Theorem 4 are coincident with the outcome of Xu et al. . Taking in Corollaries 6 and 8, the estimates on the coefficients and , are the improvement of the estimates on the first two Taylorû Maclaurin coefficients obtained in [10, Corollaries 2.3 and ]. Also, for the choices of , the results stated in Corollaries 6 and 8 would improve the bounds stated in [11, Theorems 1 and 2], respectively. Furthermore, various other interesting corollaries and consequences of our main result could be derived similarly by specializing and .
The authors would like to thank the referee for his valuable suggestions.
D. A. Brannan and J. G. Clunie, Eds., Aspects of Contemporary Complex Analysis, Academic Press, London, UK, 1980.
T. S. Taha, Topics in Univalent Function Theory [Ph.D. thesis], University of London, London, UK, 1981.
S. Prema and B. S. Keerthi, “Coefficient bounds for certain subclasses of analytic functions,” Journal of Mathematical Analysis, vol. 4, no. 1, pp. 22–27, 2013.View at: Google Scholar