- Open Access
Coefficient estimates for new subclasses of Ma-Minda bi-univalent functions
© Tang et al.; licensee Springer 2013
- Received: 9 November 2012
- Accepted: 14 June 2013
- Published: 8 July 2013
In this paper, we introduce and investigate two new subclasses and of Ma-Minda bi-univalent functions defined by using subordination in the open unit disk . For functions belonging to these new subclasses, we obtain estimates for the initial coefficients and . The results presented in this paper would generalize those in related works of several earlier authors.
- analytic and univalent functions
- bi-univalent functions
- coefficient estimates
Dedicated to Professor Hari M Srivastava
A function is said to be bi-univalent in D if both f and are univalent in D. Let σ denote the class of bi-univalent functions defined in the unit disk D. In 1967, Lewin  first introduced the class σ of bi-univalent functions and showed that for every . Subsequently, Branan and Clunie  conjectured that for . Later, Netanyahu  proved that . The coefficient estimate problem for each of () is still an open problem.
Brannan and Taha  (see also ) introduced certain subclasses of a bi-univalent function class σ similar to the familiar subclasses and of starlike and convex functions of order α (), respectively (see ). Thus, following Brannan and Taha  (see also ), a function is in the class of strongly bi-starlike functions of order α () if both functions f and are strongly starlike functions of order α. The classes and of bi-starlike functions of order α and bi-convex functions of order α, corresponding (respectively) to the function classes and , were also introduced analogously. For each of the function classes and , they found non-sharp estimates on the first two Taylor-Maclaurin coefficients and (for details, see [6, 7]).
An analytic function f is subordinate to an analytic function g, written , if there is an analytic function w with such that . If g is univalent, then if and only if and . Ma and Minda  unified various subclasses of starlike and convex functions for which either of the quantities or is subordinate to a more general superordinate function. For this purpose, they considered an analytic function φ with positive real part in the unit disk D, , , and φ maps D onto a region starlike with respect to 1 and symmetric with respect to the real axis. The classes and of Ma-Minda starlike and Ma-Minda convex functions are respectively characterized by or . A function f is bi-starlike of Ma-Minda type or bi-convex of Ma-Minda type if both f and are respectively Ma-Minda starlike or convex. These classes are denoted respectively by and . Recently, Srivastava et al. , Frasin and Aouf  and Caglar et al.  introduced and investigated various subclasses of bi-univalent functions and found estimates on the coefficients and for functions in these classes. Very recently, Ali et al. , Kumar et al. , Srivastava et al.  and Xu et al. unified and extended some related results in [7, 10–12, 17] by generalizing their classes using subordination.
Motivated by Ali et al.  and Kumar et al. , we investigate the estimates for the initial coefficients and of bi-univalent functions of Ma-Minda type belonging to the classes and defined in Section 2. Our results generalize several well-known results in [10–14] and these are also pointed out.
With this assumption on φ, we now introduce the following subclasses of Ma-Minda bi-univalent functions.
(, , ) (see Caglar et al. [, Definition 2.1]);
(, , ) (see Caglar et al. [, Definition 3.1]);
() (see Kumar et al. [, Definition 1.1]);
() (see Kumar et al. [, Definition 2.1]);
(see Ali et al. [, p.345]);
(, ) (see Frasin and Aouf [, Definition 2.1]);
(, ) (see Frasin and Aouf [, Definition 3.1]);
() (see Srivastava et al. [, Definition 1]);
() (see Srivastava et al. [, Definition 2]).
For functions in the class , the following estimates are obtained.
which yields the desired estimate on as asserted in (2.5).
Now, we consider the bounds on according to μ.
Thus, from (2.26), (2.29) and (2.30), we obtain the desired estimate on given in (2.6). This completes the proof of Theorem 2.1. □
Putting and in Theorem 2.1, we respectively get the following Corollaries 2.1 and 2.2.
where the function g is defined by (2.4).
For functions in the class , the following estimates are derived.
which yields the desired estimate on as described in (2.31).
which easily leads to the desired estimate (2.32) on . □
Taking in Theorem 2.2, we obtain the following corollary.
Corollary 2.3 [, Theorem 2.3]
Further, for , putting and in Corollary 2.3, respectively, we have the following Corollaries 2.4 and 2.5.
Corollary 2.4 [, Corollary 2.1]
Corollary 2.5 [, Corollary 2.2]
in Corollaries 2.4 and 2.5, we obtain the results of Brannan and Taha [, Theorems 2.1, 3.1 and 4.1, respectively].
The present investigation was partly supported by the Natural Science Foundation of People’s Republic of China under Grant 11271045, the Higher School Doctoral Foundation of People’s Republic of China under Grant 20100003110004 and the Natural Science Foundation of Inner Mongolia of People’s Republic of China under Grant 2010MS0117. The authors are thankful to the referees for their careful reading and making some helpful comments which have essentially improved the presentation of this paper.
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