Coefficients Bounds for Certain Subclass of Biunivalent Functions Associated with Ruscheweyh -Differential Operator
We introduce in our present investigation a new subclass of analytic and biunivalent functions associated with Ruscheweyh -differential operator in open unit disk . We use the Faber polynomial expansions to find th coefficients bounds of class of bisubordinate functions and also find initial coefficient estimates.
Let denote the class of all function which is analytic in the open unit disk and has the Taylor series expansion of the following form:By we mean the class of all functions in which are univalent in The Koebe one-quarter theorem  states that the image of under every function from contains a disk of radius It is well known that every univalent function has an inverse which is defined as whereA function is said to be biunivalent in if both and are univalent in .
If the functions and are analytic in , then is said to be subordinate to , written as if there exists a Schwarz function , analytic in , with and , such that Let denote the class of analytic and biunivalent functions in given by the Taylor-Maclaurin series expansion (1). Here we give few examples of functions in the class such that However, the famous Koebe function is not in ; for more details we refer to . For , the class of biunivalent analytic functions was first introduced and studied by Lewin where it was proved to show that . Brannan and Clunie  proved that . Netanyahu  showed that Brannan and Taha  introduced certain subclass of the biunivalent functions class . For a brief history and interesting examples of biunivalent functions we refer to [3, 5–11].
The Faber polynomials introduced by Faber . Gong  and Schiffer  demonstrated the significance of the Faber polynomials in mathematical sciences, especially in geometric function theory. In the literature, there are only a few works determining the general coefficient bounds for the analytic biunivalent functions given by (1) using Faber polynomial expansions. A very little is known about the bounds of Maclaurin’s series coefficient for by using a Faber polynomials. For more study about Faber polynomials we refer to [15–21].
Using the technique of convolution, Ruscheweyh  defined the operator on the class of analytic functions as For , we obtain The expression is called an th-order Ruscheweyh derivative of and the symbol stands for Hadamard product (or convolution).
For and , , the number is defined in  as For any nonnegative integer the -number shift factorial is defined as We have . Throughout in this paper we will assume to be fixed number between and .
The -derivative operator or -difference operator for is defined as It can easily be seen that for and The -generalized Pochhammer symbol for and is defined as and, for , let -gamma function be defined as For Ruscheweyh -differential operator was defined by Aldweby and Darus  (see also ), as If , equality (17) implies which is the well known recurrent formula for Ruscheweyh differential operator.
in the present paper we introduce new subclass of the function class , involving Ruscheweyh -differential operator By using Faber polynomial coefficient techniques we determine estimates for the general coefficient bounds for and also estimates on the coefficients and for functions in the new subclass of function class . Several related classes are also considered, and connections to earlier known results are also defined.
Definition 1. A function , , , , and ; we introduce a new class of biunivalent functions as if and only if whereand is defined by (3).
On specializing the parameters , , and , one can state the various new subclasses as illustrated in the following definition.
Definition 4. For , and A function, , is in the class if the following conditions are satisfied: where , are given by (20) and is defined by (3).
It is well known thatSpecial Cases(i)For and , the class ; see .(ii)For , , and , the class ; see .(iii)For , , , and , the class ; see .(iv)For , , and , the class ; see .
Lemma 5 (see ). Let the Schwarz function be given by then
2. Main Results
Using the Faber polynomial expansion of functions of the form (1), the coefficients of its inverse map may be expressed as  given by wheresuch that with is a homogeneous polynomial in the variables ; see . In particular, the first three terms of are in general, for any and , an expansion of (for details we refer [16, 29]), which is as follows: where and, by , while , and the sum is taken over all nonnegative integer satisfying Evidently, , or, equivalently, while , and the sum is taken over all nonnegative integer satisfying It is clear that ; the first and last polynomials are ,
Theorem 6. For , , , and If , if , , then
Proof. For the function of the form (1), we haveand, for its inverse map , we havewhere is given by (16) and
Since both function and its inverse map are in , by the definition of subordination there exist two Schwarz functions and , where , We have wherewhereIn general [15, 16] for any and , an expansion of ,where is a homogeneous polynomial of degree in the variables
For the coefficients of the Schwarz functions and , and .
Comparing the corresponding coefficients of (35) and (37), we have Similarly, corresponding to coefficients of (36) and (39), we haveNote that, for , , we have Taking the absolute values of (44) and (45), we haveBy using , , from (20) and from (16) in (47), we havewhich completes the proof of theorem.
By putting , in Theorem 6, we have the following corollary.
Corollary 7. For , , and , if and if , , thenFor , , in Theorem 6, we have the following corollary.
Corollary 8. For , , if and if , , thenFor , in Theorem 6, we obtain the following corollary.
Corollary 11. For , if and if , , then
For , , in Theorem 6, we obtain the following corollary.
Corollary 12 (see ). For and , if and if , , then
Theorem 13. For , , , and , if , then
Proof. Replacing by and in (44) and (45), respectively, we have From (56) and (58) we haveUsing (20) and (16) in (61), we haveAdding (57) and (59) we have Taking absolute values of both sides of (63) and applying the estimates , of Lemma 5 and , , we haveusing (20) and (16) in (64), we haveNow, in order to find , we subtract (59) from (57) and we have Using (60) in (66), we haveTaking the modulus of (67), we haveBy using the estimates , , of Lemma 5, and , , in (68), we have and using (56) in (69) we haveUsing (20) and (16) in (70), we haveAgain using (64) in (69) we haveFrom (59) we have Using , of Lemma 5 and (20), (16), on (73), we have
For , in Theorem 13, we obtain the following corollary.
Corollary 14. For , , and , if , then
For and , in Theorem 13, we obtain the following corollary.
Corollary 15. For , , if , then
For , , in Theorem 13, we obtain the following corollary.
Corollary 16 (see ). For , , and , if , then
For , , , in Theorem 13, we obtain the following corollary.
Corollary 17 (see ). For and , if , then
Corollary 18 (see ). For , if , then
For , , , in Theorem 13, we obtain the following corollary.
Corollary 19 (see ). For , , and , if , then
The present address of Zahid Shareef is Division of Engineering, Higher Colleges of Technology, P.O. Box 4114, Fujairah, UAE.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work is supported by MOHE Grant FRGS/1/2016/STG06/UKM/01/1.
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