Abstract
We introduce second Hankel determinant of biunivalent analytic functions associated with -pseudo-starlike function in the open unit disc subordinate to a starlike univalent function whose range is symmetric with respect to the real axis.
1. Introduction
Let be the class of all analytic functions of the formin the open unit disc . Let be the subclass of consisting of univalent functions. Let be the family of analytic functions in such that and of the formFor any two functions and analytic in , we say that the function is subordinate to in and we write it as , if there exists an analytic function , in with , such that . In view of Koebe theorem, every function has an inverse , defined byIn fact, the inverse function is given byA function is said to be biunivalent in if both and are univalent in . Let denote the class of all biunivalent functions defined in the unit disc . We notice that is nonempty. The behavior of the coefficients is unpredictable when the biunivalency condition is imposed on the function . In 1967, Lewin [1] introduced the class of biunivalent functions and investigated second coefficient in Taylor-Maclaurin series expansion for every . Subsequently, in 1967, Brannan and Clunie [2] introduced bistarlike functions and biconvex functions similar to the familiar subclasses of univalent functions consisting of strongly starlike, strongly convex, starlike, and convex functions and so on and obtained estimates on the initial coefficients conjectured that for bistarlike functions and for biconvex functions. Only the last estimate is sharp; equality occurs only for or its rotation. Since then, various subclasses of biunivalent functions class were introduced and nonsharp estimates on the first two coefficients and in Taylor-Maclaurin series expansion were found in several investigations. The coefficient estimate problem for each of is still an open problem. In 1976, Noonan and Thomas [3] defined th Hankel determinants of for and which is stated as follows:Easily one can observe that is a special case of the well known Fekete-Szegö functional where is real, for . Now for , , we get second Hankel determinantIn particular, sharp upper bounds on were obtained by the authors of articles [4–6] for various subclasses of analytic and univalent functions. In 2013, Babalola [7] determined the second Hankel determinant with Fekete-Szegö parameter for some subclasses of analytic functions. Let be an analytic function with positive real part in such that , which is symmetric with respect to the real axis. Such a function has a Maclaurin series expansion of the form
Researchers like Duren [8], Singh [9], and so on have studied various subclasses of usual known Bazilevi function of order denoted by which satisfy the geometric condition , where is nonnegative real number, different ways of perspectives of convexity, radii of convexity and starlikeness, inclusion properties, and so on. The class reduces to the starlike function and bounded turning function whenever and , respectively. This class is extended to which satisfy the geometric condition , where is nonnegative real number and . Recently, Babalola [7] defined new subclass -pseudo-starlike functions of order which satisfy the condition , and is denoted by . Babalola [7] proved that all pseudo-starlike functions are Bazilevi of type , order , and univalent in the open unit disc . For we note that functions in are defined by which is a product combination of geometric expressions for bounded turning and starlike functions. Note that the singleton subclass of contains the identity map. In 2016, Joshi et al. [10] defined two new subclasses of biunivalent functions using pseudo-starlike functions, one is class of strong -bi-pseudo-starlike functions of order and other is -bi-pseudo-starlike functions of order in the open unit disc. Many researchers [11–15] have estimated the second Hankel determinants for some subclasses of biunivalent functions. Motivated by the above-mentioned work, in this paper we have introduced -bi-pseudo-starlike functions subordinate to a starlike univalent function whose range is symmetric with respect to the real axis and estimated second Hankel determinants.
Definition 1. A function is said to be in the class , , if it satisfies the following conditions:where is an extension of to .(1)If , then the class reduces to the class and satisfies the following conditions: where is an extension of to .(2)If , , then the class reduces to the class and satisfies the following conditions: where is an extension of to .(3)If , then the class reduces to the class , and satisfies following conditions: where is an extension of to .(4)If , then the class reduces to the class of bistarlike functions and satisfies the following conditions: where is an extension of to .
Several choices of would reduce the class to some well known subclasses of .(1)For the function given by , , the class reduces to the class and satisfies the following conditions: where is an extension of to and this class is called class of bistarlike function of order .(2)For the function given by , the class reduces to the class and satisfies the following conditions: where is an extension of to and this class is called class of bistarlike function.
2. Preliminary Lemmas
Let denote the class of functions consisting of , such thatwhich are analytic in the open unit disc and satisfy for any .
Lemma 2 (see [8]). If , then for each and the inequality is sharp for the function .
Lemma 3 (see [16]). The power series for given in (14) converges in the open unit disc to a function in if and only if the Toeplitz determinantsand are all nonnegative. They are strictly positive except for , , real, and , for , where ; in this case for and for .
We may assume without any restriction that , on using Lemma 3 for and , respectively, we havewhich is equivalent toIf we consider the determinantwe get the following inequality:From (17) and (19), it is obtained thatfor some , .
Another required result is the optimal value of quadratic expression. Standard computations show that
3. Main Results
Theorem 4. If and is of the form (1) then we have the following. (1)(2)(3)
Proof. Since , there exist two Schwartz functions , in with , and , such thatDefine two functions , such thatThenThen (22) becomesNow equating the coefficients in (25)Now from (26) and (29)Now from (27) and (30)Now from (28) and (31)and with the help of the above Lemma 2, we get the required results.
Theorem 5. If is of the form (1) then
Proof. Now adding (27) and (30), we get thatNow from (26) and (29), we get thatNow from (36) and (37)where which completes the proof of the theorem.
Theorem 6. If is of the form (1) then we have the following. (1)If , then (2)If , or , then .(3), ; thenwhere
Proof. Using the values of from the above theorem, one can obtainAccording to Lemma 3 we get thatFor some , with , using (42), we haveSince , . Letting we may assume without any restriction that Thus for and , we obtainwhereNow we need to maximize in the closed square for . We must investigate the maximum of according to , , and taking into account the sign of .
First, let . Since and , we conclude that . Thus the function cannot have a local maximum in the interior of the square . Now, we investigate the maximum of on the boundary of the square .
For and (similarly and ), we obtain(i) The Case . In this case and for any fixed with , it is clear that ; that is, is an increasing function. Hence for any fixed the maximum of occurs at and(ii) The Case . Since for and for any fixed with , it is clear that and so . Hence for any fixed the maximum of occurs at . Also for we obtainTaking into account the value of (48) and case (i) and case (ii), for and for any fixed with , For and (similarly and ), we obtainSimilar to the above case of , we get thatSince for , on the boundary of the square . Thus the maximum of occurs at and in the closed square .
Letting ,Substituting the values of in the above equation,LetThen , where .
Then with help of optimal value of quadratic expression, we get the required result. This completes the proof of the theorem.
Corollary 7. If and is of the form (1) then
Corollary 8. If and is of the form (1) thenwhere
Corollary 9. If and is of the form (1) thenwhere
Corollary 10. If and is of the form (1) then we have the following. (1)If , then (2)If , or , then (3), ; thenThe above result is obtained by taking in Theorem 6, which is the second Hankel determinant of bistarlike function.
Corollary 11. If and is of the form (1) then
Corollary 12. If and is of the form (1) then
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.