Abstract
The object of the present investigation is to solve the Fekete-Szegö problem and determine the sharp upper bound to the second Hankel determinant for a new class of analytic functions involving the Carlson-Shaffer operator in the unit disk. We also obtain a sufficient condition for normalized analytic functions in the unit disk to be in this class.
1. Introduction and Preliminaries
Let be the class of functions of the form which are analytic in the open unit disk .
A function is said to be starlike of order , if
Similarly, a function is said to be convex of order , if By usual notations, we write these classes of functions by and , respectively. We denote and , the familiar subclasses of starlike, convex functions in .
Furthermore, let denote the class of analytic functions normalized by such that in .
For functions and , analytic in , we say that is subordinate to , written as or , if there exists a Schwarz function , which (by definition) is analytic in with , , and , . Furthermore, if the function is univalent in , then we have the following equivalence relation (cf., e.g., [1]; see also [2]):
For functions analytic in , we define the Hadamard product (or convolution) of and by Note that is also analytic in .
Carlson and Shaffer [3] defined the linear operator in terms of the incomplete beta function by where and denotes the Pochhammer symbol (or shifted factorial) given, in terms of the Gamma function , by
If is given by (1), then it follows from (7) that We note that for (i);(ii);(iii);(iv), the well-known Ruscheweyh derivative [4] of ;(v), the well-known Owa-Srivastava fractional differential operator [5]. We also observe that and .
With the aid of the linear operator , we introduce a subclass of as follows.
Definition 1. A function is said to be in the class , if it satisfies the condition
It follows from (12) and the definition of subordination that a function satisfies the following subordination relation: We further note that if , then the function lies in the region bounded by the right half of the lemniscate of Bernoulli given by
Noonan and Thomas [6] defined the th Hankel determinant of a sequence of real or complex numbers by This determinant has been studied by several authors including Noor [7] with the subject of inquiry ranging from the rate of growth of (as ) to the determination of precise bounds with specific values of and for certain subclasses of analytic functions in the unit disc .
For , , , and , the Hankel determinant simplifies to
The Hankel determinant was considered by Fekete and Szegö [8] and we refer to as the second Hankel determinant. It is known [9] that if given by (1) is analytic and univalent in , then the sharp inequality holds. For a family of functions in of the form (1), the more general problem of finding the sharp upper bounds for the functionals is popularly known as Fekete-Szegö problem for the class . The Fekete-Szegö problem for various known subclasses of univalent functions (i.e., starlike, convex, close-to-convex, etc.) has been completely settled [8, 10–12]. Recently, Janteng et al. [13, 14] have obtained the sharp upper bounds to the second Hankel determinant for the family of functions in whose derivatives have positive real part in . For initial work on this class of functions, one may refer to the work of MacGregor [15].
In our present investigation, we follow the techniques adopted by Libera and Złotkiewicz [16, 17] to solve the Fekete-Szegö problem and also determine the sharp upper bound to the second Hankel determinant for the class .
To establish our main results, we will need the following lemma for functions belonging to the class .
Lemma 2. Let the function , given by (4), be a member of the class . Then for some complex numbers satisfying and . The estimates in (17) and (18) are sharp for the functions defined in by
We note that the estimate (17) is contained in [9]; the estimate (18) is due to Ma and Minda [18], whereas the results in (19) are obtained by Libera and ZĹ‚otkiewicz [17] (see also [16]).
2. Main Results
Unless otherwise mentioned, we assume throughout the sequel that .
Now, we solve the Fekete-Szegö problem for the class .
Theorem 3. If the function , given by (1), belongs to the class , then for any The estimate is sharp.
Proof. From (13), it follows that where is analytic and satisfies the conditions and in . Setting we see that . From the above expression, we get so that, by (22), we get Now, it is easily seen that Differentiating the series expansion of given by (1) with respect to and comparing the coefficients of , , and in (26), we deduce that Thus, by using (27) and (28), we get which with the aid of (18) yields the required estimate (21). The estimate (21) is sharp for the function defined in by
Remark 4. If the function , given by (1), belongs to the class , then it follows at once from (27) that and Theorem 3 gives . The inequality for is sharp when is defined by and the estimate for is sharp for the function defined by
For the case , Theorem 3 reduces to the following result.
Corollary 5. Let . If the function , given by (1), belongs to the class , then The estimate is sharp for the function defined in by
Putting and in Corollary 5, we get the following.
Corollary 6. If the function , given by (1), satisfies the subordination relation then The estimate is sharp for the function defined in by
In the following theorem, we find the sharp upper bound to the second Hankel determinant for the class .
Theorem 7. If and the function , given by (1), belongs to the class , then The estimate in (39) is sharp for the function , given by (33).
Proof. From (27), (28), and (29), we deduce that Since the functions and are in the class simultaneously, we assume without loss of generality that . For convenience of notation, we write . Now, by using (19) in (40), we get for some and for some . Applying the triangle inequality in (41) and replacing by in the resulting equation, we get We next maximize the function on the closed rectangle . Since for and , the function cannot have a maximum value in the interior on the closed rectangle . Therefore, for fixed , where Setting we note that either or Since , we further observe that . Thus, the maximum value of is attained at so that the upper bound in (42) corresponds to and from which we get the assertion of the theorem.
Letting and in Theorem 7, we get the following.
Corollary 8. If the function , given by (1), satisfies the condition (36), then and the estimate is sharp for the function defined by
Next, we find the sharp upper bound for the fourth coefficient of functions belonging to the class .
Theorem 9. If the function , given by (1), belongs to the class , then The estimate is sharp.
Proof. Using (19) in (29) and following the lines of proof of Theorem 7, we deduce that
for some and . By an application of the triangle inequality in the above expression followed by replacement of by in the resulting equation, we obtain
We next maximize the function on the closed rectangle . Since
for and , it follows that cannot have a maximum value in the interior of the closed rectangle . Thus, for fixed ,
where
We further note that
for or . Since , the function attains its maximum value at . Thus, the upper bound of the function corresponds to . Putting in (52), we get our desired estimate (50).
The estimate in (50) is sharp for the function defined by
Letting and in Theorem 9, we obtain the following.
Corollary 10. If the function , given by (1), satisfies the condition (36), then and the estimate is sharp for the function defined by
Finally, we obtain a sufficient condition for a function in to be in the class .
Theorem 11. Let . If satisfies then and the result is the best possible.
Proof. Setting
and choosing the principal branch in (62), we see that is analytic in with . Taking the logarithmic differentiation in (62) and using the identity (11) in the resulting equation, we deduce that
We claim that for all . Otherwise, there exists a point such that
Letting and applying Jack’s lemma [19], we have
Using (65) in (63), we get
which contradicts the hypothesis (60). Thus, we conclude that for all and the assertion of the theorem follows from (62).
To see that the result is the best possible, we consider the principal branch of the function defined by
Then, it follows from (67) that
On differentiating the above expression logarithmically followed by the use of the identity (11), we obtain
The proof of the theorem is thus completed.
For in Theorem 11, we have the following.
Corollary 12. If satisfies then . The result is the best possible.
Remark 13. Further, by specializing the parameters and , one can obtain interesting subclasses of involving the various operators discussed in the introduction and the corresponding results obtained here can be extended to these classes.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.