Fekete-Szegö Type Coefficient Inequalities for Certain Subclass of Analytic Functions and Their Applications Involving the Owa-Srivastava Fractional Operator
A new subclass of analytic functions is introduced. For this class, firstly the Fekete-Szegö type coefficient inequalities are derived. Various known or new special cases of our results are also pointed out. Secondly some applications of our main results involving the Owa-Srivastava fractional operator are considered. Thus, as one of these applications of our result, we obtain the Fekete-Szegö type inequality for a class of normalized analytic functions, which is defined here by means of the Hadamard product (or convolution) and the Owa-Srivastava fractional operator.
1. Introduction and Definitions
Let denote the class of functions of the form which are analytic in the unit disk Also let denote the subclass of consisting of univalent functions in .
Fekete and Szegö  proved a noticeable result that the estimate holds for . The result is sharp in the sense that for each there is a function in the class under consideration for which equality holds.
The coefficient functional on represents various geometric quantities as well as in the sense that this behaves well with respect to the rotation; namely,
In fact, rather than the simplest case when we have several important ones. For example, represents , where denotes the Schwarzian derivative Moreover, the first two nontrivial coefficients of the th root transform of with the power series (1) are written by so that where
Thus it is quite natural to ask about inequalities for corresponding to subclasses of . This is called Fekete-Szegö problem. Actually, many authors have considered this problem for typical classes of univalent functions (see, e.g., [1–12]).
For two functions and , analytic in , we say that the function is subordinate to in , and we write if there exists a Schwarz function , analytic in , with such that In particular, if the function is univalent in , the above subordination is equivalent to
Let be an analytic function with which maps the open unit disk onto a star-like region with respect to and is symmetric with respect to the real axis.
This paper contains analogues of (3) for the following classes of analytic functions.
Definition 1. Let A function is said to be in the class if it satisfies the following subordination condition: where is defined to be the same as above for .
Remark 2. (i) If we set
in Definition 1, then we have the class
which consists of functions satisfying
This class was introduced by Bansal .(ii)If we set
in Definition 1, then we have a new class which consists of functions satisfying Taking in (25), we have the class which consists of functions satisfying This class was introduced by Bansal .(iii)If we set in (25), then we have the class which consists of functions satisfying This class was introduced by Murugusundaramoorthy and Magesh . The subclass was studied by MacGregor .
We denote by the class of the analytic functions in with
We will need the following lemmas.
Lemma 3 (see ). If with , then
When or , equality holds true if and only if is or one of its rotations. If , then equality holds true if and only if is or one of its rotations. If , then the equality holds true if and only if
or one of its rotations. If , then the equality holds true if and only if is the reciprocal of one of the functions such that the equality holds true in the case when .
Although the above upper bound is sharp, in the case when , it can be further improved as follows:
Lemma 4 (see ). Let with . Then for any complex number and the result is sharp for the functions given by
2. Fekete-Szegö Problem for the Function Class
Theorem 5. Let Also let where If given by (1) belongs to the function class , then where If , then Furthermore, if , then Each of these results is sharp.
Proof. Since , we have where From (46), we have Since is univalent and , the function is analytic and has a positive real part in . Also we have Thus by (47) and (49) we get Taking into account (50), we obtain where The assertion of Theorem 5 now follows by an application of Lemma 3. On the other hand, using (51) for the values of , we have Similarly, for the values of , we get To show that the bounds asserted by Theorem 5 are sharp, we define the following functions: with by and the functions and , with by respectively. Then, clearly, the functions , , and . We also write If or , then the equality in Theorem 5 holds true if and only if is or one of its rotations. When , then the equality holds true if and only if is or one of its rotations. If , then the equality holds true if and only if is or one of its rotations. If , then the equality holds true if and only if is or one of its rotations.
By making use of Lemma 4, we immediately obtain the following Fekete-Szegö type inequality.
Theorem 6. Let Also let where If given by (1) belongs to the function class , then for any complex number The result is sharp.
If we set in Theorem 6, then we have So we get the following corollary.
Corollary 9. Let Also let If given by (1) belongs to the function class , then for any complex number The result is sharp.
Putting in Corollary 9, we obtain the following corollary.
Also putting , , , and in Corollary 9, we obtain the following corollary.
Corollary 11. Let . If given by (1) belongs to the function class , then for any complex number
3. Applications to Analytic Functions Defined by Using Fractional Calculus Operators and Convolution
The subject of fractional calculus (i.e., calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so. For the applications of the results given in the preceding sections, we first introduce the class , which is defined by means of the Hadamard product (or convolution) and a certain operator of fractional calculus, known as the Owa-Srivastava operator (see, e.g., [18, 19]).
Definition 12. The fractional integral of order is defined, for a function , by where the function is analytic in a simply connected domain of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when .
Definition 13. The fractional derivative of order is defined, for a function , by where is constrained, and the multiplicity of is removed, as in Definition 12.
Definition 14. Under the hypotheses of Definition 13, the fractional derivative of order is defined, for a function , by
Using Definitions 12, 13, and 14, fractional derivatives, and fractional integrals, Owa and Srivastava  introduced the operator defined by This operator is known as the Owa-Srivastava operator. In terms of the Owa-Srivastava operator defined by (81), we now introduce the function class in the following way: Note that the function class is a special case of the function class when Now suppose that Since we can obtain the coefficient estimates for functions in the class from the corresponding estimates for functions in the class . By applying Theorem 5 to the following Hadamard product (or convolution): we get the following theorem after an obvious change of the parameter .
Theorem 15. Let Also let where If given by (1) belongs to the function class , then where Each of these results is sharp.
Theorem 16. Let Also let where If given by (1) belongs to the function class , then where Each of these results is sharp.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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