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 ourmain 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.


Introduction and Definitions
Let A denote the class of functions of the form which are analytic in the unit disk Also let S denote the subclass of A consisting of univalent functions in U.
Fekete and Szegö [1] proved a noticeable result that the estimate holds for  ∈ S. 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  ∈ A represents various geometric quantities as well as in the sense that this behaves well with respect to the rotation; namely,   ( −  (  )) =  2   () ( ∈ R) .
In fact, rather than the simplest case when we have several important ones.For example, represents   (0)/6, where   denotes the Schwarzian derivative Moreover, the first two nontrivial coefficients of the th root transform ( (  )) 2 International Journal of Analysis 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 S. This is called Fekete-Szegö problem.Actually, many authors have considered this problem for typical classes of univalent functions (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12]).
For two functions  and , analytic in U, we say that the function () is subordinate to () in U, and we write if there exists a Schwarz function (), analytic in U, with such that In particular, if the function  is univalent in U, the above subordination is equivalent to Let () be an analytic function with which maps the open unit disk U onto a star-like region with respect to 1 and is symmetric with respect to the real axis.This paper contains analogues of (3) for the following classes of analytic functions.
A function  ∈ A is said to be in the class R  (, ; ) if it satisfies the following subordination condition: where () is defined to be the same as above for  ∈ U.

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 [13].
(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 [14].
We denote by P the class of the analytic functions in U with We will need the following lemmas.
Lemma 3 (see [12]). (33) or one of its rotations.If ] = 1, 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 ] = 0.
Although the above upper bound is sharp, in the case when 0 < ] < 1, it can be further improved as follows: Lemma 4 (see [17]).Let  ∈ P with () = 1 +  1  +  2  2 + ⋅ ⋅ ⋅ .Then for any complex number ] and the result is sharp for the functions given by

Fekete-Szegö Problem for the Function Class R 𝑏 (𝛼, 𝛾; 𝜑)
By making use of Lemma 3, we first prove the Fekete-Szegö type inequalities asserted by Theorem 5 below.
Also let where If () given by (1) belongs to the function class R  (, ; ), where Each of these results is sharp.
Proof.Since  ∈ R  (, ; ), we have where International Journal of Analysis From (46), we have Since () is univalent and ℎ() ≺ (), the function is analytic and has a positive real part in U. 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  1 ≤  ≤ To show that the bounds asserted by Theorem 5 are sharp, we define the following functions: with by International Journal of Analysis 5 and the functions   () and   () (0 ≤  ≤ 1), with By making use of Lemma 4, we immediately obtain the following Fekete-Szegö type inequality.
Also let where If () given by (1) belongs to the function class R  (, ; ), then for any complex number The result is sharp.Taking  = 2 + 1 in Theorem 6, we have the following corollary.
Corollary 8 (see [13]).Let Also let where If () given by (1) belongs to the function class R   (), 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 R  (, ; , ), then for any complex number        3 −  ( The result is sharp.
Corollary 10 (see [13]).Let Also let If () given by (1) belongs to the function class R   (, ), then for any complex number        3 −  ( The result is sharp.

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 R   (, ; ), 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 ( − ) −1 is removed by requiring log( − ) to be real when  −  > 0.
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 [20]

Remark 7 .
The coefficient bounds for | 2 | and | 3 | are special cases of those asserted by Theorem 5.