Second Hankel Determinant for a Class of Analytic Functions Defined by Fractional Derivative

Recommended by Vladimir Mityushev By making use of the fractional differential operator Ω λ z due to Owa and Srivastava, a class of analytic functions R λ α, ρ 0 ≤ ρ ≤ 1, 0 ≤ λ < 1, |α| < π/2 is introduced. The sharp bound for the nonlinear functional |a 2 a 4 − a 2 3 | is found. Several basic properties such as inclusion, subordination, integral transform, Hadamard product are also studied.


Introduction
Let A denote the class of functions analytic in the open unit disc and let A 0 be the class of functions f in A given by the normalized power series Also let S, S * β , CV β , and K denote, respectively, the subclasses of A 0 consisting of functions which are univalent, starlike of order β, convex of order β cf. 1 , and close-to-convex cf. 2 in U.In particular, S * 0 S * and CV 0 CV are the familiar classes of starlike and convex functions in U cf. 2 .
Given f and g in A, the function f is said to be subordinate to g in U if there exits a function ω ∈ A satisfying the conditions of the Schwarz Lemma such that f z g ω z , z ∈ U .We denote the subordination by f z ≺ g z z ∈ U or f ≺ g in U.

International Journal of Mathematics and Mathematical Sciences
It is well known 2 that if g is univalent in U, then f ≺ g in U is equivalent to f 0 g 0 and f U ⊂ g U .
For the functions f and g given by the power series their Hadamard product or convolution , denoted by f * g, is defined by By making use of the Hadamard product, Carlson-Shaffer 3 defined the linear operator L a, c : where and λ k is the Pochhammer symbol or shifted factorial defined in terms of the gamma function by It can be readily verified that L a, a a / ∈ Z − 0 is the identity operator; the operators L a, b , L c, d commute, where b, d/ ∈Z − 0 , that is, and the transitive property, that is, holds.Each of the following definitions will also be required in our present investigation.
Definition 1.1 cf. 4, 5 , see also 6 .Let the function f be analytic in a simply connected region of the z-plane containing the origin.The fractional derivative of f of order λ is defined by where the multiplicity of z − ζ λ is removed by requiring log z − ζ to be real when z − ζ > 0.
A. K. Mishra and P. Gochhayat 3 Using Definition 1.1 and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava 5 introduced the fractional differintegral operator Ω λ z : A 0 → A 0 defined by and Definition 1.2 cf.7 .For the function f given by 1.2 and q ∈ N : {1, 2, 3, . ..}, the qth Hankel determinant of f is defined by We now introduce the following class of functions.

1.16
Let P be the family of functions p ∈ A satisfying p 0 1 and R p z > 0 z ∈ U .It follows from 1.15 that where α is real, |α| < π/2, and p z ∈ P. We note that and the class R λ ρ has been studied in 8 .
It is well known cf. 2 that for f ∈ S and given by 1.2 , the sharp inequality |a 3 − a 2 2 | ≤ 1 holds.This corresponds to the Hankel determinant with q 2 and n 1.For a given family F of functions in A 0 , the more general problem of finding sharp estimates for |μa 2  2 − a 3 | μ ∈ R or μ ∈ C is popularly known as the Fekete-Szeg ö problem for F. The Fekete-Szeg ö problem for the families S, S * , CV, K has been completely solved by many authors including 9-12 .
In the present paper, we consider the Hankel determinant for q 2 and n 2 and we find the sharp bound for the functional |a 2 a 4 − a 2  3 | f ∈ R λ α, ρ .We also obtain some basic properties of the class R λ α, ρ .Our investigation includes a recent result of Janteng et al. 13 .We also generalize some results of Ling and Ding 8 .

Preliminaries
To establish our results, we recall the following.Lemma 2.1 see 2 .Let the function p ∈ P and be given by the series Then, the sharp estimate Lemma 2.2 cf.14, page 254 , see also 15 .Let the function p ∈ P be given by the power series 2.1 .Then, for some x, |x| ≤ 1, and for some z, |z| ≤ 1.

Main results
We prove the following.

3.4
Therefore, 3.4 yields Since the functions p z and p e iθ z , θ ∈ R are members of the class P simultaneously, we assume without loss of generality that c 1 > 0. For convenience of notation, we take c 1 c c ∈ 0, 2 .Using 2.3 along with 2.4 , we get

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An application of triangle inequality and replacement of |x| by μ give 4  12 : F c, μ say ,
The choice of α 0 yields what follows.
Corollary 3.2.Let the function f given by 1.2 be a member of the class R λ ρ .Then,

3.14
Equality holds for the function

3.20
Hence, f z ∈ R μ α, ρ , and the proof of Theorem 3.4 is complete.
Proof.Since the Hadamard product is associative and commutative, we have Therefore, Now applying Lemma 2.5, we get R e iα Ω λ z f * g z z > ρ cos α.
Then, the function I f defined by the integral transform Proof.The Integral transform I f can be written in terms of Carlson-Shaffer operator as