On certain subclasses of analytic functions associated with the Carlson – Shaffer operator

The object of the present paper is to solve Fekete–Szegö problem and determine the sharp upper bound to the second Hankel determinant for a certain class R(a, c, A,B) of analytic functions in the unit disk. We also investigate several majorization properties for functions belonging to a subclass  ̃ R(a, c, A,B) of R(a, c, A,B) and related function classes. Relevant connections of the main results obtained here with those given by earlier workers on the subject are pointed out.


Introduction and preliminaries. Let
A function f ∈ A is said to be starlike function of order α and convex function of order α, respectively, if and only if Re{zf (z)/f (z)} > α and Re{1 + (zf (z)/f (z))} > α for 0 ≤ α < 1 and for all z ∈ U.By usual notations, we denote these classes of functions by S (α) and C (α) (0 ≤ α < 1), respectively.We write S (0) = S and C (0) = C , the familiar subclasses of starlike functions and convex functions in U. Furthermore, a function f ∈ A is said to be in the class R(α), if it satisfies the inequality: Note that R(α) is a subclass of close-to-convex functions of order α (0 ≤ α < 1) in U. We write R(0) = R, the familiar class functions whose derivatives have a positive real part in U.
Let P denote the class of analytic functions of the form such that Re{φ(z)} > 0 in U.
For functions f and g, analytic in the unit disk U, we say the f is said to be subordinate to g, written as f ≺ g or f (z) ≺ g(z) (z ∈ U), if there exists an analytic function ω in U with ω(0) = 0, |ω(z)| ≤ |z| (z ∈ U) and f (z) = g(ω(z)) for all z ∈ U.In particular, if g is univalent in U, then we have the following equivalence (see [20]): Following MacGregor [19], we say that f is majorized by g in U and write (1.4) f (z) g(z) (z ∈ U), if there exists a function ψ, analytic in U such that |ψ(z)| ≤ 1 and (1.5) f (z) = ψ(z)g(z) (z ∈ U).
For the functions f and g given by the power series their Hadamard product (or convolution), denoted by f g is defined as We note that f g is analytic in U.
By making use of the Hadamard product, Carlson-Shaffer [3] defined the linear operator L (a, c) : where If f ∈ A is given by (1.1), then it follows from (1.7) that We note that for , the well-known Owa-Srivastava fractional differential operator [25].We also observe that With the aid of the linear operator L (a, c), we introduce a subclass of A as follows: Definition 1.1.For the fixed parameters A, B (−1 ≤ B < A ≤ 1), a > 0 and c > 0, a function f ∈ A is said to be in the class R λ (a, c, A, B), if it satisfies the following subordination relation: Using the identity (1.9) in (1.10), it follows that By suitably specializing the parameters a, c, λ, A and B, we obtain the following subclasses of A .
We note that R(0, α, β) = R(α, β) (0 ≤ α < 1, 0 < β ≤ 1), the class studied by Juneja and Mogra [10], which in turn give the class considered in [2] for Next, we define a subclass of T as follows: Definition 1.2.For the fixed parameters A, B (−1 ≤ B < A ≤ 1, 1 ≤ B < 0), a > 0 and c > 0, a function f ∈ T is said to be in the class R λ (a, c, A, B), if it satisfies the following subordination relation: In view of (1.9), it is easily seen that the subordination relation (1.11) is equivalent to If we set h(z) = L (a, c)f (z)/z, then the above expression further reduces to We write Noonan and Thomas [23] defined the q-th Hankel determinant of the function f ∈ A given by (1.1) as This determinant has been studied by several authors with the subject of inquiry ranging from the rate of growth of H q (n) (as n → ∞) [24] to the determination of precise bounds with specific values of n and q for certain subclasses of analytic functions in the unit disc U.
For n = 1, q = 2 = 1 and n = q = 2, the Hankel determinant simplifies to We refer to H 2 (2) as the second Hankel determinant.It is known [4] that if f given by (1.1) is analytic and univalent in U, then the sharp inequality For a family F of functions in A of the form (1.1), the more general problem of finding the sharp upper bounds for the functionals ) is popularly known as Fekete-Szegö problem for the class F .The Fekete-Szegö problem for the known classes of univalent functions, that is, starlike functions, convex functions and closeto-convex functions has been completely settled [5,11,12,13].Recently, Janteng et al. [8,9] have obtained the sharp upper bounds to the second Hankel determinant H 2 (2) for the family R.For initial work on the class R, one may refer to the paper by MacGregor [17].
A majorization properties for the class of starlike functions of complex order γ and the class of convex functions of complex order γ (γ ∈ C * ) has been investigated by Altintaş et al. [1] and MacGregor [19] has also studied the same problem for the classes S and C , respectively.Recently, Goyal and Goswami [6], and Goyal et al. [7] generalized these results for different function classes.
In the present article, by following the techniques devised by Libera and Złotkiewicz [14,15], we solve the Fekete-Szegö problem and also determine the sharp upper bound to the second Hankel determinant for the class R λ (a, c, A, B).We also investigate several majorization properties for certain subclasses of analytic functions in the unit disk U. Relevant connections of the results presented here with those obtained in earlier works are also mentioned.
To establish our main results, we shall need the following lemmas.
Lemma 1.1 ([4]).Let the function φ, given by (1.3) be a member of the class P. Then and the estimate is sharp for the function

Lemma 1.2 ([16]
).If the function φ, given by (1.3) belongs to the class P, then for any γ ∈ C and the result is sharp for the functions given by Lemma 1.3 ([15], see also [14]).If the function φ, given by (1.3) belongs to the class P, then x and for some complex numbers x, z satisfying |x| ≤ 1 and |z| ≤ 1.

Hankel determinant for the class R λ (a, c, A, B).
Unless otherwise mentioned, we assume throughout the sequel that Now, we determine the sharp upper bound for the functional |a 3 − γa 2 2 | (γ ∈ R) for functions belonging to the class R λ (a, c, A, B).Theorem 2.1.If γ ∈ R and the function f , given by (1.1) belongs to the class R λ (a, c, A, B), then where The estimate in (2.1) is sharp.
Proof.From (1.10), we have , where the function φ, given by (1.3) belongs to the class P. Writing the series expansion of 2), and comparing the like powers of z in the resulting equation, we deduce that and Using (2.3) and (2.4), we obtain and with the aid of Lemma 1.2, the above expression yields (2.6) If γ < ρ 1 , then which in view of (2.6) implies the first case of the estimate in (2.1).In the case ρ 1 ≤ γ ≤ ρ 2 , we obtain Thus, from (2.6), we get the second case of the estimate in (2.1).Finally, for γ > ρ 2 , we deduce that which again with the aid of (2.6) gives the third case of the estimate in (2.1).
It is easily seen that the estimate for the first and third cases in (2.1) are sharp for the function f , defined in U by 1+Bz , λ > 0. The estimate for the second case in (2.1) is sharp for the function f , defined in U by 1+Bz 2 , λ > 0, where the function 3 F 2 is defined by (1.6). where The estimate is sharp for the functions f , defined in U by Letting a = 2, c = 1, A = 1 − 2α (0 ≤ α < 1) and B = −1 in Theorem 1, we get Corollary 2.2.If γ ∈ R and the function f , given by (1.1) belongs to the class R λ (α), then The estimate is sharp for the functions f , defined in U by , λ > 0.
In the following theorem, we find the sharp upper bound to the second Hankel determinant for the class R λ (a, c, A, B).
If the function f , given by (1.1) belongs to the class R λ (a, c, A, B), then The estimate in (2.9) is sharp.
Proof.Assuming that f , given by (1.1) belongs to the class R λ (a, c, A, B) and using (2.3), (2.4) and (2.5), we deduce that (2.10) where Since the functions φ(z) and φ(e iθ z) (θ ∈ R), defined by (1.3) are in the class P simultaneously, we assume without loss of generality that p 1 > 0.
For convenience, we write p 1 = p (0 ≤ p ≤ 2).Now, by using Lemma 1.3 in (2.10), we get (2.11) for some complex numbers x (|x| ≤ 1) and z (|z| ≤ 1).Applying the triangle inequality in (2.11) and upon replacing |x| by y in the resulting expression, we get (2.12) we have where A routine calculation yields Thus, F (p) = 0 implies that either p = 0 or which is not true.We, further observe that occurs at p = 0. Thus, the upper bound of (2.12) corresponds to p = 0 and y = 1, from which we get the estimate in (2.9).It is easily seen that the estimate (2.9) is sharp for the function f , given by (2.7) and thus the proof of Theorem 2.2 is completed.
Corollary 2.3.If a ≥ c > 0, ac − 2a + 5c + 2 ≥ 0 and the function f , given by (1.1) belongs to the class R a,c (α), then The estimate is sharp for the function f , defined by and the estimate is sharp for the function f , defined by Taking a = 2, c = 1, A = 1 − 2α and B = −1 in Theorem 2.2, we get the following result, which in turn yields the corresponding work of Mishra and Kund [21] for λ = 0, and the work of Janteng et al. [8] for λ = α = 0. Corollary 2.5.If the function f , given by (1.1) belongs to the class R λ (α), then and the estimate is sharp for the function f , defined in U by , λ > 0.

Majorization properties.
We prove the following lemmas, which will be used in our investigation of majorization properties for the class R λ (a, c, A, B).Proof.It follows from (1.11) that which upon substituting the series expansion of L (a, c)f (z)/z and Letting z → 1 − through real values in the above expression, we find that Since a ≥ c > 0, b n+1 ≥ 0 and Re(λ) ≥ 0, the above inequality implies that This completes the proof of Lemma 3.1.
Similarly, we have and the proof of Lemma 3.2 is completed.

Now, we prove
Theorem 3.1.Let the function g be in the class T .If a ≥ c > 0, the function h ∈ T satisfies and L (a, c)g L (a, c)h in U, then where r(λ, a, c, A, B) is the root of the cubic equation Proof.From (3.3), by using Lemma 3.2, we get for |z| = r < 1 Since L (a, c)g(z) L (a, c)h(z) in U, we have by (1.5) where r(λ, α) is the root of the cubic equation In the special case λ = 0, Corollary 3.1 simplifies to the following result.
where r(α) is the root of the cubic equation With the aid of the following inclusion relation [27,Theorem 7] we get the following result from Corollary 3.2.
where r(a, A, B) is the root of the cubic equation Proof.From the definition of subordination, it follows from (3.10) that where ω is analytic in U with ω(0) = 0 and |ω(z)| < 1 for all z ∈ U. Now, making use of the the identity (1.9) for the function g in (3.13), we deduce that Since L (a, c)f is majorized by L (a, c)g in U, we have where ψ is analytic in U and satisfies |ψ(z)| ≤ 1 in U. Differentiating the above expression with respect to z, using the identity (1.9) for both the functions f and g in the resulting equation, we obtain A be the class of functions f of the form (1.1) f (z) = z + ∞ n=2 a n z n which are analytic in the open unit disk U = {z ∈ C : |z| < 1}.Also, let T denote the subclass of A consisting of functions of the form (1.2) g(z) = z − ∞ n=2 b n z n (b n ≥ 0).

Corollary 3 . 2 .
Let the function g be in the class T .If the function h ∈ S (α) ∩ T and g h in U, then