Hankel Determinants for New Subclasses of Analytic Functions Related to a Shell Shaped Region †

: Using the operator L ac deﬁned by Carlson and Shaffer, we deﬁned a new subclass of analytic functions ML ac ( λ ; ψ ) deﬁned by a subordination relation to the shell shaped function ψ ( z ) = z + √ 1 + z 2 . We determined estimate bounds of the four coefﬁcients of the power series expansions, we gave upper bound for the Fekete–Szeg˝o functional and for the Hankel determinant of order two for f ∈ ML ac ( λ ; ψ ) .


Introduction
Let H(D) be the class of functions which are analytic in the open unit disk D := {z ∈ C : |z| < 1}, and also let A be the subset of H(D) comprising of functions f (z) = z + ∞ ∑ n=2 a n z n , z ∈ D. (1) Let f i (z) = ∞ ∑ n=0 a n,i z n (i = 1,2) which are analytic in D, then the well-known Hadamard (or convolution) product of f 1 and f 2 is given by a n,1 a n,2 z n , z ∈ D.
For two functions f , g ∈ H(D), we say that f is subordinate to g, denoted by f ≺ g, if there exists a Schwarz function ϑ ∈ H(D) with |ϑ(z)| < 1, z ∈ D, and ϑ(0) = 0, such that f (z) = g(ϑ(z)) for all z ∈ D. In particular, if g is univalent in D, then the following equivalence relationship holds true: f (z) ≺ g(z) ⇔ f (0) = g(0) and f (D) ⊂ g(D).
Let P be the well-known class of Carathéodory functions that is a set of functions φ ∈ H(D) with the power series expansion φ(z) = 1 + p 1 z + p 2 z 2 + . . . , z ∈ D, (2) and such that Re φ(z) > 0 for all z ∈ D.
It is well-known (see Duren [2]) that, if f is given by (1) and is univalent in D, then H 2,1 ( f ) ≤ 1 occurs, and this result is sharp. The determinant H q,n has also been measured by many authors. For example, the rate of growth of H q,n ( f ) as n → ∞ for functions f ∈ A with bounded boundary was determined. In [3], it has been shown, a fraction of two bounded analytic functions with its Laurent series around the origin having integral coefficients, is rational. The Hankel determinant of meromorphic functions, (see [4]), and various properties of these determinants can be found in [5]). In 1966, the Hankel determinant of areally mean p-valent functions, univalent functions, and starlike functions were extensively studied by Pommerenke [6]. Lately, several authors have investigated H 2,1 of innumerable subclasses of univalent and multivalent functions and, for more details on Hankel determinants, one may refer [1,[6][7][8][9][10][11][12][13][14]. For T ⊂ A, a problem of finding a sharp (best possible) upper bound of a 3 − µ a 2 2 for the subclass T is generally called Fekete-Szegő problem for the subclass T , where µ is a real or a complex number. There are some well known subclasses of univalent functions, such that the starlike functions, convex functions, and close-to-convex functions, for which the problem of finding sharp upper bounds for the functional a 3 − µ a 2 2 was completely solved (see [15][16][17][18]). For the family of analytic functions R := { f ∈ A : Re f (z) > 0, z ∈ D}, Janteng et al. [19] have found the sharp upper bound to |H 2,2 ( f )|. For initial work on the class R, one may refer to the article of MacGregor [20].
The concept of shell-like domains gained importance in the recent times and it was introduced by Sokół and Paprocki [21]. Recently, for ψ(z) = z + √ 1 + z 2 , Raina and Sokół [22] have widely studied and found some coefficient inequalities for f ∈ S (ψ) if it satisfies the subordination condition that z f (z)/ f (z) ≺ ψ(z), and these results are further improved by Sokół and Thomas [23], the Fekete-Szegő inequality for f ∈ C(ψ) were obtained and, in view of the Alexander result between the class S * (ψ) and C(ψ), the Fekete-Szegő inequality for functions in S * (ψ) were also obtained. The function ψ(z) := z + √ 1 + z 2 maps the unit disc D onto a shell shaped region on the right half plane, and it is analytic and univalent on D. The range ψ(D) is symmetric respecting the real axis and ψ(z) is a function with positive real part in D, with ψ(0) = ψ (0) = 1. Moreover, it is a starlike domain with respect to the point ψ(0) = 1 (see [24]), such as Figure 1 shows. Definition 1. [22] Let f ∈ A be normalized by f (0) = f (0) − 1 = 0 in the unit disc D. We denote by S * (ψ) the class of analytic functions and satisfying the condition that where the branch of the square root is chosen to be the principal one that is ψ(0) = 1. where is the incomplete beta function, and (t) n denotes the Pochhammer symbol (or the shifted factorial) defined in terms of the Gamma function by (t) n := For f ∈ A is given by (1) and by (3), one can get the Carlson and Shaffer operator and Remark 1. Next, we will emphasize a few special cases of the operator L(a, c), as follows: (v) L 2 2−δ f (z) =: Ω δ z f (z), 0 ≤ δ < 1 is the well-known Owa-Srivastava fractional differential operator of f [27].
Motivated by the articles of Raina and Sokół [22], Sokół and Thomas [23], Dziok and Raina [28], and Raina et al. [29], using the concept of subordination and the linear operator L a c , we define a new subclass of A denoted by ML a c (λ; ψ). For this subclass, we obtained coefficient inequalities, Fekete-Szegő inequality, and upper bound for the Hankel determinant |H 2 (2)|.
We define a new subclass ML a c (λ; ψ) of A as below: , with a ∈ C and c ∈ C \ Z − 0 , denote the subclass of functions f ∈ A that satisfies the subordination condition where the branch of the square root is chosen to be the principal one that is ψ(0) = 1.
In the following remark, we prove that ML a c (λ; ψ) is non-empty.
Considering the circular transformation with 0 ≤ λ ≤ 1, and assuming that 0 ≤ A ≤ 1/2, we obtain that Ψ λ maps the unit disc D onto the open disc that is symmetric respecting the real axes connecting the points Ψ λ (−1) and Ψ λ (1). If α = c 4a , then A = 1/4, and for λ = 1, λ = 0, and λ = 1/2, using the MAPLE TM software we get the next images of D by Ψ λ like in the Figure 2: , for λ = 1, λ = 0, and λ = 1/2. It follows that there exist values of the Now, by suitably specializing the parameter λ, we define the new subclasses of ML a c (λ; ψ) as remarked below: SL a c (ψ) denote the subclass of A, the members of which are given by (1) and satisfy the subordination condition (ii) For λ = 1, let ML a c (1, ψ) =: RL a c (ψ) denote the subclass of A, members of which are of the form (1) and if it satisfy the condition (L a c f (z)) ≺ z + 1 + z 2 .
(iii) For the special case for a = c, let ML(λ; ψ) := ML c c (λ; ψ), members of which are given by (1) and satisfy the subordination In the all of the above subordinations, and throughout the whole paper, the branch of the square root is chosen at the principal one, which is ψ(0) = 1, and a ∈ C, c ∈ C \ Z − 0 . Using the techniques of Libera and Zlotkiewicz [11] and Koepf [17], combined with the help of MAPLE TM software, we find Fekete-Szegő inequality and Hankel determinant for the function of the class ML a c (λ; ψ).

Preliminaries
To establish our main results, we recall the followings lemmas. The first lemma is the well-known Carathéodory's lemma (see also [30] Corollary 2.3): Lemma 1. [31] If p ∈ P and given by (2), then |p k | ≤ 2, for all k ≥ 1, and the result is best possible for The next lemma gives us a majorant for the coefficients of the functions of the class P, and more details may be found in [32] (Lemma 1): Lemma 2. [33] Let φ ∈ P be given by (2). Then, The result is sharp for the functions given by When υ < 0 or υ > 1, the equality holds if and only if φ is 1 + z 1 − z or one of its rotations. If 0 < v < 1, then equality holds if and only if φ is 1 + z 2 1 − z 2 or one of its rotations. If υ = 0, the equality holds if and only if or one of its rotations. If υ = 1, the equality holds if and only if φ is the reciprocal of one of the functions such that the equality holds in the case of υ = 0. Although the above upper bound is sharp, when 0 < υ < 1, it can be improved as follows: and We also need the following result: Lemma 4. [33] Let φ ∈ P given by (2). Then, and for some complex numbers x, z satisfying |x| ≤ 1 and |z| ≤ 1.

Coefficient Bounds and Fekete-Szegő Inequality
In our first result, we will determine coefficient bounds for f ∈ ML a c (λ; ψ), and this tends to solve the Fekete-Szegő problem for the subclass ML a c (λ; ψ).
Theorem 2. If f ∈ ML a c (λ; ψ) is of the form (1), then, for any µ ∈ C, we have Proof. If f ∈ ML a c (λ; ψ) is of the form (1), from (17) and (18), we get Taking the modules for the both sides of the above relation, with the aid of the inequality (7) of Lemma 2, we easily get the required estimate.
For a = c, the above theorem reduces to the following special case: Corollary 1. If f ∈ ML(λ; ψ) is given by (1) then, for any µ ∈ C, we have Remark 4. If f ∈ ML(λ; ψ) is given by (1) then, for the special case µ = 1, we get If we take µ ∈ R in Theorem 2, we get the next special case: If the function f ∈ ML a c (λ; ψ) is given by (1), If δ 3 ≤ µ < δ 2 , then where These results are sharp.
From the assumptions, using the second above equality, it follows that ν ∈ R. We have ν ≥ 1 is equivalent to µ ≥ δ 2 , and ν ≤ 0 is equivalent to µ ≤ δ 1 . Then, taking the modules for both sides of the above equality, with the aid of the inequality (8) of Lemma 3, we obtain the first estimates of Theorem 3. where and 16|A| = G(2, t), 4|C| = G(0, 1).

Remark 5.
By suitably specializing the parameter λ, one can deduce the above results for the subclasses of SL a c (λ; ψ), and RL a c (λ; ψ), which are defined, respectively, in Remark 3 (i) and (ii). Furthermore, by taking a = c, we can easily state the result for the function class ML(λ, ψ) given in Remark 3 (iii). The details involved may be left as an exercise for the interested reader.