Upper Bound of Second Hankel determinant for generalized Sakaguchi type spiral-like functions

abstract: In this paper, the authors introduce a generalized Sakaguchi type spiral-like function class S(λ, β, s, t) and obtain sharp upper bound to the second Hankel determinant |H2(1)| for the function f in the above class. Relevances of the main result are also briefly indicated.


Let
U := {z : z ∈ C and |z| < 1} be the unit disk in the complex z-plane.Let A be the class of functions f of the form: which are analytic in U and satisfy the following normalization condition: Further, by S we shall denote the class of all functions f in A which are univalent in U.
A function f ∈ A is said to be λ-spiral starlike function of order β, denoted by SP (λ, β) if and only if the following inequality holds true: For β = 0, the class SP (λ, 0) reduces to S p (λ) which has been studied by Spacek [22].Observed that for λ = 0, S p (0) = S * , the familiar class of starlike functions L. Jena and T. Panigrahi in U.
Recently, Frasin [9] introduced and studied a generalized Sakaguchi type function class S(α, s, t) as follows.A function f (z) ∈ A is said to be in the class S(α, s, t) if it satisfies for some α (0 ≤ α < 1), s, t ∈ C, s = t and for all z ∈ U.
The qth Hankel determinant for q ≥ 1 and n ≥ 1 is stated by Noonan and Thomas [19] as A good amount of literature is available about the importance of Hankel determinant.It is useful in the study of power series with integral coefficients (see [3]), meromorphic functions (see [25]) and also singularities (see [5]).Noor (see [20]) determined the rate of growth of H q (n) as n −→ ∞ for the functions in S with a bounded boundary while Ehrenborg (see [7]) studied the Hankel determinant of exponential polynomials.For q = 2, n = 1, a 1 = 1 and q = 2, n = 2, the Hankel determinant simplifies respectively to It is well-known [6] that for f ∈ S and given by (1.1), the sharp inequality |a 3 − a 2 2 | holds.Fekete-Szegö (see [8]) then further generalized the estimate |a 3 − µa 2  2 | with µ real and f ∈ S. For a given family F of the functions in A, the sharp upper bound for the nonlinear functional | is popularly known as the second Hankel determinant.Second Hankel determinant for various subclasses of analytic functions were obtained by various authors.For details, (see [1,2,11,12,13,14,15,18]).Following the techniques devised by Libera and Zlotkiewicz (see [16,17]), in the present paper, the authors determine a sharp upper bound of the second Hankel determinant |H 2 (1)| for the function f belonging to the class S(λ, β, s, t).

Preliminaries
Let P denote the class of functions normalized by which are regular in U and satisfying ℜ {p(z)} > 0 for every z ∈ U.Here p(z) is called caratheòdory function (see [6]).
To investigate the main result, we need the following lemmas.

Main Result
Theorem 3.1.Let the function f given by (1.1) be in the class S(λ, β, s, t).Then The estimate in (3.1) is sharp.
Proof: Let the function f (z) given by (1.1) be in the class S(λ, β, s, t).Then from the Definition 1.1, there exists an analytic function p ∈ P in the unit disk U with p(0) = 1 and ℜ(p(z)) > 0 such that Replacing f (tz), f (sz), f ′ (sz) and p(z) by their equivalent series in (3.2), after simplification, we obtain Equating the coefficients of z, z 2 and z 3 in (3.3), we get Substituting the values of a 2 , a 3 and a 4 from (3.4) in the second Hankel functional |a 2 a 4 − a 2 3 | for the function f ∈ S(λ, β, s, t), we obtain Making use of the result |xa + yb| ≤ |x||a| + |y||b|, where x, y, a and b are real numbers and |e −inλ | = 1, where n is a real number, after simplification, we obtain where Substituting the values of c 2 and c 3 from Lemma 2.2 in the right hand side of (3.6), we have Making use of well-known fact that |z| < 1 in (3.8), upon simplification gives Using the values of d 1 , d 2 , d 3 and d 4 given in (3.7), after simplification, we obtain ) and L. Jena and T. Panigrahi , where a, b ≥ 0 in the right hand side of (3.13), upon simplification, we obtain From (3.12) and (3.14), it follows that Upper Bound of Second Hankel determinant Substituting the values from the relation (3.10), (3.11) and (3.15) in the right hand side of (3.9), we get Choosing c 1 = c ∈ [0, 2], applying triangle inequality and replacing |x| by µ on the right hand sides of (3.16), we obtain where tiating on both sides of (3.18) partially with respect to µ, we get For 0 < µ < 1, for fixed c, 0 < c < 2, we observe from (3.19) that ∂H ∂µ > 0.
Therefore,  | for the functions f ∈ A belonging to the class S(λ, β, s, t).We conclude this paper by remarking that the above theorem include several previously established results for particular values of the parameters λ, β, s, t.For example, taking s = 1, t = 0 and β = 0 in Theorem 3.1 we get the result due to Krishna and Reddy (see [15]).Further, by letting s = 1, t = 0, β = 0 and λ = 0 in Theorem 3.1, we obtain the result |a 2 a 4 − a 2  3 | ≤ 1.This result is sharp and coincides with that of Janteng et al. (see [13]).Now we are working on to find the sharp upper bound for the above function class using third Hankel determinant.