Properties of Sakaguchi Kind Functions Associated with Bessel Function

The aim of the paper is to obtain the First Hankel Determinant and the Second Hankel determinant. We shall make use of few lemmas which are based on Caratheodory’s class of analytic functions. We establish a new Sakaguchi class of univalent function, further we estimate the sharp bound for initial coefﬁcients a 2 and a 3 using the Bessel function expan-sion. We have discussed about the coefﬁcient a 4 as well for the Second Hankel Determinant. The results are obtained for Sakaguchi kind. Our results travel along exploring the stages of Hankel Determinants. Various types of technologies like wire, optical or other electromagnetic systems are used for the transmission of data in one device to another. Filters play an important role in the process that can remove disorted signals. By using different parameter values for the function belongs to Sakaguchi kind of functions the Low pass ﬁlter and High pass ﬁlter can be designed and that can be done by the coefﬁcient estimates.


Introduction
Let A denote the class of analytic functions of the form: a κ ξ κ , ξ ∈ U = {ξ ∈ C : |ξ| < 1} (1) and S be the subclass of A which are univalent in U. If κ ∈ A is represented as: then, the Hadamard product of f and κ is constructed as follows: If the function F is univalent in U, then the following holds (see [1] and [2]): The infinite series is given by: where Γ denotes the Gamma function. [3] the normalized Bessel function of the first kind g ν : U → C defined by (see also [4] - [6]) For the strict inquality 0 < q < 1, g ν is defined by; using (4), the next two products are obtained: (i) The q-shifted fractional for a positive integer k is given by; ii) The q-generalised Pochhammer symbol for a positive number r is defined by; For the conditions ν > 0, λ > −1, and 0 < q < 1, we can define the function I λ ν,q : U → C by; The Hankel determinants H ξ (1) = a 3 − a 2 2 and H ξ (2) = a 2 a 4 − a 2 3 are discussed.

Remark 1.
A simple reckoning shows that: By making using of idealogy of q-derivative, we instigate the linear operator N λ ν,q : A → A defined by: with the conditions (ν > 0, λ > −1, 0 < q < 1), where Proof. (i)To prove we consider RHS and arrive at the LHS Applying limit, we get

Properties of Sakaguchi Kind Functions Associated with Bessel Function
Now, we bring in the class of functions M λ ν,q (ρ, t, Ψ) as follows is analytic in U and satisfies: In this paper, we obtain the Fekete-Szego inequalities and Second Hankel Determinant for the function of the class M λ ν,q (ρ, t, Ψ).
Proof. When f (ξ) ∈ M λ ν,q (ρ, t, Ψ),then we observe that there exists a Schwarz function W , which is analytic in U, with W (0) = 0 and |W (ξ)| < 1, z ∈ D, such that: Since W is a Schwarz function, it follows that the function p 1 defined by; belongs to P. By defining the function p by: In view of (9) and (10), we have: By making use of (10), we easily obtain: and from (12), we obtain On the other hand, from (11), according to (5), it follows that and combning d 1 , d 2 and d 3 values, we have Therefore, and by making use of the Lemma 1, the upcomming results are discussed.

Conclusion
By setting the values of ρ, t and u n we bring out the interesting coefficient inequality and subordination techniques, for the subclasses of M λ ν,q (ρ, t, Ψ). The investigation of initial coefficient bounds Fekete-Szego inequality and Second Hankel determinant for various subclasses can be a scope of future research for disorted signals.