Certain Subclass of m -Valent Functions Associated with a New Extended Ruscheweyh Operator Related to Conic Domains

The main object of the present paper is to introduce certain subclass of m -valent functions associated with a new extended Ruscheweyh linear operator in the open unit disk. Also, we investigate a number of geometric properties including coe ﬃ cient estimates and the Fekete – Szegö type inequalities for this subclass. Several known consequences of the main results are also pointed out. ,


Introduction
Let AðmÞ denote the class of functions of the next form: which are analytic and m-valent in the open unit disc D = f ξ ∈ ℂ : jξj < 1g, and let Að1Þ = A. Also, let f , g be analytic in D, and the function f ðξÞ is said to be subordinate to gðξÞ if there exists a function ωðξÞ analytic in D with ωð0Þ = 0 and |ωðξÞ | <1, ξ ∈ D, such that f ðξÞ = gðωðξÞÞ. In such a case, we write f ðξÞ ≺ gðξÞ. If g is univalent function, then f ðξÞ ≺ gðξÞ if and only if f ð0Þ = gð0Þ and f ðDÞ ⊂ gðDÞ (see [1,2] and [3]).
For functions f ðξÞ given by (1) and gðξÞ is defined by and the Hadamard product or convolution of f ðξÞ and gðξÞ is defined by For v ∈ ℂ, k ∈ ℝ, and n ∈ ℕ, the Pochhammer k-symbol ðvÞ n,k is given by (see [4]) We define the function ϕ m ðδ, k ; ξÞ by Corresponding to the function ϕ m ðδ, k ; ξÞ, we consider a linear operator D δ+mk−k : AðmÞ ⟶ AðmÞðδ>−mk, k > 0Þ which is defined by means of the following Hadamard product (or convolution): It is easily verified from (6) that We note that (1) For k = 1, the operator D δ+mk−k f ðξÞ reduced to the differential operator D δ+m−1 f ðξÞ introduced by Goel and Sohi [5] (see also [6,7] and [8]) (2) For m = 1, we obtain the k-Ruscheweyh derivative (3) For k = m = 1, the operator D δ+mk−k f ðξÞ reduced tothe well-familiar Ruscheweyh operator D δ ([10]) By using the linear operator D δ+mk−k f ðξÞ, we define the subclass β − ST m ðδ, k, bÞ of AðmÞ as follows: Geometrically, a function f ∈ β − ST m ðδ, k, bÞ if and only if takes all the values in the conic domain Ω β = ψ β ðDÞ, where or equivalently, The boundary ∂Ω β of the above set becomes the imaginary axis when β = 0, a hyperbola when 0 < β < 1, a parabola when β = 1, and an ellipse when 1 < β < ∞. The functions ψ β ðξÞ are defined by where t is chosen such that k = cosh ðπR ′ ðtÞ/4RðtÞÞ, and RðtÞ is the Legendre's complete elliptic integral of the first kind and R ′ ðtÞ the complementary integral of RðtÞ (see [11,12] and [13]).
Lemma 3 [22]. Let hðξÞ = 1 + ∑ ∞ n=1 c n ξ n ∈ P , i.e., let h be analytic in D and satisfy RfhðξÞg > 0 for ξ in D; then, the following sharp estimate holds The result is sharp for the functions given by 3 Journal of Function Spaces Lemma 4 [22]. If hðξÞ = 1 + ∑ ∞ n=1 c n ξ n ∈ P , then or one of its rotations. If ν = 1, the equality holds if and only if g is the reciprocal of one of the functions such that equality holds in the case of ν = 0.
Also, the above upper bound is sharp, and it can be improved as follows when 0 < ν < 1: In this paper, we investigate a coefficient estimates and the familiar Fekete-Szegö type inequalities for the subclass β − ST m ðδ, k, bÞ.
Proof. Suppose the inequality (23) holds. Also, let us assume We have Now consider The last expression is bounded by 1 if (23) holds. This completes the proof of Theorem 5. ☐ The result is sharp for the function Putting m = 1 in Theorem 5, we obtain the following corollary.

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Proof. Let where is analytic function in D, and it can be written as Comparing the coefficients of ξ n+m−1 on both sides Taking absolute on both sides and then applying the coefficient estimates jc n j ≤ L 1 (see [13]), we have We apply the mathematical induction on (36) so for n = 2 , and this shows that result is true for n = 2. Now for n = 3, and using (37), we obtain which is true for n = 3. Let us assume that (31) is true for n = t , that is, Consider Therefore, the result is true for n = t + 1. Consequently, using mathematical induction, we proved that the result holds true for all nðn ≥ 2Þ. This completes the proof of Theorem 8. ☐ Putting m = 1 in Theorem 8, we obtain the following corollary.
Corollary 9. If f ∈ β − ST ðδ, k, bÞ, then and for all n = 3, 4, 5, ⋯, Theorem 10. Let f ∈ β − ST m ðδ, k, bÞ. Then., f ðDÞ contains an open disk of radius Proof. Let w 0 ≠ 0 be a complex number such that f ðξÞ ≠ w 0 for ξ ∈ D. Then, Now using Theorem 8, we have 5 Journal of Function Spaces and hence This completes the proof of Theorem 10. ☐ Putting m = 1 in Theorem 10, we obtain the following corollary.
Corollary 11. Let f ∈ β − ST ðδ, k, bÞ. Then, f ðDÞ contains an open disk of radius Theorem 12. Let f ∈ β − ST m ðδ, k, bÞ with the form (1). Then, for a complex number μ, we have Proof. If f ∈ β − ST m ðδ, k, bÞ, then there exists a Schwarz function w, with wð0Þ = 0 and jwðξÞj < 1 such that Let h ∈ P be a function defined by This gives Using (53) in (50), we obtain For any complex number μ, we have Then (55) can be written as where Now, taking absolute value on both sides and using Lemma 3, we obtain the required result. ☐ Putting m = 1 in Theorem 12, we obtain the following corollary.

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Ethical Approval
This article does not contain any studies with human participants or animals performed by any of the authors.