Certain subclass of Meromorphic function associated with Wright function

Lemma 1. Suppose that the sequence {An}∞ n=1 is defined by A1 = (1 −2λ) cos θ(1 + γ(1 −2λ)) (1 −λ)ϕ1(α, β) , An+1 = 2 cos θ(1 + γ(1 −2λ)) (n + 2)(1 −λ)ϕn+1(α, β) × " 1 −2λ + n X k=1 ϕk(α, β)(1 −λ + kλ)Ak

Theorem 1. Let f ∈Σ(θ, λ, γ). Then the integral representation of Wα,βf(z) is given by Wα,βf(z) = z−1 exp Z z 0 1 t (1 −λ)A(t) 1 −λA(t) + 1  dt  , (z ∈D∗), (2.1) where w is an analytic function in D with w(0) = 0 and |w(z)| < 1, and A(t) is defined as

Theorem 2. Suppose η ∈C with |η| = 1 and η ̸= 1. Then f ∈Σ(θ, λ, γ) if and only if f(z) ∗ " (1 −2λ)(1 −η)

Theorem 3. Suppose ϵ is a real number such that 0 ≤ϵ < 1. If f ∈Σ satisfies the condition

Corollary 1. If f ∈Σ satisfies the inequality

Theorem 4. Assume that f ∈Σ(θ, λ, γ). Then |a1| ≤(1 −2λ) cos θ(1 + γ(1 −2λ)) (1 −λ)ϕ1(α, β) , |an| ≤ Λ(1 −2λ) (1 −λ)nϕn(α, β) × n−1 Y k=1 (k + 1)(1 −λ) + 2(1 −λ + kλ)Λ k + 2 .

Theorem 5. Let f ∈Σ(θ, λ, γ) and let {An}∞ n=1 be the sequence of coefficient bounds such that |an| ≤An for all n ∈N. Then the operator Wα,βf(z) is: (i) Meromorphically starlike of order ρ (0 ≤ρ < 1) in the disk |z| < r1, where r1 is the largest value satisfying the equation: ∞ X n=1 n + 2 −ρ 1 −ρ  ϕnAnrn+1 ≤1. (3.3) (ii) Meromorphically convex of order ρ (0 ≤ρ < 1) in the disk |z| < r2, where r2 is the largest value satisfying the equation:

Definition 1. [9] A meromorphic function F(z) of the form F(z) = 1 z + P∞ n=1 cnzn is said to be meromorphically starlike of order ρ (0 ≤ρ < 1) in the disk |z| < r if it satisfies the analytic condition: −ℜ zF ′(z) F(z) 

Definition 2. [9] A meromorphic function F(z) of the form F(z) = 1 z + P∞ n=1 cnzn is said to be meromorphically convex of order ρ (0 ≤ρ < 1) in the disk 0 < |z| < r if it satisfies the analytic condition: −ℜ  1 + zF ′′(z) F ′(z)