Starlikeness Properties of a New Integral Operator for Meromorphic Functions

Definition 1.1. Let n ∈ ℕ,   γ_i > 0,   i ∈ {1,2, 3, …, n}. We define the integral operator $$ by

For the sake of simplicity, from now on we will write $$ instead of $$.

By $$, we denote the class of functions f ∈ Σ such that

In order to derive our main results, we have to recall here the following preliminary results.

Lemma 1.2 (see [2].)Suppose that the function Ψ :   ℂ² → ℂ satisfies the following condition:

If the function p(z) = 1 + p₁z + ⋯ is analytic in 𝕌 and then,

Proposition 1.3 (see [3].)If f ∈ Σ satisfying

then,

Theorem 2.1. Let f_i ∈ ∑,   γ_i > 0 for all    i ∈ {1, …, n}. If

then $$ belongs to Σ^*(0).

Proof. On successive differentiation of $$, which is defined in (1.5), we get

Then from (2.2), we obtain By multiplying (2.3) with z yield, That is equivalent to Or We can write the left-hand side of (2.6), as the following: We define the regular function p in 𝕌 by and p(0) = 1. Differentiating p(z) logarithmically, we obtain From (2.7), (2.8), and (2.9), we obtain Let us put From (2.1), (2.10), and (2.11), we obtain Now, we proceed to show that Indeed, from (2.11), we have Thus, from (2.12), (2.14), and by using Lemma 1.2, we conclude that ℜ{p(z)} > 0, and so that is, $$ is starlike of order 0.

Theorem 2.2. For i ∈ {1, …, n}, let γ_i > 0 and f_i ∈ Σ_k(α_i)  (0 ≤ α_i < 1). If $$ be the integral operator given by (1.5) and

Then $$ belong to Σ^*(μ), where $$.

Proof. Following the same steps as in Theorem 2.1, we obtain

Taking the real part of both terms of the last expression, we have Since f_i ∈ Σ_k(α_i),    for   i ∈ {1, …, n}, we receive Therefore, Using (2.16), (2.20), and applying Proposition 1.3, we get $$, where $$.

Corollary 2.3. For i ∈ {1, …, n}, let γ_i > 0 and f_i ∈ Σ_k(α)  (0 ≤ α < 1). If

$$ be the integral operator given by (1.5) and Then $$ is starlike of order $$.

Theorem 2.4. For i ∈ {1, …, n}, let γ_i > 0 and $$. If

$$ be the integral operator given by (1.5) and Then $$ is starlike of order $$.

Proof. Following the same steps as in Theorem 2.1, we obtain

We calculate the real part from both terms of the above equality and obtain Since $$ for all i ∈ {1, …, n}, the above relation then yields Because $$, we obtain that Using (2.24), (2.28) and applying Proposition 1.3, we get $$ is a starlike function of order $$

Corollary 2.5. For i ∈ {1, …, n}, let γ_i > 0 and $$. If

 $$ be the integral operator given by (1.5) and Then $$ is starlike of order $$.

Corollary 2.6. Let γ > 0, and $$. If

ℋ_γ(z) be the integral operator, Then ℋ_γ(z) is starlike of order 1 − (1 − β)γ.