Abstract
In the present article, we define a new subclass of pseudo-type meromorphic bi-univalent functions class $Σ'$ of complex order $γ\in \mathbb{C}\backslash \{0\}$ and investigate the initial coefficient estimates $|b_0|, |b_1|$ and $|b_2|.$ Further we mention several new or known consequences of our result.
Results & Lemmas (6)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 2.1.
Lemma 2.1. [9] If Φ ∈P, the class of all functions with ℜ(Φ(z)) > 0, (z ∈∆) then |ck| ≤2, for each k, where Φ(z) = 1 + c1z + c2z2 + · · ·…
Lemma 2.1. [9] If Φ ∈P, the class of all functions with ℜ(Φ(z)) > 0, (z ∈∆) then |ck| ≤2, for each k, where Φ(z) = 1 + c1z + c2z2 + · · · for z ∈∆. Define the functions p and q in P given by p(z) = 1 + u(z) 1 −u(z) = 1 + p1 z + p2 z2 + · · · and q(z) = 1 + v(z) 1 −v(z) = 1 + q1 z + q2 z2 + · · · .
Theorem 2.1.
Theorem 2.1. Let g be given by (1.5) in the class Pγ Σ′(λ, µ, φ). Then |b0| ≤ |γ||B1| |µ −µλ −λ|, (2.1) |b1| ≤ |γ| 2|µ −λ −2µλ| s 4|(B1…
Theorem 2.1. Let g be given by (1.5) in the class Pγ Σ′(λ, µ, φ). Then |b0| ≤ |γ||B1| |µ −µλ −λ|, (2.1) |b1| ≤ |γ| 2|µ −λ −2µλ| s 4|(B1 −B2)2| + 4|B2 1| + 8|B1(B1 −B2)| + |µ(µ −1)(1 −λ) + 2λ|2|γ2B4 1| |µ −µλ −λ|4 (2.2)
Theorem 2.2.
Theorem 2.2. Let g be given by (1.5) in the class Pγ Σ′(µ, φ). Then |b0| ≤|γ| |B1|, (2.21) |b1| ≤ |γ| |1 + µ| q |(B1 −B2)2| + |B2 1| +…
Theorem 2.2. Let g be given by (1.5) in the class Pγ Σ′(µ, φ). Then |b0| ≤|γ| |B1|, (2.21) |b1| ≤ |γ| |1 + µ| q |(B1 −B2)2| + |B2 1| + 2|B1(B1 −B2)| + |γ|2 |B4 1|. (2.22) and |b2| ≤ |γ|
Theorem 2.3.
Theorem 2.3. Let g be given by (1.5) in the class PΣ′(µ, φ). Then |b0| ≤|B1|, |b1| ≤ 1 |1 + µ| q |(B1 −B2)2| + |B2 1| + 2|B1(B1 −B2)| + |B4…
Theorem 2.3. Let g be given by (1.5) in the class PΣ′(µ, φ). Then |b0| ≤|B1|, |b1| ≤ 1 |1 + µ| q |(B1 −B2)2| + |B2 1| + 2|B1(B1 −B2)| + |B4 1|. and |b2| ≤ 1 |1 + 2µ| |B1| + 2|B2 −B1| + |B1 −2B2 + B3| + |B1|3 where µ ≥1, z, w ∈∆∗.
Corollary 3.1.
Corollary 3.1. Let g be given by (1.5) in the class Pγ Σ′(λ, µ, 1+z 1−z α) ≡Pγ Σ′(λ, µ, α). Then |b0| ≤ 2|γ|α |µ −µλ −λ|, (3.1) |b1| ≤…
Corollary 3.1. Let g be given by (1.5) in the class Pγ Σ′(λ, µ, 1+z 1−z α) ≡Pγ Σ′(λ, µ, α). Then |b0| ≤ 2|γ|α |µ −µλ −λ|, (3.1) |b1| ≤ 2|γ|α |µ −λ −2λµ| s (α −2)2 + |µ(µ −1)(1 −λ) + 2λ|2|γ2|
Corollary 3.2.
Corollary 3.2. Let g be given by (1.5) in the class Pγ Σ′ λ, µ, 1+(1−2β)z 1−z ≡Pγ Σ′(λ, µ, β). Then |b0| ≤2|γ|(1 −β) |µ −µλ −λ|, (3.4)…
Corollary 3.2. Let g be given by (1.5) in the class Pγ Σ′ λ, µ, 1+(1−2β)z 1−z ≡Pγ Σ′(λ, µ, β). Then |b0| ≤2|γ|(1 −β) |µ −µλ −λ|, (3.4) |b1| ≤ 2|γ|(1 −β) |µ −λ −2λµ| s
Function classes studied:
Related Papers