Ma-Minda φ-classes studied in this paper:
Abstract
The purpose of this article is to introduce a new subclass of analytic and
bi-univalent functions, in associated with sigmoid function and to investigate the up-
per bounds for |a2| and |a3|, where a2, a3 are the initial Taylor-Maclaurin coefficients.
Further, we obtain the Fekete-Szeg¨o inequalities for this subclass of the bi-univalent
function class Σ. We also give several illustrative examples of the bi-univalent function
class which we introduce here.
Results & Lemmas (8)
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 1.1.
Lemma 1.1. [4] Let h be a sigmoid function and (1.4) Φ(z) = 2h(z) = 1 + ∞ X m=1 (−1)m 2m ∞ X n=1 (−1)n n! zn
Lemma 1.1. [4] Let h be a sigmoid function and (1.4) Φ(z) = 2h(z) = 1 + ∞ X m=1 (−1)m 2m ∞ X n=1 (−1)n n! zn
Lemma 1.2.
Lemma 1.2. [4] Let Φm,n(z) = 2h(z) = 1 + ∞ X m=1 (−1)m 2m ∞ X n=1 (−1)n n! zn !m
Lemma 1.2. [4] Let Φm,n(z) = 2h(z) = 1 + ∞ X m=1 (−1)m 2m ∞ X n=1 (−1)n n! zn !m
Lemma 1.3.
Lemma 1.3. [4] If Φ(z) ∈P is starlike, then f is a normalized univalent function of the form (1.1). Setting m = 1, Fadipe-Joseph et al. [4]…
Lemma 1.3. [4] If Φ(z) ∈P is starlike, then f is a normalized univalent function of the form (1.1). Setting m = 1, Fadipe-Joseph et al. [4] remarked that Φ(z) = 1 + ∞ X n=1 cnzn,
Lemma 1.4.
Lemma 1.4. [11] If p ∈P, then |pi| ≦2 for each i, where P is the family of all functions p, analytic in D, for which Re p(z) > 0 (z ∈D),…
Lemma 1.4. [11] If p ∈P, then |pi| ≦2 for each i, where P is the family of all functions p, analytic in D, for which Re{p(z)} > 0 (z ∈D), where p(z) = 1 + p1z + p2z2 + · · · (z ∈D). In particular, the equality holds for all n for the next function p(z) = 1 + z 1 −z = 1 + 2 ∞ X n=1 zn.
Theorem 2.1.
Theorem 2.1. Let f be assumed as in (1.1) and f ∈Gκ,δ,ϑ Σ (Φ). Then |a2| ≤min F1, F2, F3, where F1 = 1 2 (ϑ + κ + 2δ), F2 = s 2 (2ϑ + κ)(κ…
Theorem 2.1. Let f be assumed as in (1.1) and f ∈Gκ,δ,ϑ Σ (Φ). Then |a2| ≤min{F1, F2, F3}, where F1 = 1 2 (ϑ + κ + 2δ), F2 = s 2 (2ϑ + κ)(κ + 1) + 12δ and F3 =
Theorem 2.2.
Theorem 2.2. Let f be given by (1.1) and f ∈Gκ,δ,ϑ Σ (Φ). Then for ν ∈R, a3 −νa2 2 ≤ ( 1 2(2ϑ+κ+6δ); 0 ≤|h(ν)| ≤ 1 4(2ϑ+κ+6δ) 4 |h(ν)|;…
Theorem 2.2. Let f be given by (1.1) and f ∈Gκ,δ,ϑ Σ (Φ). Then for ν ∈R, a3 −νa2 2 ≤ ( 1 2(2ϑ+κ+6δ) ; 0 ≤|h(ν)| ≤ 1 4(2ϑ+κ+6δ) 4 |h(ν)| ; |h(ν)| ≥
Corollary 3.1.
Corollary 3.1. Let f(z) = z + ∞ P n=2 anzn be in the class N κ,ϑ Σ (Φ). Then |a2| ≤min F1, F2, F3, where F1 = 1 2 (ϑ + κ), F2 = s
Corollary 3.1. Let f(z) = z + ∞ P n=2 anzn be in the class N κ,ϑ Σ (Φ). Then |a2| ≤min{F1, F2, F3}, where F1 = 1 2 (ϑ + κ), F2 = s
Corollary 3.2.
Corollary 3.2. Let f(z) = z + ∞ P n=2 anzn be in the class Mδ,ϑ Σ (Φ). Then |a2| ≤min F1, F2, F3,
Corollary 3.2. Let f(z) = z + ∞ P n=2 anzn be in the class Mδ,ϑ Σ (Φ). Then |a2| ≤min{F1, F2, F3},
Definitions (1)
Def 1.1.
Definition 1.1. A function f ∈Σ of the form (1.1) belongs to the class Gκ,δ,ϑ Σ (Φ), κ ≥0, ϑ ≥1, δ ≥0, if the following conditions are…
Definition 1.1. A function f ∈Σ of the form (1.1) belongs to the class Gκ,δ,ϑ Σ (Φ), κ ≥0, ϑ ≥1, δ ≥0, if the following conditions are satisfied: (1 −ϑ) f(z) z κ + ϑf ′(z)
Function classes studied:
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