Abstract
In the present investigation, we find estimates on the coef-
ficients |a2| and |a3| for functions in the function class SΣ (λ, φ) . The
results presented in this paper improve or generalize the recent work of
Magesh and Yamini [15].
1
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.
Theorem 1
Theorem 1 Let f given by (1) be in the class SΣ (λ, φ). Then |a2| ≤ B1 √B1 (12λ4 −28λ3 + 15λ2 + 2λ + 1) B2 1 −(1 + 3λ −2λ2)2 B2 + (1…
Theorem 1 Let f given by (1) be in the class SΣ (λ, φ) . Then |a2| ≤ B1 √B1 (12λ4 −28λ3 + 15λ2 + 2λ + 1) B2 1 −(1 + 3λ −2λ2)2 B2 + (1 + 3λ −2λ2)2 B1 (7) and |a3| ≤ ⎧ ⎪ ⎪ ⎪ ⎪
Corollary 1
Corollary 1 Let f given by (1) be in the class S∗ Σ (φ). Then |a2| ≤ B1 √B1 B2 1 −B2 + B1 and |a3| ≤ ⎧ ⎪ ⎪ ⎪ ⎪
Corollary 1 Let f given by (1) be in the class S∗ Σ (φ). Then |a2| ≤ B1 √B1 B2 1 −B2 + B1 and |a3| ≤ ⎧ ⎪ ⎪ ⎪ ⎪
Corollary 2
Corollary 2 If let φ (z) = 1 + z 1 −z α = 1 + 2αz + 2α2z2 +... (0 < α ≤1), then inequalities (7) and (8) become |a2| ≤ 2α |20λ4 −44λ3 +…
Corollary 2 If let φ (z) = 1 + z 1 −z α = 1 + 2αz + 2α2z2 + ... (0 < α ≤1) , then inequalities (7) and (8) become |a2| ≤ 2α |20λ4 −44λ3 + 25λ2 −2λ + 1| α + (1 + 3λ −2λ2)2 (19)
Corollary 3
Corollary 3 Let f given by (1) be in the class S∗ Σ (α), (0 < α ≤1). Then |a2| ≤ 2α √ α + 1 (21) and |a3| ≤ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪
Corollary 3 Let f given by (1) be in the class S∗ Σ (α) , (0 < α ≤1) . Then |a2| ≤ 2α √ α + 1 (21) and |a3| ≤ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪
Corollary 4
Corollary 4 If let φ (z) = 1 + (1 −2α) z 1 −z = 1 + 2 (1 −α) z + 2 (1 −α) z2 + · · · (0 < α ≤1), then inequalities (7) and (8) become |a2|…
Corollary 4 If let φ (z) = 1 + (1 −2α) z 1 −z = 1 + 2 (1 −α) z + 2 (1 −α) z2 + · · · (0 < α ≤1) , then inequalities (7) and (8) become |a2| ≤ 2 (1 −α) 2 (1 −α) (12λ4 −28λ3 + 15λ2 + 2λ + 1) −(1 + 3λ −2λ2)2 + (1 + 3λ −2λ2)2 (23)
Corollary 5
Corollary 5 Let f given by (1) be in the class S⋆ Σ (α), (0 ≤α < 1). Then |a2| ≤ 2 (1 −α) 1 + |1 −2α| (25) and |a3| ≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨
Corollary 5 Let f given by (1) be in the class S⋆ Σ (α) , (0 ≤α < 1) . Then |a2| ≤ 2 (1 −α) 1 + |1 −2α| (25) and |a3| ≤ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨
Function classes studied:
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