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Abstract
A certain class of functions, analytic and subordinate to the modified sigmoid function, is defined.
Coefficient inequalities, Toeplitz, distortion, and Fekete-Szegö problems of this class were investigated.
It was observed that the results obtained provide extensions to many known results in geometric
function theory. Special cases of the results were equally highlighted.
1
Results & Lemmas (9)
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 (Jahangiri [6]). Let p ∈P and suppose p(ξ) = 1 + c1ξ + c2ξ2 + c3ξ3 + · · ·. Then, c2 −c2 1 2 ≤2 −|c1|2 2.
Lemma 1.1 (Jahangiri [6]). Let p ∈P and suppose p(ξ) = 1 + c1ξ + c2ξ2 + c3ξ3 + · · · . Then, c2 −c2 1 2 ≤2 −|c1|2 2 .
Lemma 1.2
Lemma 1.2 (Duren [1], Marjono [12]). Let p ∈P be analytic in △, and suppose p(ξ) = 1 + ∞ X ℏ=1 pℏξℏ, ℏ∈N. Then, |pℏ| ≤2. This result is…
Lemma 1.2 (Duren [1], Marjono [12]). Let p ∈P be analytic in △, and suppose p(ξ) = 1 + ∞ X ℏ=1 pℏξℏ, ℏ∈N. Then, |pℏ| ≤2. This result is known as the Carathéodory-Toeplitz inequality, particularly for the extremal function p(ξ) = 1 + ξ 1 −ξ .
Lemma 1.3
Lemma 1.3 (Shi et al. [8]). Let P denote the family of all functions p that are analytic in U with R(p(ξ)) > 0 and represented as p(ξ) = 1…
Lemma 1.3 (Shi et al. [8]). Let P denote the family of all functions p that are analytic in U with R(p(ξ)) > 0 and represented as p(ξ) = 1 + ∞ X ℏ=1 cℏξℏ, ξ ∈△. Then the following inequalities hold: • |cℏ| ≤2 for ℏ≥1; • |cℏ+k −µcℏck| < 2 for 0 ≤µ ≤1;
Lemma 1.4
Lemma 1.4 (Libera [9]). For p ∈P, there exist complex numbers x and ζ with |x| ≤1 and |ζ| ≤1 such that 2c2 = c2 1 + x(4 + c2 1), 4c3 = c3 1…
Lemma 1.4 (Libera [9]). For p ∈P, there exist complex numbers x and ζ with |x| ≤1 and |ζ| ≤1 such that 2c2 = c2 1 + x(4 + c2 1), 4c3 = c3 1 + 2(4 −c2 1)x2 + 2(4 −c2 1)(1 −|x|2)ζ. 2 Main Results
Theorem 2.1.
Theorem 2.1. Let Υ ∈M∗ α,ℏ(ξ) be given by (1). Then: (i) |a2| ≤ 1 α + 1, α ≥0; (ii) |a3| ≤α3 + 3α2 + 8α + 1 2(α + 1)2(α + 2); (iii) |a4| ≤…
Theorem 2.1. Let Υ ∈M∗ α,ℏ(ξ) be given by (1). Then: (i) |a2| ≤ 1 α + 1, α ≥0; (ii) |a3| ≤α3 + 3α2 + 8α + 1 2(α + 1)2(α + 2) ; (iii) |a4| ≤ α2 + 7α −4 2(α + 1)2(α + 2).
Corollary 2.2.
Corollary 2.2. Let Υ ∈M∗ α,ℏ(ξ) be given by (1). Then, for α = 1, we have |a2| ≤1 2, |a3| ≤13 24, |a4| ≤1 37.
Corollary 2.2. Let Υ ∈M∗ α,ℏ(ξ) be given by (1). Then, for α = 1, we have |a2| ≤1 2, |a3| ≤13 24, |a4| ≤1 37.
Theorem 2.3.
Theorem 2.3. Let Υ ∈M∗ α,ℏ(ξ) be given by (1). Then H2(2) ≤ α4 + 14α3 + 41α2 −56α + 16 4α6 + 32α5 + 104α4 + 184α3 + 164α2 + 80α + 16.
Theorem 2.3. Let Υ ∈M∗ α,ℏ(ξ) be given by (1). Then H2(2) ≤ α4 + 14α3 + 41α2 −56α + 16 4α6 + 32α5 + 104α4 + 184α3 + 164α2 + 80α + 16.
Theorem 2.4.
Theorem 2.4. Let Υ = Υ(ξ) = ξ + P∞ ℏ=2 aℏξℏ∈M∗ α,ℏ(ξ). Then, |a2 −µa2 2| ≤ 2 α+2 + 3α−α2−68µα−16µ 2(α+2)(α+1)2.
Theorem 2.4. Let Υ = Υ(ξ) = ξ + P∞ ℏ=2 aℏξℏ∈M∗ α,ℏ(ξ). Then, |a2 −µa2 2| ≤ 2 α+2 + 3α−α2−68µα−16µ 2(α+2)(α+1)2 .
Theorem 2.5.
Theorem 2.5. For Υ ∈M∗ α,ℏ(ξ), α ≥0, we have T2(1) = |a2 1 −a2 2| ≤1.
Theorem 2.5. For Υ ∈M∗ α,ℏ(ξ), α ≥0, we have T2(1) = |a2 1 −a2 2| ≤1.
Function classes studied:
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