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Ma-Minda φ-classes studied in this paper:
Results & Lemmas (9)
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Lemma 1.
Lemma 1. ([9]) For a function p ∈P of the form p (z) = 1 + ∞ P n=1 pnzn, z ∈E, the sharp inequality |pn| ⩽2 holds for each n ⩾1. Equality…
Lemma 1. ([9]) For a function p ∈P of the form p (z) = 1 + ∞ P n=1 pnzn, z ∈E, the sharp inequality |pn| ⩽2 holds for each n ⩾1. Equality holds for the function p (z) = 1+z 1−z.
Lemma 2.
Lemma 2. ([10]) Let p ∈P of the form p (z) = 1 + ∞ P n=1 pnzn and µ ∈C. Then |pn −µpkpn−k| ⩽2max 1, |2µ −1|, 1 ⩽k ⩽n −1. If |2µ −1| ⩾1,…
Lemma 2. ([10]) Let p ∈P of the form p (z) = 1 + ∞ P n=1 pnzn and µ ∈C. Then |pn −µpkpn−k| ⩽2max {1, |2µ −1|} , 1 ⩽k ⩽n −1. If |2µ −1| ⩾1, then the inequality is sharp for the function p (z) = 1+z 1−z or its rotations. If |2µ −1| < 1, then the inequality is sharp for the function p (z) = 1+zn 1−zn or its rotations.
Lemma 3. · coeff
Lemma 3. ([6]) Let p ∈P of the form p (z) = 1 + ∞ P n=1 pnzn, z ∈E and α, β, γ ∈ℜ. Then αp13 −βp1p2 + γp3 ⩽2 |α| + 2 |β −2α| + 2 |α −β +…
Lemma 3. ([6]) Let p ∈P of the form p (z) = 1 + ∞ P n=1 pnzn, z ∈E and α, β, γ ∈ℜ. Then αp13 −βp1p2 + γp3 ⩽2 |α| + 2 |β −2α| + 2 |α −β + γ| . 3. Main results This section is devoted to the proof of our main results. We will now determine the upper bounds of the Taylor coefficients, logarithmic coefficients, and Hankel and Toeplitz determinants of logarithmic coefficients, respectively, as follows: 3.1. Taylor coefficients
Theorem 1.
Theorem 1. If f is of the form (1) belongs to S∗ SC (sin z), then |a2| ⩽1 2, |a3| ⩽1 2, |a4| ⩽1 4, and |a5| ⩽1 2.
Theorem 1. If f is of the form (1) belongs to S∗ SC (sin z) , then |a2| ⩽1 2, |a3| ⩽1 2, |a4| ⩽1 4, and |a5| ⩽1 2.
Theorem 2.
Theorem 2. If f is of the form (1) belongs to S∗ SC (sin z), then |γ1| ⩽1 4, |γ2| ⩽1 4, |γ3| ⩽1 8, and |γ4| ⩽7 16.
Theorem 2. If f is of the form (1) belongs to S∗ SC (sin z) , then |γ1| ⩽1 4, |γ2| ⩽1 4, |γ3| ⩽1 8, and |γ4| ⩽7 16.
Theorem 3.
Theorem 3. If f ∈S∗ SC (sin z) and has the series representation (1), then |H2,1 (Ff/2)| ⩽ 87 1024.
Theorem 3. If f ∈S∗ SC (sin z) and has the series representation (1), then |H2,1 (Ff/2)| ⩽ 87 1024.
Theorem 4.
Theorem 4. If f ∈S∗ SC (sin z) and has the series representation (1), then |H2,2 (Ff/2)| ⩽33 256.
Theorem 4. If f ∈S∗ SC (sin z) and has the series representation (1), then |H2,2 (Ff/2)| ⩽33 256.
Theorem 5.
Theorem 5. If f ∈S∗ SC (sin z), then γ12 −γ22 ⩽65 256.
Theorem 5. If f ∈S∗ SC (sin z) , then γ12 −γ22 ⩽65 256.
Theorem 6.
Theorem 6. If f ∈S∗ SC (sin z), then |T2,2 (γn)| ⩽11 32.
Theorem 6. If f ∈S∗ SC (sin z) , then |T2,2 (γn)| ⩽11 32.
Definitions (1)
Def 1.
Definition 1. Let S∗ SC (sin z) be the class of functions defined by zf′ (z) h (z) ≺φ (z), z ∈E, (2) where h (z) = f(z)−f(−z) 2 and φ (z) =…
Definition 1. Let S∗ SC (sin z) be the class of functions defined by zf′ (z) h (z) ≺φ (z) , z ∈E, (2) where h (z) = f(z)−f(−z) 2 and φ (z) = 1 + sin z. It is observed that the classes S∗ SC and S∗
Function classes studied:
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