Abstract
We determine coefficient estimates for α-spiral functions of order ϱ with
respect to N-symmetric points (|α| < π/2, 0 ≤ϱ < 1 and N is a positive integer).
Sharp coefficient bounds are also obtained for functions of the form f(z)−t, where t is a
positive integer and f(z) is an α-spiral function of order ϱ. Using this we deduce coefficient
estimates for inverses of univalent α-spiral and meromorphic univalent α-spiral functions
with vanishing early coefficients.
Results & Lemmas (10)
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Theorem 2.1.
Theorem 2.1. If f(z) = z + P∞ n=2 anzn ∈SN(α, ϱ), then (2.8) |amN+1| ≤ m−1 Y j=0 2(1 −ϱ) cos αe−iα N + j
Theorem 2.1. If f(z) = z + P∞ n=2 anzn ∈SN(α, ϱ), then (2.8) |amN+1| ≤ m−1 Y j=0 2(1 −ϱ) cos αe−iα N + j
Corollary 2.1.
Corollary 2.1. If f(z) = z + P∞ n=2 anzn ∈S∗ N(ϱ), then |amN+1| ≤ m−1 Y j=0 2 −2ϱ N + j (j + 1) for m = 1, 2,..., and |amN+p| ≤1
Corollary 2.1. If f(z) = z + P∞ n=2 anzn ∈S∗ N(ϱ), then |amN+1| ≤ m−1 Y j=0 2 −2ϱ N + j (j + 1) for m = 1, 2, . . . , and |amN+p| ≤1
Corollary 2.2.
Corollary 2.2. If f(z) = z + P∞ n=2 anzn ∈CN(ϱ), then |amN+1| ≤ 1 mN + 1 m−1 Y j=0 2 −2ϱ N + j (j + 1) for m = 1, 2,..., and
Corollary 2.2. If f(z) = z + P∞ n=2 anzn ∈CN(ϱ), then |amN+1| ≤ 1 mN + 1 m−1 Y j=0 2 −2ϱ N + j (j + 1) for m = 1, 2, . . . , and
Theorem 3.1.
Theorem 3.1. Suppose f(z) = z + P∞ n=p+1 anzn ∈S(α, ϱ), and for integral t ≥1, let f(z)−t = z−t + ∞ X γ=−t+p a(−t) γ zγ, 0 < |z| < 1. Then…
Theorem 3.1. Suppose f(z) = z + P∞ n=p+1 anzn ∈S(α, ϱ), and for integral t ≥1, let f(z)−t = z−t + ∞ X γ=−t+p a(−t) γ zγ, 0 < |z| < 1. Then (3.19) |a(−t) γ
Lemma 3.1.
Lemma 3.1. If p, t and q are positive integers, then for 0 ≤ϱ < 1, we have 4t(1 −ϱ) cos2 α t(1 −ϱ) + q−1 X m=1 [t(1 −ϱ) −mp] m−1 Y j=0…
Lemma 3.1. If p, t and q are positive integers, then for 0 ≤ϱ < 1, we have 4t(1 −ϱ) cos2 α t(1 −ϱ) + q−1 X m=1 [t(1 −ϱ) −mp] m−1 Y j=0 |(2t/p)(1 −ϱ) cos αe−iα −j| j + 1
Lemma 3.2.
Lemma 3.2. If p, t and q are positive integers and 0 ≤ϱ < 1, then (γ + t)2 ≥(qp)2(−γ −ϱt) t(1 −ϱ) −qp for γ ≥−t + qp and t(1 −ϱ) −(q −1)p…
Lemma 3.2. If p, t and q are positive integers and 0 ≤ϱ < 1, then (γ + t)2 ≥(qp)2(−γ −ϱt) t(1 −ϱ) −qp for γ ≥−t + qp and t(1 −ϱ) −(q −1)p ≥0. The proof of Lemma 3.2 is straightforward and hence omitted. P r o o f o f T h e o r e m 3.1. For 0 < |z| < 1, let q(z) = eiα −t · z[f(z)−t]′ f(z)−t = eiα zf ′(z) f(z) and q(0) can be defined so that q(z) is continuous at z = 0. Moreover, let (3.20) w(z) = q(z) −eiα
Theorem 3.2.
Theorem 3.2. If F(w) = w + P∞ n=p+1 Anwn ∈S(α, 0)−1, then |An| ≤ mp n(n −1) m−1 Y j=0 |(2n/p) cos αe−iα −j| j + 1 for mp + 1 ≤n ≤(m +…
Theorem 3.2. If F(w) = w + P∞ n=p+1 Anwn ∈S(α, 0)−1, then |An| ≤ mp n(n −1) m−1 Y j=0 |(2n/p) cos αe−iα −j| j + 1 for mp + 1 ≤n ≤(m + 1)p, m = 1, 2, . . .
Theorem 3.3.
Theorem 3.3. If G(w) = w + P∞ n=p−1 Bnw−n ∈Σ(α, 0)−1, then |Bn| ≤ mp n(n + 1) m−1 Y j=0 |(2n/p) cos αe−iα −j| j + 1 for mp −1 ≤n ≤(m +…
Theorem 3.3. If G(w) = w + P∞ n=p−1 Bnw−n ∈Σ(α, 0)−1, then |Bn| ≤ mp n(n + 1) m−1 Y j=0 |(2n/p) cos αe−iα −j| j + 1 for mp −1 ≤n ≤(m + 1)p −2, m = 1, 2, . . . For α = 0, Theorems 3.2 and 3.3 give the following results of Poole [10].
Corollary 3.1.
Corollary 3.1. If F(w) = w + P∞ n=p+1 Anwn ∈S∗−1, then |An| ≤ mp n(n −1) m−1 Y j=0 (2n/p) −j j + 1 for mp + 1 ≤n ≤(m + 1)p, m = 1, 2,...
Corollary 3.1. If F(w) = w + P∞ n=p+1 Anwn ∈S∗−1, then |An| ≤ mp n(n −1) m−1 Y j=0 (2n/p) −j j + 1 for mp + 1 ≤n ≤(m + 1)p, m = 1, 2, . . .
Corollary 3.2.
Corollary 3.2. If G(w) = w + P∞ n=p−1 Bnw−n ∈Σ∗−1, then |Bn| ≤ mp n(n + 1) m−1 Y j=0 (2n/p) −j j + 1 for mp −1 ≤n ≤(m + 1)p, m = 1,…
Corollary 3.2. If G(w) = w + P∞ n=p−1 Bnw−n ∈Σ∗−1, then |Bn| ≤ mp n(n + 1) m−1 Y j=0 (2n/p) −j j + 1 for mp −1 ≤n ≤(m + 1)p, m = 1, 2, . . . Moreover, when n = mp+1, Theorem 3.2 gives a result in [5]. Similarly, when n = mp −1, Theorem 3.3 gives a result in [5]. References
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