Abstract
We give coefficient characterizations for analytic functions with nega-
tive coefficients to be in subclasses of uniformly starlike and uniformly convex families.
This leads to extremal properties and neighborhood criteria.
Results & Lemmas (6)
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Theorem 1.
Theorem 1. (a) If P∞ n=2[2n −(α + 1)] |an| ≤1 −α, −1 ≤α < 1, then f of the form (1) is in Sp(α). (b) If P∞ n=2 n[2n −(α + 1)] |an| ≤1 −α,…
Theorem 1. (a) If P∞ n=2[2n −(α + 1)] |an| ≤1 −α, −1 ≤α < 1, then f of the form (1) is in Sp(α). (b) If P∞ n=2 n[2n −(α + 1)] |an| ≤1 −α, then f of the form (1) is in UCV (α).
Theorem 2.
Theorem 2. (a) A necessary and sufficient condition for f of the form (2) to be in TSp(α), −1 ≤α < 1, is that P∞ n=2[2n−(α+1)]an ≤1−α. (b) A…
Theorem 2. (a) A necessary and sufficient condition for f of the form (2) to be in TSp(α), −1 ≤α < 1, is that P∞ n=2[2n−(α+1)]an ≤1−α. (b) A necessary and sufficient condition for f of the form (2) to be in TV (α) is that P∞ n=2 n[2n −(α + 1)]an ≤1 −α.
Theorem 3.
Theorem 3. For −1 ≤α < 1, TSp(α) = T ∗¡ 1+α 2 ¢ and TV (α) = C ¡ 1+α 2 ¢.
Theorem 3. For −1 ≤α < 1, TSp(α) = T ∗¡ 1+α 2 ¢ and TV (α) = C ¡ 1+α 2 ¢ .
Corollary 1.
Corollary 1. The extreme points of TSp(α), −1 ≤α < 1, are f1(z)=z and fn(z) = z − 1−α 2n−(α+1)zn, n = 2, 3,.... The extreme points of TV…
Corollary 1. The extreme points of TSp(α), −1 ≤α < 1, are f1(z)=z and fn(z) = z − 1−α 2n−(α+1)zn, n = 2, 3, . . . . The extreme points of TV (α) are f1(z) = z and fn(z) = z − 1−α n(2n−(α+1))zn, n = 2, 3, . . . .
Corollary 2.
Corollary 2. (a) If f ∈TSp(α), −1 ≤α < 1, then r −1 −α 3 −αr2 ≤|f(z)| ≤r + 1 −α 3 −αr2, 1 −2(1 −α) 3 −α r ≤|f ′(z)| ≤1 + 2(1 −α) 3 −α r,…
Corollary 2. (a) If f ∈TSp(α), −1 ≤α < 1, then r −1 −α 3 −αr2 ≤|f(z)| ≤r + 1 −α 3 −αr2, 1 −2(1 −α) 3 −α r ≤|f ′(z)| ≤1 + 2(1 −α) 3 −α r, |z| = r. (b) If f ∈TV (α), then r − 1 −α 2(3 −α)r2 ≤|f(z)| ≤r + 1 −α 2(3 −α)r2, 1 −1 −α
Theorem 4.
Theorem 4. For −1 ≤α < 1 and −1 ≤β ≤(3α −1)/(3 −α), Nδ(TSp(α)) ⊂Sp(β) when δ = 3α−1−β(3−α) 2(3−α).
Theorem 4. For −1 ≤α < 1 and −1 ≤β ≤(3α −1)/(3 −α), Nδ(TSp(α)) ⊂Sp(β) when δ = 3α−1−β(3−α) 2(3−α) .
Function classes studied:
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