Results & Lemmas (11)
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Theorem 2.1.
Theorem 2.1. Let the function f = h + g to be so that h and g are given by (1.1). Furthermore, let (2.1) ∞ ∑ n=2 λn (1 −αλ) −α (1 −λ) 1 −α…
Theorem 2.1. Let the function f = h + g to be so that h and g are given by (1.1). Furthermore, let (2.1) ∞ ∑ n=2 λn (1 −αλ) −α (1 −λ) 1 −α |an| + ∞ ∑ n=1 µn (1 −αλ) + α (1 −λ) 1 −α |bn| ≤1,
Corollary 2.2.
Corollary 2.2. Let the function f = h + g to be so that h and g are given by (1.1). Furthermore, let (2.4) ∞ ∑ n=2 λn −α 1 −α |an| + ∞ ∑…
Corollary 2.2. Let the function f = h + g to be so that h and g are given by (1.1). Furthermore, let (2.4) ∞ ∑ n=2 λn −α 1 −α |an| + ∞ ∑ n=1 µn + α 1 −α |bn| ≤1 where 0 ≤α < 1, n (1 −α) ≤λn −α and n (1 −α) ≤µn + α for n ≥2 then f is sense-preserving, harmonic univalent in U and f ∈SH (ϕ, ψ; α).
Theorem 2.3.
Theorem 2.3. Let the functions f = h + g be so that h and g are given by (1.3). Then f ∈TSH (ϕ, ψ; α; λ) if and only if (2.5) ∞ ∑ n=2 λn (1…
Theorem 2.3. Let the functions f = h + g be so that h and g are given by (1.3). Then f ∈TSH (ϕ, ψ; α; λ) if and only if (2.5) ∞ ∑ n=2 λn (1 −αλ) −α (1 −λ) 1 −α |an| + ∞ ∑ n=1 µn (1 −αλ) + α (1 −λ) 1 −α |bn| ≤1,
Corollary 2.4.
Corollary 2.4. Let the functions f = h + g be so that h and g are given by (1.3). Then f ∈TSH (ϕ, ψ; α; 0) if and only if ∞ ∑ n=2 λn −α 1…
Corollary 2.4. Let the functions f = h + g be so that h and g are given by (1.3). Then f ∈TSH (ϕ, ψ; α; 0) if and only if ∞ ∑ n=2 λn −α 1 −α |an| + ∞ ∑ n=1 µn + α 1 −α |bn| ≤1. where 0 ≤α < 1, n (1 −α) ≤λn −α and n (1 −α) ≤µn + α for n ≥2.
Corollary 2.5.
Corollary 2.5. Let the function f = h + g be so that h and g are given by (1.3). Then f ∈TSH ( z (1 −z)2, z (1 −z)2; α; 0 ) if and only if…
Corollary 2.5. Let the function f = h + g be so that h and g are given by (1.3). Then f ∈TSH ( z (1 −z)2 , z (1 −z)2 ; α; 0 ) if and only if ∞ ∑ n=2 n −α 1 −α |an| +
Corollary 2.6.
Corollary 2.6. Let the function f = h + g be so that h and g are given by (1.3). Then f ∈TSH ( z + z2 (1 −z)3, z (1 −z)2; α; 0 )
Corollary 2.6. Let the function f = h + g be so that h and g are given by (1.3). Then f ∈TSH ( z + z2 (1 −z)3 , z (1 −z)2 ; α; 0 )
Theorem 2.7.
Theorem 2.7. Let f ∈TSH (ϕ, ψ; α; λ) and A ≤λn (1 −αλ) −α (1 −λ), A ≤µn (1 −αλ) + α (1 −λ) for n ≥2. Then for |z| = r < 1 we have |f(z)|…
Theorem 2.7. Let f ∈TSH (ϕ, ψ; α; λ) and A ≤λn (1 −αλ) −α (1 −λ) , A ≤µn (1 −αλ) + α (1 −λ) for n ≥2. Then for |z| = r < 1 we have |f(z)| ≤(1 + |b1|) r + (1 −α A −µ1 (1 −αλ) + α (1 −λ) A |b1| ) r2, |z| = r < 1 and |f(z)| ≥(1 −|b1|) r − (1 −α A
Corollary 2.8.
Corollary 2.8. Let f ∈TSH (ϕ, ψ; α; λ) and A ≤λn (1 −αλ) −α (1 −λ), A ≤ µn (1 −αλ) + α (1 −λ) for n ≥2 where A = min λ2 (1 −αλ) −α (1 −λ),…
Corollary 2.8. Let f ∈TSH (ϕ, ψ; α; λ) and A ≤λn (1 −αλ) −α (1 −λ), A ≤ µn (1 −αλ) + α (1 −λ) for n ≥2 where A = min {λ2 (1 −αλ) −α (1 −λ) , µ2 (1 −αλ) + α (1 −λ)} . Then we have { ω : |ω| < (A −1 + α A −A −µ1 (1 −αλ) −α (1 −λ) A |b1| )} ⊂f (U) .
Theorem 3.1.
Theorem 3.1. Let h1 (z) = z, hn (z) = z − 1 −α λn (1 −αλ) −α (1 −λ)zn (n ≥2) and gn (z) = z + 1 −α µn (1 −αλ) + α (1 −λ) ¯zn (n ≥1). Then f…
Theorem 3.1. Let h1 (z) = z, hn (z) = z − 1 −α λn (1 −αλ) −α (1 −λ)zn (n ≥2) and gn (z) = z + 1 −α µn (1 −αλ) + α (1 −λ) ¯zn (n ≥1) . Then f ∈TSH (ϕ, ψ; α; λ) if and only if it can be expressed as (3.1) f(z) = ∞ ∑ n=1 (xnhn (z) + yngn (z))
Theorem 4.1.
Theorem 4.1. If f ∈TSH (ϕ, ψ; α; λ) and F ∈TSH (ϕ, ψ; α; λ) then f ∗F ∈TSH (ϕ, ψ; α; λ).
Theorem 4.1. If f ∈TSH (ϕ, ψ; α; λ) and F ∈TSH (ϕ, ψ; α; λ) then f ∗F ∈TSH (ϕ, ψ; α; λ) .
Theorem 4.2.
Theorem 4.2. The class TSH (ϕ, ψ; α; λ) is closed under convex combinations.
Theorem 4.2. The class TSH (ϕ, ψ; α; λ) is closed under convex combinations.
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