Results & Lemmas (6)
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Theorem 1.
Theorem 1. For 0 ≤α < 1 and f = h + g let ∞ X n=2 [n]m q ([n]q −α)|an| + ∞ X n=1 [n]m q ([n]q + α)|bn| ≤1 −α (7) where h and g are,…
Theorem 1. For 0 ≤α < 1 and f = h + g let ∞ X n=2 [n]m q ([n]q −α)|an| + ∞ X n=1 [n]m q ([n]q + α)|bn| ≤1 −α (7) where h and g are, respectively, given by (2) and (4). Then i) f is harmonic univalent in D and f ∈Hm q (α) if the inequality (7) holds.
Corollary 2.
Corollary 2. For m = 0, Theorem 1 yields the results obtained by the author in ( [11], Theorems 1 and 2). This can be easily verified since…
Corollary 2. For m = 0, Theorem 1 yields the results obtained by the author in ( [11], Theorems 1 and 2). This can be easily verified since D0 qf(z) = D0 qh(z) + (−1)0D0qg(z) = h(z) + g(z).
Corollary 3.
Corollary 3. For q→1−, Theorem 1 yields the results obtained in ( [12], Theorems 1 and 2) since lim q→1−Dm q f(z) = lim q→1− n Dm q h(z) +…
Corollary 3. For q→1−, Theorem 1 yields the results obtained in ( [12], Theorems 1 and 2) since lim q→1−Dm q f(z) = lim q→1− n Dm q h(z) + (−1)mDm q g(z) o = z +
Theorem 4.
Theorem 4. f ∈clcoH m q (α) if and only if f(z) = ∞ X n=1 Xnhn(z) + Yngn(z) , (8) where h1(z) = z, hn(z) = z − 1−α [n]m
Theorem 4. f ∈clcoH m q (α) if and only if f(z) = ∞ X n=1 Xnhn(z) + Yngn(z) , (8) where h1(z) = z, hn(z) = z − 1−α [n]m
Theorem 5.
Theorem 5. If f ∈H m q (α), then for |z| = r < 1 we have the distortion bounds (1−b1)r− 1 [2]m q 1 −α [2]q −α −1 + α [2]q −αb1 …
Theorem 5. If f ∈H m q (α), then for |z| = r < 1 we have the distortion bounds (1−b1)r− 1 [2]m q 1 −α [2]q −α −1 + α [2]q −αb1 r2≤|f(z)|≤(1+b1)r+ 1 [2]m q
Corollary 6.
Corollary 6. If f ∈H m q (α), then ω: |ω| < [2]m+1 q −1 −([2]m q −1)α [2]m q ([2]q −α) 1 −[2]q −α [2]q + αb1 ⊂f(D).
Corollary 6. If f ∈H m q (α), then ω : |ω| < [2]m+1 q −1 −([2]m q −1)α [2]m q ([2]q −α) 1 −[2]q −α [2]q + αb1 ⊂f(D).
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