Abstract
Making use of a linear operator, we introduce certain subclass of meromor-
phically univalent functions in the punctured unit disk and study its properties including
some inclusion results, coefficient and distortion problems. Our result generalize many
results known in the literature.
Results & Lemmas (7)
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Lemma 1.1
Lemma 1.1(Eenigenburg et al. [5]). Let h be convex univalent in U with h(0) = 1 and ℜ(kh(z) + ν) > 0, (k, ν ∈C; z ∈U). If q(z) is analytic…
Lemma 1.1(Eenigenburg et al. [5]). Let h be convex univalent in U with h(0) = 1 and ℜ(kh(z) + ν) > 0, (k, ν ∈C; z ∈U). If q(z) is analytic in U with q(0) = 1, then the subordination q(z) + zq′(z) kq(z) + ν ≺h(z) (z ∈U), implies that q(z) ≺h(z) (z ∈U). 2. Main results We begin with the following :
Theorem 2.1.
Theorem 2.1. Let λ ≥0 and a > 0. Then (1) M λ+1(a, c; h) ⊂M λ(a, c; h) for (h ∈N and ℜ(h(z)) < 2 + λ, z ∈U); (2) M λ(a, c; h) ⊂M λ(a + 1,…
Theorem 2.1. Let λ ≥0 and a > 0. Then (1) M λ+1(a, c; h) ⊂M λ(a, c; h) for (h ∈N and ℜ(h(z)) < 2 + λ, z ∈U); (2) M λ(a, c; h) ⊂M λ(a + 1, c; h) for (h ∈N and ℜ(h(z)) < 1 + a, z ∈U).
Corollary 2.1.
Corollary 2.1. Let λ ≥0, a > 0 and −1 < B < A ≤1. Then (1) M λ+1(a, c; A, B) ⊂M λ(a, c; A, B), if A −B 1 + B < 1 + λ; (2) M λ(a, c; A, B)…
Corollary 2.1. Let λ ≥0, a > 0 and −1 < B < A ≤1 . Then (1) M λ+1(a, c; A, B) ⊂M λ(a, c; A, B), if A −B 1 + B < 1 + λ; (2) M λ(a, c; A, B) ⊂M λ(a + 1, c; A, B), if A −B 1 + B < a.
Theorem 2.2.
Theorem 2.2. Let 0 < α < µ and h ∈N be so that ℜ h(z) < µ α. If f ∈ M λ(a, c; h), then the function g(z) defined by (2.4) Lλ(a, c)g(z) = µ…
Theorem 2.2. Let 0 < α < µ and h ∈N be so that ℜ{h(z)} < µ α. If f ∈ M λ(a, c; h), then the function g(z) defined by (2.4) Lλ(a, c)g(z) = µ −α zµ Z z 0 tµ−1[ Lλ(a, c)f(z)]αdt 1 α ,
Corollary 2.2.
Corollary 2.2. Let 0 < α < µ and −1 < B < A ≤1 be so that 1 + A 1 + B < µ α. If f ∈M λ(a, c; A, B), then the function g(z) defined by (2.7)…
Corollary 2.2. Let 0 < α < µ and −1 < B < A ≤1 be so that 1 + A 1 + B < µ α. If f ∈M λ(a, c; A, B), then the function g(z) defined by (2.7) Lλ(a, c)g(z) = µ −α zµ Z z 0 tµ−1[ Lλ(a, c)f(z)]α 1 α , is also in the same class M λ(a, c; A, B).
Theorem 2.3.
Theorem 2.3. Let λ > −1, −1 ≤B < A ≤1, and a > 0, c > 0. If f(z) = z−1 + P∞ n=1 anzn−1 ∈M λ(a, c; A, B), then (2.8) |an| ≤(A −B)n(a)n…
Theorem 2.3. Let λ > −1, −1 ≤B < A ≤1, and a > 0, c > 0. If f(z) = z−1 + P∞ n=1 anzn−1 ∈M λ(a, c; A, B), then (2.8) |an| ≤(A −B)n(a)n (c)n(λ + 1)n (n = 1, 2, 3, · · · ). When B = −1, and A = 1 −2α (0 ≤α < 1), the result is sharp for the function given by f(z) = L(1, λ + 1) L(a, c) 1 z(1 −z)2(1−α) .
Theorem 2.4.
Theorem 2.4. Let a > 0, c > 0, −1 ≤B < A ≤1 and λ > −1. If f(z) = z−1 + P∞ n=1 anzn−1 ∈M λ(a, c; A, B), then (2.11) |f(z)| ≤ L(a, c) L(1, λ…
Theorem 2.4. Let a > 0, c > 0, −1 ≤B < A ≤1 and λ > −1. If f(z) = z−1 + P∞ n=1 anzn−1 ∈M λ(a, c; A, B), then (2.11) |f(z)| ≤ L(a, c) L(1, λ + 1) 1 r F(A −B, 1; 1; r) (|z| < r),