Abstract
The purpose of the present paper is to introduce a new subclass of meromor-
phic multivalent functions defined by using a linear operator associated with the general-
ized hypergeometric function. Some properties of this class are established here by using
the principle of differential subordination and convolution in geometric function theory.
Results & Lemmas (8)
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Lemma 1
Lemma 1([15]). Let f ∈K, g ∈S∗, where S∗and K denote the subclasss of univalent functions consisting of starlike and convex functions in U.…
Lemma 1([15]). Let f ∈K, g ∈S∗, where S∗and K denote the subclasss of univalent functions consisting of starlike and convex functions in U. Then for each analytic function h in U, (2.1) (f ∗hg)(U) (f ∗g)(U) ⊆coh(U), where coh(U) denotes the closed convex hull of h(U).
Lemma 2
Lemma 2([15]). Let either 0 < a ≤c and c ≥2 when a, c are real, or Re(a+c) ≥3, Re(a) ≤Re(c) and Im(a) = Im(c) when a,c are complex. Then…
Lemma 2([15]). Let either 0 < a ≤c and c ≥2 when a, c are real, or Re(a+c) ≥3, Re(a) ≤Re(c) and Im(a) = Im(c) when a,c are complex. Then the function (2.2) f(z) = ∞ ∑ n=0 (a)n (c)n zn+1 (z ∈U), belongs to the class K of convex functions. Now we give two inclusion relationships for the class of meromorphic functions defined by (1.11).
Theorem 1.
Theorem 1. Let 0 < ai ≤αi, ∀ai, αi, bj ∈ C Z− 0, (i = 1, · · ·, q, j = 1, · · ·, s), h ∈N and (2.3) max z∈U (Re[h(z)]) < 1 + 1 p −η (z ∈U).…
Theorem 1. Let 0 < ai ≤αi, ∀ai, αi, bj ∈ C \ Z− 0 , (i = 1, · · · , q, j = 1, · · · , s), h ∈N and (2.3) max z∈U (Re[h(z)]) < 1 + 1 p −η (z ∈U). Further suppose that either ai, αi are real such that αi ≥2 or ai, αi are complex such that Re(ai + αi) ≥3, Re(ai) ≤Re(αi) and Im(ai) = Im(αi). Then (2.4) T λ
Theorem 2.
Theorem 2. Let 0 < bj ≤βj, ∀ai, αi, bj, βj ∈C Z− 0, ( i = 1, · · ·, q, j = 1, · · ·, s), h ∈Nand h satisfies (2.3). If bj, βj are real such…
Theorem 2. Let 0 < bj ≤βj, ∀ai, αi, bj, βj ∈C \ Z− 0 , ( i = 1, · · · , q, j = 1, · · · , s), h ∈Nand h satisfies (2.3). If bj, βj are real such that βj ≥2 or if bj, βj are complex such that Re(bj + βj) ≥3, Re(bj) ≤Re(βj) and Im(bj) = Im(βj). Then T λ ai,βj(η, p, h) ⊂T λ ai,bj(η, p, h).
Corollary 3.
Corollary 3. Let 0 < ai ≤αi and 0 < bj ≤βj, ∀ai, αi, bj βj ∈C Z− 0, (i = 1, · · ·, q, j = 1, · · ·, s), h(z) = 1 + Az 1 + Bz and 1 + A 1 +…
Corollary 3. Let 0 < ai ≤αi and 0 < bj ≤βj, ∀ai, αi, bj βj ∈C \ Z− 0 , (i = 1, · · · , q, j = 1, · · · , s), h(z) = 1 + Az 1 + Bz and 1 + A 1 + B < 1 + 1 p −η where −1 ≤ B < A ≤1. Further suppose that either ai, αi, bj, βj are real such that αi ≥2and βj ≥2 or ai, αi, bj, βj are complex such that Re(ai + αi) ≥3, Re(bj + βj) ≥3, Re(ai) ≤Re(αi), Im(ai) = Im(αi), Re(bj) ≤Re(βj) and Im(bj) = Im(βj). Then T λ ai,βj(η, p, 1 + Az 1 + Bz ) ⊂T λ ai,bj(η, p, 1 + Az 1 + Bz ) ⊂T λ
Theorem 3.
Theorem 3. Let f ∈T λ ai,bj(η, p, h) and the function h satisfies (2.3). Then (3.1) |ak| ≤(k + 1)Πq i=1(ai)k (λ + p)kΠs j=1(bj)k, k = 0, 1,…
Theorem 3. Let f ∈T λ ai,bj(η, p, h) and the function h satisfies (2.3). Then (3.1) |ak| ≤(k + 1)Πq i=1(ai)k (λ + p)kΠs j=1(bj)k , k = 0, 1, · · · .
Theorem 4.
Theorem 4. A function f belongs to the class T λ ai,bj(η, p, h) if and only if there exists a Schwarz function w(z) such that (4.1) f(z) =…
Theorem 4. A function f belongs to the class T λ ai,bj(η, p, h) if and only if there exists a Schwarz function w(z) such that (4.1) f(z) = [ ∞ ∑ k=0 Πq i=1(ai)k (λ + p)kΠs j=1(bj)k zk−p ] ∗
Theorem 5.
Theorem 5. Let λ ≥0, ϕ ∈K, h ∈N and h satisfies (2.3). Then f ∈T λ ai,bj(η, p, h) ⇒[z−(p+1)ϕ] ∗f ∈T λ ai,bj(η, p, h).
Theorem 5. Let λ ≥0, ϕ ∈K, h ∈N and h satisfies (2.3). Then f ∈T λ ai,bj(η, p, h) ⇒[z−(p+1)ϕ] ∗f ∈T λ ai,bj(η, p, h).