Abstract
In this paper, we consider a new subclass of analytic and bi-univalent functions associated with q-Ruscheweyh differential
operator in the open unit disk U. For functions belonging to the class Σq(λ, φ), we obtain estimates on the first two Taylor-
Maclaurin coefficients. Further, we derive another subclass of analytic and bi-univalent functions as a special consequences of
the results.
Results & Lemmas (5)
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Lemma 1.3
Lemma 1.3 ([20]). Let the function p ∈P be given by the following series: p(z) = 1 + p1z + p2z2 + p3z3 + · · ·, (z ∈U). The sharp estimate…
Lemma 1.3 ([20]). Let the function p ∈P be given by the following series: p(z) = 1 + p1z + p2z2 + p3z3 + · · · , (z ∈U). The sharp estimate given by |pn| ⩽2, (n ∈N), holds true. 2. A set of main results For functions f in the class Σq(λ, ϕ), the following result is obtained.
Theorem 2.1.
Theorem 2.1. Let f ∈Σq(λ, ϕ) be of the form (1.2). Then |a2| ⩽ B 3 2 1 q |q[λ + 1]q qλB2 1 + q[λ + 1]q(B1 −B2) |, (2.1)
Theorem 2.1. Let f ∈Σq(λ, ϕ) be of the form (1.2). Then |a2| ⩽ B 3 2 1 q |q[λ + 1]q qλB2 1 + q[λ + 1]q(B1 −B2) | , (2.1)
Corollary 3.2.
Corollary 3.2. Let the function f ∈Σ1 q(λ, β) be of the form (1.1). Then |a2| ⩽ s 2(1 −β) qλ+1[λ]q + q2λ+1, and |a3| ⩽ 2(1 −β) q[λ]q + qλ+1…
Corollary 3.2. Let the function f ∈Σ1 q(λ, β) be of the form (1.1). Then |a2| ⩽ s 2(1 −β) qλ+1[λ]q + q2λ+1 , and |a3| ⩽ 2(1 −β) q[λ]q + qλ+1 2(1 −β)([λ]q + qλ[2]q) + [λ]q + qλ [λ + 1]q[λ + 2]q ! .
Corollary 3.4.
Corollary 3.4. Let the function f given by f ∈Σ2 q(β):= Σ1 q(0, β) be of the form (1.1). Then |a2| ⩽ s 2(1 −β) q, and |a3| ⩽2(1 −β) q 2(1…
Corollary 3.4. Let the function f given by f ∈Σ2 q(β) := Σ1 q(0, β) be of the form (1.1). Then |a2| ⩽ s 2(1 −β) q , and |a3| ⩽2(1 −β) q 2(1 −β) + 1 1 + q
Corollary 3.6.
Corollary 3.6. Let the function f ∈Σ3 q(λ, α) be of the form (1.1). Then |a2| ⩽ 2α p |q[λ + 1]q[2αqλ + q[λ + 1]q(1 −α)]|, and |a3 ⩽ 2α…
Corollary 3.6. Let the function f ∈Σ3 q(λ, α) be of the form (1.1). Then |a2| ⩽ 2α p |q[λ + 1]q[2αqλ + q[λ + 1]q(1 −α)]| , and |a3 ⩽ 2α q[λ]q + qλ+1 2α [λ + 1]q +
Function classes studied:
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