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Abstract
The main objective of the present paper is to define a subclass Qq(λ, µ, A, B) of analytic
functions by using subordination along with the newly defined q-analogue of Choi-Saigo-
Srivastava operator. Such results as coefficient estimates, integral representation, linear
combination, weighted and arithmetic means, and radius of starlikeness for this class are
derived.
Mathematics Subject Classification (2010). 30C45, 30C50
Results & Lemmas (8)
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Lemma 1.2.
Lemma 1.2. [17] Let −1 ≤B2 ≤B1 < A1 ≤A2 ≤1. Then 1 + A1z 1 + B1z ≺1 + A2z 1 + B2z. Throughout this paper, we assume that λ > −1, µ > 0, 0 <…
Lemma 1.2. [17] Let −1 ≤B2 ≤B1 < A1 ≤A2 ≤1. Then 1 + A1z 1 + B1z ≺1 + A2z 1 + B2z . Throughout this paper, we assume that λ > −1, µ > 0, 0 < q < 1 and −1 ≤B < A ≤1, unless otherwise stated. We also suppose that all coefficients an of f are real positive numbers. 2. Main results
Theorem 2.1.
Theorem 2.1. Let f ∈A and be of the form (1.1). Then f ∈Qq (λ, µ, A, B) if and only if ∞ X n=2 [n, q] (1 −B) −1 + A [µ, q]n−1 [1 + λ, q]n−1…
Theorem 2.1. Let f ∈A and be of the form (1.1). Then f ∈Qq (λ, µ, A, B) if and only if ∞ X n=2 {[n, q] (1 −B) −1 + A} [µ, q]n−1 [1 + λ, q]n−1 an < A −B. (2.1)
Theorem 2.2.
Theorem 2.2. Let f ∈Qq (λ, µ, A, B). Then Iq λ,µf(z) = exp Z z 0 1 t 1 −Aϕ(t) 1 −Bϕ(t) dq(t) , where |ϕ(z)| < 1.
Theorem 2.2. Let f ∈Qq (λ, µ, A, B) . Then Iq λ,µf(z) = exp Z z 0 1 t 1 −Aϕ(t) 1 −Bϕ(t) dq(t) , where |ϕ(z)| < 1.
Theorem 2.3.
Theorem 2.3. Let fj ∈Qq (λ, µ, A, B) and have the form fj(z) = z + ∞ X k=1 ak,jzk (j = 1, 2, 3,..., l). Then F ∈Qq (λ, µ, A, B), where F(z)…
Theorem 2.3. Let fj ∈Qq (λ, µ, A, B) and have the form fj(z) = z + ∞ X k=1 ak,jzk (j = 1, 2, 3, . . . , l). Then F ∈Qq (λ, µ, A, B), where F(z) = l X j=1 cjfj(z) with l X j=1
Theorem 2.4.
Theorem 2.4. If f and g belong to Qq (λ, µ, A, B), then their weighted mean hj (j ∈N) is also in Qq (λ, µ, A, B), where hj is defined by…
Theorem 2.4. If f and g belong to Qq (λ, µ, A, B) , then their weighted mean hj (j ∈N) is also in Qq (λ, µ, A, B) , where hj is defined by hj(z) = (1 −j) f(z) + (1 + j) g(z) 2 . (2.3)
Theorem 2.5.
Theorem 2.5. Let fj with j = 1, 2,..., α (α ∈N) belong to the class Qq (λ, µ, A, B). Then the arithmetic mean h of fj given by h(z) = 1 α α…
Theorem 2.5. Let fj with j = 1, 2, ..., α (α ∈N) belong to the class Qq (λ, µ, A, B). Then the arithmetic mean h of fj given by h(z) = 1 α α X j=1 fj(z) (2.4) also belongs to the class Qq (λ, µ, A, B).
Theorem 2.6.
Theorem 2.6. Let f ∈Qq (λ, µ, A, B). Then f ∈S∗(γ), for |z| < r1, where r1 = (1 −γ) [n, q] (1 −B) −1 + A [µ,q]n−1 [1+λ,q]n−1 (n −γ) (A…
Theorem 2.6. Let f ∈Qq (λ, µ, A, B). Then f ∈S∗(γ) , for |z| < r1, where r1 = (1 −γ) {[n, q] (1 −B) −1 + A} [µ,q]n−1 [1+λ,q]n−1 (n −γ) (A −B) 1 n−1 .
Theorem 2.7.
Theorem 2.7. Let −1 ≤B2 ≤B1 < A1 ≤A2 ≤1 and Iq λ+1,µf(z) ̸= 0 in E. If ([λ + 1, q]) Iq λ,µf(z) qλIq λ+1,µf(z) −[λ, q] qλ ≺1 + A1z 1 + B1z.…
Theorem 2.7. Let −1 ≤B2 ≤B1 < A1 ≤A2 ≤1 and Iq λ+1,µf(z) ̸= 0 in E. If ([λ + 1, q]) Iq λ,µf(z) qλIq λ+1,µf(z) −[λ, q] qλ ≺1 + A1z 1 + B1z . Then f ∈Qq (λ + 1, µ, A2, B2) .
Function classes studied:
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