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Abstract
Two subclasses of starlike and convex functions analytic in the unit open disk using q-derivative operator have
been investigated in the present paper. The necessary and sufficient condition for the function belonging to these
classes have been obtained. We further examine various properties, such as the Hadamard product and the quasi-
Hadamard product. The coefficient estimates for the function belonging to these classes are also found.
Mathematics Subject Classification 2020: 30C45, 30C50
Results & Lemmas (8)
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Theorem 1.
Theorem 1. The function f ∈ T defined by (1.10) is in the class Kq(ψ) if and only if
Theorem 1. The function f ∈ T defined by (1.10) is in the class Kq(ψ) if and only if
Theorem 2.
Theorem 2. The function f ∈ T defined by (1.10) is in the class if and only if
Theorem 2. The function f ∈ T defined by (1.10) is in the class if and only if
Theorem 3.
Theorem 3. The function f ∈ T defined by (1.10) is in the class Kq(ψ) if and only if
Theorem 3. The function f ∈ T defined by (1.10) is in the class Kq(ψ) if and only if
Theorem 4.
Theorem 4. The function f ∈ T defined by (1.10) is in the class if and only if
Theorem 4. The function f ∈ T defined by (1.10) is in the class if and only if
Theorem 6.
Theorem 6. If the function f ∈ T defined by (1.10) satisfies the inequality ∞
Theorem 6. If the function f ∈ T defined by (1.10) satisfies the inequality ∞
Theorem 7.
Theorem 7. Let the functions fi(i = 1,2,···,m) given by (3.2), belong to the class. Then, the quasi-Hadamard product f1 ∗′ f2 ∗′ · · · ∗′…
Theorem 7. Let the functions fi(i = 1,2,···,m) given by (3.2), belong to the class . Then, the quasi-Hadamard product f1 ∗′ f2 ∗′ · · · ∗′ fm belongs to the class .
Theorem 8.
Theorem 8. Let the functions fi(i = 1,2,· · ·,m) given by (3.2), belong to the the class Kq(ψ). Then, the quasi- Hadamard product f1 ∗′ f2…
Theorem 8. Let the functions fi(i = 1,2,· · ·,m) given by (3.2), belong to the the class Kq(ψ). Then, the quasi- Hadamard product f1 ∗′ f2 ∗′ · · · ∗′ fm belongs to the class Sq(2m−1)(ψ).
Theorem 9.
Theorem 9. Let the functions fi(i = 1,2,· · ·,m) given by (3.2), belong to the the class Kq(ψ) and the functions gj(j = 1,2,· · ·,s) given…
Theorem 9. Let the functions fi(i = 1,2,· · ·,m) given by (3.2), belong to the the class Kq(ψ) and the functions gj(j = 1,2,· · ·,s) given by (3.3) belong to the class . Then, the quasi-Hadamard product f1 ∗′ f2 ∗′ · · · ∗′ fm ∗′ g1 ∗′ g2 ∗′ · · · ∗′ gs belongs to the class
Function classes studied:
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