Results & Lemmas (25)
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Lemma 1
Lemma 1: Let f ∈A satisfies (4). Then we have the following:
Lemma 1: Let f ∈A satisfies (4). Then we have the following:
Lemma 2
Lemma 2[11]: The sequence k k=1 b ∞ is a subordinating factor sequence if and only if:
Lemma 2[11]: The sequence k k=1 {b }∞ is a subordinating factor sequence if and only if:
Theorem 1
Theorem 1: Let f ∈A. If
Theorem 1: Let f ∈A . If
Corollary 1
Corollary 1: Let f ∈A be in the class * S (m,,a,b, ) λ α. Then:
Corollary 1: Let f ∈A be in the class * S (m, ,a,b, ) λ α . Then:
Theorem 2
Theorem 2: Let f ∈A. If:
Theorem 2: Let f ∈A . If:
Corollary 2
Corollary 2: Let f ∈A be in the class * C (m,,a,b, ) λ α. Then:
Corollary 2: Let f ∈A be in the class * C (m, ,a,b, ) λ α . Then:
Theorem 3
Theorem 3: Let 1 2 0 < < 1 ≤α α, 1 2 0 < < 1 ≤λ
Theorem 3: Let 1 2 0 < < 1 ≤α α , 1 2 0 < < 1 ≤λ
Theorem 4
Theorem 4: Let 1 2 0 < < 1 ≤α α, 1 2 0 < < 1 ≤λ
Theorem 4: Let 1 2 0 < < 1 ≤α α , 1 2 0 < < 1 ≤λ
Theorem 5
Theorem 5: Let * f S (m,,a,b, ) ∈ λ α. Then:
Theorem 5: Let * f S (m, ,a,b, ) ∈ λ α . Then:
Corollary 3
Corollary 3: Under the hypothesis of Theorem 5. f (z) is included in a disc centered at the origin with radius r given by:
Corollary 3: Under the hypothesis of Theorem 5. f (z) is included in a disc centered at the origin with radius r given by:
Theorem 6
Theorem 6: Let * f C (m,,a,b, ) ∈ λ α. Then:
Theorem 6: Let * f C (m, ,a,b, ) ∈ λ α . Then:
Corollary 4
Corollary 4: Under the hypothesis of Theorem 6. f (z) is included in a disc centered at the origin with radius r given by:
Corollary 4: Under the hypothesis of Theorem 6. f (z) is included in a disc centered at the origin with radius r given by:
Theorem 7
Theorem 7: Let 0f (z) = z and:
Theorem 7: Let 0f (z) = z and:
Corollary 5
Corollary 5: The extreme points of the class * S [m,,a,b, ] λ α are the functions 0f (z) = z and:
Corollary 5: The extreme points of the class * S [m, ,a,b, ] λ α are the functions 0f (z) = z and:
Theorem 8
Theorem 8: Let 0f (z) = z and: n 1 n n m n 1 (b) f (z) = z z, (n = 1,2,...) (n 1)(n 1
Theorem 8: Let 0f (z) = z and: n 1 n n m n 1 (b) f (z) = z z , (n = 1,2,...) (n 1)(n 1
Corollary 6
Corollary 6: The extreme points of the class * C [m,,a,b, ] λ α are the functions 0f (z) = z and:
Corollary 6: The extreme points of the class * C [m, ,a,b, ] λ α are the functions 0f (z) = z and:
Lemma 2
Lemma 2, this is equivalent to the following inequality:
Lemma 2, this is equivalent to the following inequality:
Corollary 7
Corollary 7: If the function f defined by (1) satisfies:
Corollary 7: If the function f defined by (1) satisfies:
Corollary 8
Corollary 8: If the function f defined by (1) satisfies:
Corollary 8: If the function f defined by (1) satisfies:
Theorem 9
Theorem 9, we have the following corollary related to the starlikeness of the fractional derivative and fractional integral operator f (z)…
Theorem 9, we have the following corollary related to the starlikeness of the fractional derivative and fractional integral operator f (z) α Ω due to Owa and Srivastava which is mentioned above.
Corollary 9
Corollary 9: If the function f defined by (1) satisfies:
Corollary 9: If the function f defined by (1) satisfies:
Corollary 10
Corollary 10: If the function f defined by (1) satisfies:
Corollary 10: If the function f defined by (1) satisfies:
Corollary 11
Corollary 11: If the function f defined by (1) satisfies:
Corollary 11: If the function f defined by (1) satisfies:
Theorem 10
Theorem 10, we have the following corollary related to the convexity of the fractional derivative and fractional integral operator f (z) α…
Theorem 10, we have the following corollary related to the convexity of the fractional derivative and fractional integral operator f (z) α Ω due to Owa and Srivastava which is mentioned above.
Corollary 12
Corollary 12: If the function f defined by (1) satisfies:
Corollary 12: If the function f defined by (1) satisfies:
Definitions (5)
Def 1
Definition 1: Let the function φ be given by:
Definition 1: Let the function φ be given by:
Def 2
Definition 2: A function f ∈A is said to be in the class S(m,,a,b, ) λ α for m∈Z, 0 λ ≥, 0 < 1 ≤α, if and only if:
Definition 2: A function f ∈A is said to be in the class S(m, ,a,b, ) λ α for m∈Z , 0 λ ≥ , 0 < 1 ≤α , if and only if:
Def 3
Definition 3: A function f ∈A is said to be in the class C(m,,a,b, ) λ α for m∈Z, 0 λ ≥, 0 < 1 ≤α, if and only if:
Definition 3: A function f ∈A is said to be in the class C(m, ,a,b, ) λ α for m∈Z , 0 λ ≥ , 0 < 1 ≤α , if and only if:
Def 4
Definition 4[10]: Let g be analytic and univalent in U. If f is analytic in U, f (0) = g(0) and f ( ) g( ) ⊂ U U, then one says that f is…
Definition 4[10]: Let g be analytic and univalent in U . If f is analytic in U , f (0) = g(0) and f ( ) g( ) ⊂ U U , then one says that f is subordinate to g in U and we write f g p
Def 5
Definition 5[10]: An infinite sequence k k=1 b ∞ of complex numbers is said to be a subordinating factor sequence if for every univalent…
Definition 5[10]: An infinite sequence k k=1 {b }∞ of complex numbers is said to be a subordinating factor sequence if for every univalent function f in K, one has: