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Abstract
In this paper, we consider a new class of analytic functions which is defined by means of a
Ruscheweyh q-differential operator. We investigated some new results such as coefficients inequalities and
other interesting properties of this class. Comparison of new results with those that were obtained in earlier
investigation are given as Corollaries.
Results & Lemmas (17)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Lemma 1.1.
Lemma 1.1. [15] Let h(z) = 1+P∞ n=1 cnzn be subordinate to H(z) = 1+P∞ n=1 Cnzn. If H(z) is univalent in E and H(E) is convex, then |cn|…
Lemma 1.1. [15] Let h(z) = 1+P∞ n=1 cnzn be subordinate to H(z) = 1+P∞ n=1 Cnzn. If H(z) is univalent in E and H(E) is convex, then |cn| ≤|C1| , n ≥1.
Lemma 1.2.
Lemma 1.2. ( [8], [10]) If q(z) = 1 + c1z + c2z2+... is an analytic function with positive real part in E, then c2 −vc2 1 ≤2 max 1, |2v…
Lemma 1.2. ( [8], [10]) If q(z) = 1 + c1z + c2z2+... is an analytic function with positive real part in E, then c2 −vc2 1 ≤2 max {1, |2v −1|} . The result is sharp for the functions q(z) = 1 + z2 1 −z2 , or q(z) = 1 + z 1 −z .
Lemma 1.3.
Lemma 1.3. [8] Let the function w ∈E be given by w(z) = c1z + c2z2 +... z ∈E. Then for every complex number v, c2 −vc2 1 ≤1 + (|v| −1)…
Lemma 1.3. [8] Let the function w ∈E be given by w(z) = c1z + c2z2 + ... z ∈E. Then for every complex number v, c2 −vc2 1 ≤1 + (|v| −1) |c1|2 .
Lemma 1.4.
Lemma 1.4. [11] Let k ∈[0, ∞) be a fixed and qk(z) = (A + 1)pk(z) −(A −1) (B + 1)pk(z) −(B −1), then qk(z) = 1 + H1(k)z + H2(k)z2 +..., z…
Lemma 1.4. [11] Let k ∈[0, ∞) be a fixed and qk(z) = (A + 1)pk(z) −(A −1) (B + 1)pk(z) −(B −1), then qk(z) = 1 + H1(k)z + H2(k)z2 + ..., z ∈E. and H1 := H1(k) = A −B 2 L1(k), H2 := H2(k) = A −B 4 {2D(k) −(B + 1)H1} L1(k) where L1(k) and D(k) are defined in (1.3) and (1.4). 2. Main Results
Theorem 2.1.
Theorem 2.1. A function f ∈A and of the form (1.1) is in the class k −USq(λ, A, B, β), if it satisfies the condition ∞ X n=2 …
Theorem 2.1. A function f ∈A and of the form (1.1) is in the class k −USq(λ, A, B, β), if it satisfies the condition ∞ X n=2 {2(k + 1) {1 −[n, q] −β[n, q][n −1, q]}
Corollary 2.1.
Corollary 2.1. A function f ∈A and of the form (1.1) is in the class k −USq(λ, 1 −2α, −1), if it satisfies the condition ∞ X n=2 (k + 1) [n,…
Corollary 2.1. A function f ∈A and of the form (1.1) is in the class k −USq(λ, 1 −2α, −1) , if it satisfies the condition ∞ X n=2 {(k + 1) [n, q] −k −α} ϕn−1 |an| ≤1 −α. When q →1, β = 0, λ = 0, then we have the following known result, proved by Noor and Sarfraz [11].
Corollary 2.2.
Corollary 2.2. A function f ∈A and of the form (1.1 is in the class k −ST (A, B), if it satisfies the condition ∞ X n=2 2(k + 1)(n −1) +…
Corollary 2.2. A function f ∈A and of the form (1.1 is in the class k −ST (A, B), if it satisfies the condition ∞ X n=2 {2(k + 1)(n −1) + |n(B + 1) −(A + 1)|} |an| ≤|B −A| .
Corollary 2.3.
Corollary 2.3. A function f ∈A and of the form (1.1) is in the class k −UST (1 −2α, −1), if it satisfies the condition ∞ X n=2 n(k + 1) −(k…
Corollary 2.3. A function f ∈A and of the form (1.1) is in the class k −UST (1 −2α, −1), if it satisfies the condition ∞ X n=2 {n(k + 1) −(k + α)} |an| ≤1 −α, where 0 ≤α < 1 and k ≥0. When λ = 0, β = 0, A = 1 −2α, B = −1 with 0 ≤α < 1 and k = 0, then we have the following known result, proved by Selverman in [17].
Corollary 2.4.
Corollary 2.4. A function f ∈A and of the form (1.1) is in the class 0 −UST (1 −2α, −1), if it satisfies the condition ∞ X n=2 n −α |an| ≤1…
Corollary 2.4. A function f ∈A and of the form (1.1) is in the class 0 −UST (1 −2α, −1), if it satisfies the condition ∞ X n=2 {n −α} |an| ≤1 −α, 0 ≤α < 1.
Theorem 2.2.
Theorem 2.2. If f(z) ∈k −USq(λ, A, B, β) and is of the form (1.1). Then |an| ≤ n−2 Y j=0 |L1(k)(A −B) −2[j, q]B| 2 [j + 1, q] q + β[j +…
Theorem 2.2. If f(z) ∈k −USq(λ, A, B, β) and is of the form (1.1). Then |an| ≤ n−2 Y j=0 |L1(k)(A −B) −2[j, q]B| 2 [j + 1, q] {q + β[j + 2, q]} ϕj+1 , n ≥2, (2.2) where L1(k) is defined by (1.3).
Corollary 2.5.
Corollary 2.5. A function f ∈A and of the form (1.1) is in the class k −ST [A, B], if it satisfies the condition |an| ≤ n−2 Y j=0 |L1(k)(A…
Corollary 2.5. A function f ∈A and of the form (1.1) is in the class k −ST [A, B] , if it satisfies the condition |an| ≤ n−2 Y j=0 |L1(k)(A −B) −2jB| 2 (j + 1) . When λ = 0, A = 1, B = −1 and β = 0 then we have the following known result, proved by Kanas and Wisniowska in [6].
Corollary 2.6.
Corollary 2.6. A function f ∈A and of the form (1.1) is in the class k −UST [A, B], if it satisfies the condition |an| ≤ n−2 Y j=0 |L1(k) +…
Corollary 2.6. A function f ∈A and of the form (1.1) is in the class k −UST [A, B] , if it satisfies the condition |an| ≤ n−2 Y j=0 |L1(k) + j| (j + 1) . When λ = 0, A = 1−2α, β = 0, B = −1 with 0 ≤α < 1, then we have the following known result, proved by Shams et al. in [18].
Corollary 2.7.
Corollary 2.7. A function f ∈A and of the form (1.1) is in the class SD(k, α), if it satisfies the condition |an| ≤ n−2 Y j=0 |L1(k)(1 −α)…
Corollary 2.7. A function f ∈A and of the form (1.1) is in the class SD(k, α), if it satisfies the condition |an| ≤ n−2 Y j=0 |L1(k)(1 −α) + j| (j + 1) . where 0 ≤α < 1 and k ≥0. When λ = 0, β = 0, k = 0, then T1(k) = 2 and we get the following known result, proved in [4]
Corollary 2.8.
Corollary 2.8. A function f ∈A and of the form (1.1) is in the class S∗[A, B], if it satisfies the condition |an| ≤ n−2 Y j=0 |(A −B) −jB|…
Corollary 2.8. A function f ∈A and of the form (1.1) is in the class S∗[A, B], if it satisfies the condition |an| ≤ n−2 Y j=0 |(A −B) −jB| (j + 1) , −1 ≤B < A ≤1. When λ = 0, β = 0, A = 1 −2α, B = −1 with 0 ≤α < 1 and k = 0, then we have the following known result, proved by Selverman in [17].
Corollary 2.9.
Corollary 2.9. A function f ∈A and of the form (1.1) is in the class S∗(α), if it satisfies the condition |an| ≤ n−2 Q j=0 (j −2α) (n −1)!,…
Corollary 2.9. A function f ∈A and of the form (1.1) is in the class S∗(α), if it satisfies the condition |an| ≤ n−2 Q j=0 (j −2α) (n −1)! , 0 ≤α < 1.
Theorem 2.3.
Theorem 2.3. Let −1 ≤B < A ≤1and 0 ≤k < ∞be fixed and let f(z) ∈k −USq(λ, A, B, β) and is of the form (1.1) Then for a complex number µ. a3…
Theorem 2.3. Let −1 ≤B < A ≤1and 0 ≤k < ∞be fixed and let f(z) ∈k −USq(λ, A, B, β) and is of the form (1.1) Then for a complex number µ. a3 −µa2 2 ≤
Theorem 2.4.
Theorem 2.4. Let 0 ≤k < ∞, −1 ≤B < A ≤1, be fixed and let f(z) ∈k −USq(λ, A, B, β) and is of the form (1.1) Then for a complex number µ. a3…
Theorem 2.4. Let 0 ≤k < ∞, −1 ≤B < A ≤1, be fixed and let f(z) ∈k −USq(λ, A, B, β) and is of the form (1.1) Then for a complex number µ. a3 −µa2 2 ≤ (A −B)L1(k) 2[2, q] {q + [3, q]β} ϕ2 max {1, |2v −1|} , where v is given by (2.17).
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