Abstract
In this paper, a new subclass Sα,n(m, l, q, λ, φ) defined by the gener-
alised derivative operator Dα,n(m, l, q, λ) is introduced. The Fekete-Szeg¨o functional
|a3 −µa2
2| of the subclass Sα,n(m, l, q, λ, φ) is obtained. Then by convolution, we
state another subclass Sα,n,g(m, l, q, λ, φ) defined by fractional derivatives. Another
set of Fekete-Szeg¨o result is determined.
2010 Mathematics Subject Classification: 30C45.
Results & Lemmas (9)
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.
Lemma 1. (Ma and Minda [21]) If p(z) = 1 + c1z + c2z2 +... is an analytic function with positive real part in U, then |c2 −νc2 1| ≤ …
Lemma 1. (Ma and Minda [21]) If p(z) = 1 + c1z + c2z2 + ... is an analytic function with positive real part in U, then |c2 −νc2 1| ≤ −4ν + 2 if ν ≤0, 2 if
Lemma 2.
Lemma 2. (Ma and Minda [21]) If p(z) = 1+c1z +c2z2 +... is an analytic function with positive real part in U, then for any complex number µ…
Lemma 2. (Ma and Minda [21]) If p(z) = 1+c1z +c2z2 +... is an analytic function with positive real part in U, then for any complex number µ |c2 −µc2 1| ≤2 max{1, |2µ −1|}. The result is sharp for the function p(z) = 1 + z 1 −z or p(z) = 1 + z2 1 −z2 . Next, we state and prove the following theorem.
Theorem 1.
Theorem 1. Let φ(z) = 1 + B1z + B2z2 + · · ·, where Bk are real with B1 > 0. Let the function f be given by (1) and belongs to the class…
Theorem 1. Let φ(z) = 1 + B1z + B2z2 + · · · , where Bk are real with B1 > 0. Let the function f be given by (1) and belongs to the class Sα,n(m, l, q, λ, φ). Then |a3 −µa2 2| ≤ B2 1(l+q)m 3α(n+1)(n+2)(l+q+2λ)m + B2(l+q)m
Theorem 2.
Theorem 2. If σ1 ≤µ ≤σ2, then in view of Lemma 1, Theorem 1 can be im- proved. Let σ3 be given by σ3 = 2α(n + 1)(l + q + λ)2m[B2 + B2 1]…
Theorem 2. If σ1 ≤µ ≤σ2, then in view of Lemma 1, Theorem 1 can be im- proved. Let σ3 be given by σ3 = 2α(n + 1)(l + q + λ)2m[B2 + B2 1] 3αB2 1(l + q)m(l + q + 2λ)m(n + 2). If σ1 ≤µ ≤σ3, then |a3 −µa2 2| + 2α(n + 1)(l + q + λ)2m 3αB2 1(l + q)m(l + q + 2λ)m(n + 2) B1 −B2 + (3αB2 1(l + q)m(l + q + 2λ)m(n + 2) + (n + 1)(l + q + λ)2mB2
Corollary 1.
Corollary 1. Let −1 ≤B < A ≤1. Let the function f be given by (1) and belongs to the class Sα,n(m, l, q, λ, (1 + Az)/(1 + Bz)). Then |a3…
Corollary 1. Let −1 ≤B < A ≤1. Let the function f be given by (1) and belongs to the class Sα,n(m, l, q, λ, (1 + Az)/(1 + Bz)). Then |a3 −µa2 2| ≤
Corollary 2.
Corollary 2. Let φ(z) = 1 + B1z + B2z2 + · · · and the function f be given by (1) and belongs to the class Sα,n(m, l, q, λ, φ). For a…
Corollary 2. Let φ(z) = 1 + B1z + B2z2 + · · · and the function f be given by (1) and belongs to the class Sα,n(m, l, q, λ, φ). For a complex number µ |a3 −µa2 2| ≤ B1(l + q)m (3α)(n + 1)(n + 2)(l + q + 2λ)m max ( 1, " −B1 − B2 B1 + µ 3α(l + q)m(n + 2)(l + q + 2λ)m 22α(n + 1)(l + q + λ)m
Theorem 2
Theorem 2 Let φ(z) = 1 + B1z + B2z2 + · · ·, where Bk are real with B1 > 0. Let the function f be given by (1) and belongs to the class…
Theorem 2 Let φ(z) = 1 + B1z + B2z2 + · · · , where Bk are real with B1 > 0. Let the function f be given by (1) and belongs to the class Sα,n,g(m, l, q, λ, φ). Then |a3 −µa2 2| ≤
Theorem 3
Theorem 3 Let φ(z) = 1 + B1z + B2z2 +...., where Bk are real with B1 > 0. Let the function f be given by (1) and belongs to the class…
Theorem 3 Let φ(z) = 1 + B1z + B2z2 + ...., where Bk are real with B1 > 0. Let the function f be given by (1) and belongs to the class Sα,n,η(m, l, q, λ, φ). Then |a3 −µa2 2| ≤
Theorem 2
Theorem 2 reduces to the work of Srivastava and Mishra (see [7]) for a class of functions for which Ωηf(z) is a parabolic starlike function…
Theorem 2 reduces to the work of Srivastava and Mishra (see [7]) for a class of functions for which Ωηf(z) is a parabolic starlike function (see [3]). Acknowledgement: The work here is supported by MOHE grant: FRGS/1/2016/STG06/UKM/01/1. 28