Abstract
The class R--,(A, B) for -1~B < A~1 and'Y > (A - 1)/(1 - B)
consisting of normalised analytic functions in the open unit disc is defined with the
help of Convolution technique. It consists of univalent starlike functions for "Y~0.
We establish containment property, integral transforms and a sufficient condition
for an analytic function to be in R--,(A,B). Using the concept of dual spaces we
find a convolution condition for a function in this class.
Results & Lemmas (8)
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Theorem 2.1.
Theorem 2.1. /七+dA, !J) C R-y(A, /3) holds for -1 < B < A < 1 and, >.. (A - 1)/(1 - B)
Theorem 2.1. /七+dA, !J) C R-y(A, /3) holds for -1 < B < A < 1 and, > .. (A - 1)/(1 - B)
Theorem 3.1.
Theorem 3.1. If g(z) is in 凡(A, B), 1;::: 0 and G(z) is defined by G(z) = (1 + c)丨 uc-lg(uz)du, for c > (A - 1)/(1- B) (3.1) 。 then G(z) is…
Theorem 3.1. If g(z) is in 凡(A, B), 1;::: 0 and G(z) is defined by G(z) = (1 + c)丨 uc-lg(uz)du, for c > (A - 1)/(1- B) (3.1) 。 then G(z) is in R-y(A, B).
Corollary 3.
Corollary 3.l(a). If g(z) is in Rn(l, -1), then the function G(z) defined by (3.1) is also in 瓦(1,-1). Again putting A = 1 2(3(0~(3 < 1) B…
Corollary 3.l(a). If g(z) is in Rn(l, -1), then the function G(z) defined by (3.1) is also in 瓦(1,-1). Again putting A = 1 2(3(0~(3 < 1) B = -1 and = 1 1 - 2a m Theorem 3.1 we obtain the following corollary that seerns to be a new result for the class R1-20(l - 頤，-1).
Corollary 3.
Corollary 3.l(b). Let g(z) be in R1-20(l - 頤，-1) then G(z) defined by 3.1 is in R 辶2a(l - 讎，-1). The Theorem 3.1 can be further…
Corollary 3.l(b). Let g(z) be in R1-20(l - 頤，-1) then G(z) defined by {3.1} is in R 辶2a(l - 讎，-1). The Theorem 3.1 can be further strengthened if one takes , and c to be non negative integers. Thus we obtain.
Theorem 3.2.
Theorem 3.2. lfg(:) is in h (A, 13) and G(:) is defined by G(z) = (11 + I) /1 u1L 一1y(-u:)c/u, n=0,1,2, 。 (3.10)
Theorem 3.2. lfg(:) is in h\(A, 13) and G(:) is defined by G(z) = (11 + I) /1 u1L 一1y(-u:)c/u, n=0,1,2, 。 (3.10)
Corollary 3.2.
Corollary 3.2. If g(z) is in R11(0) then G(z) defined by 3.10 is in R11+1(a).
Corollary 3.2. If g(z) is in R11(0) then G(z) defined by {3.10} is in R11+1(a).
Theorem 4.1.
Theorem 4.1. For a function f(z) = z + L~:i anzn in H such that if for some real number 1 2'.: 0 and A, B with 一1:SB< A:S 1, 立n + 1) + IA -…
Theorem 4.1. For a function f(z) = z + L~:i anzn in H such that if for some real number 1 2'.: 0 and A, B with 一1 :SB< A :S 1, 立n + 1) + IA - Bnl} C('Y, n)lanl~A - B holds, n=2 (4.1)
Theorem 5.1.
Theorem 5.1. A function f(z) in JI is in the class R-y(A, B) (1~0) if and only if Jl.,(A, B) is the dual of 瓦(A,B).
Theorem 5.1. A function f(z) in JI is in the class R-y(A, B) (1~0) if and only if Jl.,(A, B) is the dual of 瓦(A,B).
Function classes studied:
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