A Generalized Class of Close-to-Convex Functions
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Authors: P. KAUR AND S. S. BILLING
DOI: 10.46793/KgJMat2004.533K
Abstract:
Let ℋαϕ(β) denote the class of functions f, analytic in the open unit disk ???? which satisfy the condition
where α, β are pre-assigned real numbers and ϕ(z) is a starlike function. The special cases of the class ℋαϕ(β) have been studied in literature by different authors. In 2007, Singh et al. [?] studied the class ℋαz(β) and they established that functions in ℋαz(β) are univalent for all real numbers α, β satisfying the condition α ≤ β < 1 and the result is sharp in the sense that constant β cannot be replaced by a real number smaller than α. Singh et al. [?] in 2005, proved that for 0 < α < 1 functions in class ℋαz(α) are univalent. In 1975, Al-Amiri and Reade [?] showed that functions in class ℋαz(0) are univalent for all α ≤ 0 and also for α = 1 in ????. In the present paper, we prove that members of the class ℋαϕ(β) are close-to-convex and hence univalent for real numbers α, β and for a starlike function ϕ satisfying the condition β + α − 1 < αℜ
≤ β < 1.
Keywords:
Analytic function, univalent function, close-to-convex function
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