### 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 ℋ_{α}^{ϕ}(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 ℋ_{α}^{ϕ}(β) have been studied in literature by diﬀerent authors. In 2007, Singh et al. [?] studied the class ℋ_{α}^{z}(β) and they established that functions in ℋ_{α}^{z}(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 ℋ_{α}^{z}(α) are univalent. In 1975, Al-Amiri and Reade [?] showed that functions in class ℋ_{α}^{z}(0) are univalent for all α ≤ 0 and also for α = 1 in ????. In the present paper, we prove that members of the class ℋ_{α}^{ϕ}(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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