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In this paper, we generalize the main results of [Effros EG, (2009) Proc Natl Acad. Sci USA 106:1006–1008]. Namely, we provide the necessary and sufficient conditions for jointly convexity of perspective functions and generalized perspective functions.

Assume that ∆ is the open unit disk in the complex plane and A is the class of normalized analytic functions in ∆. In this paper we introduce and study the classwhere 0 ≤ α ≤ 1 and ≺ is the subordination relation. Some properties of this class like differential subordination, coefficients estimates and Fekete-Szegö inequality associated with the k-th root transform are considered.2000 Mathematics Subject Classification. 30C45.

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We consider a family of analytic and normalized functions with the property that zf (z)/f (z) (or 1 + zf (z)/f (z)) lies in a domain bounded by a right branch of a hyperbola ρ = ρ(s) = (2 cos ϕ s)-s , where 0 < s ≤ 1 and |ϕ| < (π s)/2. A comprehensive characteristic of that families and relations with the well known families of univalent functions are presented. Some relevant examples are indicated.

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