Abstract
Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α1, β1), g ∈ R(α2, β2) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β1), g ∈ R(0, β2) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles.
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Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.
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Ponnusamy, S., Singh, V. Convolution properties of some classes of analytic functions. J Math Sci 89, 1008–1020 (1998). https://doi.org/10.1007/BF02358538
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DOI: https://doi.org/10.1007/BF02358538