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Differential sobordination and Bazilevič functions

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Abstract

LetM(z)=z n +…,N(z)=z n +… be analytic in the unit disc Δ and let λ(z)=N(z)/zN′(z). The classical result of Sakaguchi-Libera shows that Re(M′(z)/N′(z))<0 implies Re(M(z)/N(z))>0 in Δ whenever Re(λ(z))>0 in Δ. This can be expressed in terms of differential subordination as follows: for anyp analytic in Δ, withp(0)=1,p(z)+λ(z)zp′(z)<1+z/1−z impliesp(z)<1+z/1−z, for Reλ(z)>0,z∈Δ.

In this paper we determine different type of general conditions on λ(z),h(z) and ϕ(z) for which one hasp(z)+λ(z)zp′(z)<h(z) impliesp(z)<ϕ(z)<h(z) z∈Δ. Then we apply the above implication to obtain new theorems for some classes of normalized analytic funotions. In particular we give a sufficient condition for an analytic function to be starlike in Δ.

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  1. S Ponnusamy

    Present address: Department of Mathematics, University of Helsinki, Hallituskatu 15, PO Box 4, FIN 00014, Helsinki, Finland

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  1. School of Mathematics, SPIC Science Foundation, 92, G. N. Chetty Road, 600 017, Madras, India

    S Ponnusamy

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Ponnusamy, S. Differential sobordination and Bazilevič functions. Proc. Indian Acad. Sci. (Math. Sci.) 105, 169–186 (1995). https://doi.org/10.1007/BF02880363

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  • DOI: https://doi.org/10.1007/BF02880363

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