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A linear operator and associated families of meromorphically multivalent functions. (English) Zbl 0997.30009

Let Σ p denote the class of functions f(z) which are analytic and p-valent in the punctured unit disk

𝒰 * =𝒰{0},𝒰={z:|z|<1}·

For the given real numbers a,c-c we can define a linear operator

p (a,c)f(z):=φ p (a,c;z)*f(z),(fΣ p )

where * is a convolution (Hadamard product) and φ p (a,c;z) is a special function defined as follows

φ p (a,c;z):=z -p + k=1 (a) k (c) k z k-p ·

For the given fixed parameters p, a, c, A, B, -1B<A1, we say that a function fΣ p is in the class a,c (p;A,B) if it also satisfies the inequality

z( p (a,c)f(z)) ' +p p (a,c)f(z) Bz( p (a,c)f(z)) ' +Ap p (a,c)f(z)<1forz𝒰·

In this paper some properties of the classes a,c (p;A,B) and the operators p (a,c)f are investigated. Among others it is proved: Theorem. If ap(A-B) B+1, then a+1,c (p;A,B) a,c (p;A,B).

30C45Special classes of univalent and multivalent functions
30C50Coefficient problems for univalent and multivalent functions