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

Let \(\Sigma_p\) denote the class of functions \(f(z)\) which are analytic and \(p\)-valent in the punctured unit disk \[ {\mathcal U}^*={\mathcal U}\setminus\{0\},\quad {\mathcal U}= \{z:|z|< 1\}. \] For the given real numbers \(a,c-c\not\in\mathbb{N}\) we can define a linear operator \[ {\mathcal L}_p(a, c) f(z):= \phi_p(a,c;z)* f(z),\quad(f\in\Sigma_p) \] where \(*\) is a convolution (Hadamard product) and \(\phi_p(a,c;z)\) is a special function defined as follows \[ \phi_p(a,c;z):= z^{-p}+ \sum^\infty_{k=1} {(a)_k\over (c)_k} z^{k- p}. \] For the given fixed parameters \(p\), \(a\), \(c\), \(A\), \(B\), \(-1\leq B< A\leq 1\), we say that a function \(f\in\Sigma_p\) is in the class \({\mathcal H}_{a,c}(p;A,B)\) if it also satisfies the inequality \[ \Biggl|{z({\mathcal L}_p(a,c) f(z))'+ p{\mathcal L}_p(a,c) f(z)\over Bz({\mathcal L}_p(a, c) f(z))'+ Ap{\mathcal L}_p(a, c) f(z)}\Biggr|< 1\quad\text{for }z\in{\mathcal U}. \] In this paper some properties of the classes \({\mathcal H}_{a,c}(p;A,B)\) and the operators \({\mathcal L}_p(a,c)f\) are investigated. Among others it is proved: Theorem. If \(a\geq {p(A-B)\over B+1}\), then \({\mathcal H}_{a+1,c}(p;A,B)\subset{\mathcal H}_{a,c}(p;A,B)\).


30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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