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Zbl 0997.30009
Liu, Jinlin; Srivastava, H.M.
A linear operator and associated families of meromorphically multivalent functions.
(English)
[J] J. Math. Anal. Appl. 259, No.2, 566-581 (2001). ISSN 0022-247X

Let $\Sigma_p$ denote the class of functions $f(z)$ which are analytic and $p$-valent in the punctured unit disk $${\cal U}^*={\cal U}\setminus\{0\},\quad {\cal U}= \{z:|z|< 1\}.$$ For the given real numbers $a,c-c\not\in\bbfN$ we can define a linear operator $${\cal L}_p(a, c) f(z):= \phi_p(a,c;z)* f(z),\quad(f\in\Sigma_p)$$ 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 $p$, $a$, $c$, $A$, $B$, $-1\le B< A\le 1$, we say that a function $f\in\Sigma_p$ is in the class ${\cal H}_{a,c}(p;A,B)$ if it also satisfies the inequality $$\Biggl|{z({\cal L}_p(a,c) f(z))'+ p{\cal L}_p(a,c) f(z)\over Bz({\cal L}_p(a, c) f(z))'+ Ap{\cal L}_p(a, c) f(z)}\Biggr|< 1\quad\text{for }z\in{\cal U}.$$ In this paper some properties of the classes ${\cal H}_{a,c}(p;A,B)$ and the operators ${\cal L}_p(a,c)f$ are investigated. Among others it is proved: Theorem. If $a\ge {p(A-B)\over B+1}$, then ${\cal H}_{a+1,c}(p;A,B)\subset{\cal H}_{a,c}(p;A,B)$.
[Jan Stankiewicz (Rzeszów)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions
30C50 Coefficient problems for univalent and multivalent functions

Keywords: meromorphic function; multivalent; Hadamard product

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