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

Let ${{\Sigma }}_{p}$ denote the class of functions $f\left(z\right)$ which are analytic and $p$-valent in the punctured unit disk

${𝒰}^{*}=𝒰\setminus \left\{0\right\},\phantom{\rule{1.em}{0ex}}𝒰=\left\{z:|z|<1\right\}·$

For the given real numbers $a,c-c\notin ℕ$ we can define a linear operator

${ℒ}_{p}\left(a,c\right)f\left(z\right):={\phi }_{p}\left(a,c;z\right)*f\left(z\right),\phantom{\rule{1.em}{0ex}}\left(f\in {{\Sigma }}_{p}\right)$

where $*$ is a convolution (Hadamard product) and ${\phi }_{p}\left(a,c;z\right)$ is a special function defined as follows

${\phi }_{p}\left(a,c;z\right):={z}^{-p}+\sum _{k=1}^{\infty }\frac{{\left(a\right)}_{k}}{{\left(c\right)}_{k}}{z}^{k-p}·$

For the given fixed parameters $p$, $a$, $c$, $A$, $B$, $-1\le B, we say that a function $f\in {{\Sigma }}_{p}$ is in the class ${ℋ}_{a,c}\left(p;A,B\right)$ if it also satisfies the inequality

$\left|\frac{z{\left({ℒ}_{p}\left(a,c\right)f\left(z\right)\right)}^{\text{'}}+p{ℒ}_{p}\left(a,c\right)f\left(z\right)}{Bz{\left({ℒ}_{p}\left(a,c\right)f\left(z\right)\right)}^{\text{'}}+Ap{ℒ}_{p}\left(a,c\right)f\left(z\right)}\right|<1\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}z\in 𝒰·$

In this paper some properties of the classes ${ℋ}_{a,c}\left(p;A,B\right)$ and the operators ${ℒ}_{p}\left(a,c\right)f$ are investigated. Among others it is proved: Theorem. If $a\ge \frac{p\left(A-B\right)}{B+1}$, then ${ℋ}_{a+1,c}\left(p;A,B\right)\subset {ℋ}_{a,c}\left(p;A,B\right)$.

##### MSC:
 30C45 Special classes of univalent and multivalent functions 30C50 Coefficient problems for univalent and multivalent functions