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A certain family of meromorphically multivalent functions. (English) Zbl 0959.30010
Summary: Let $\Sigma_p$ denote the class of meromorphic $p$-valued functions $f(z)=z ^{-p}+ \displaystyle\sum^\infty _{k=0} a_kz^{k-p+1}$ which are analytic in the punctured disk $U$; $0<|z|\leq 1$. Further let $D^{n+p-1} f(z)$ be the Hadamard product (or convolution) of $f(z) \in \Sigma_p$ with $z^{-p} (1-z)^{-(n+p)}$, $n>-p$, $p$ an integer. A class of meromorphically $p$-valent functions $\Omega_{n,p} (A,B,\alpha)(-1 \leq B<A\le 1$, $0\le\alpha<p)$ in $U$ is defined as a subclass of $\Sigma_p$ for which $-z^{p+1}(D^{n+p-1}f(z))'$ is subordinate to $[p+\{pB+(p-\alpha) (A-B)\}z]/ (1+Bz)$ for $z$ in $U$. This is an extension of a class of functions in $\Sigma_p$ studied by {\it B. A. Uralegaddi} and {\it C. Somanatha} [Tamkang J. Math. 23, No. 3, 223-231 (1992; Zbl 0769.30012)]. The major results relate to the inclusion property of $\Omega_{n,p}(A,B,\alpha)$, coefficient estimates and convexity of the class. It is shown that $\Omega_{n+1,p} (A,B,\alpha) \subset \Omega_{n,p}(A,B,\alpha)$; that if $f(z)$ and $g(z)$ belong to $\Omega_{n,p}(A,B,\alpha)$ and $0\le \beta\le 1$, then $\beta f+(1- \beta)g$ belongs to the class; and that $$|a_k|\leq \frac{(p-\alpha) (A-B)}{(k-p+1) \delta(n,k)},$$ where $\delta(n,k)= {n+p+k \choose n+k}$. This estimate is sharp.

MSC:
30C55General theory of univalent and multivalent functions
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References:
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