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 \(\Omega_{n,p} (A,B,\alpha)(-1 \leq B<A\leq 1\), \(0\leq\alpha<p)\) 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 B. A. Uralegaddi and 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 \(0\leq \beta\leq 1\), 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.


30C55 General theory of univalent and multivalent functions of one complex variable


Zbl 0769.30012
Full Text: DOI


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