Subclasses of meromorphic functions with missing coefficients (Q1330814)

Let \(\Omega\) denote the family of analytic functions of the form \(f(z)= {1\over z}+ \sum_{n=1}^ \infty a_ n z^ n\) \((0<z<1)\). Let \(\Omega(p)\) denote the family of such functions that satisfy the conditions \(a_ j=0\) \((0\leq j\leq p-1)\) and \(\Omega^ + (p)\) denote the subfamily of \(\Omega(p)\) each of which satisfies the conditions \(a_{p+n} >0\) \((n=0,1,2,\dots)\). Moreover let \(\Omega(p, \alpha, \beta, \gamma)\) denote the subfamily of \(\Omega(p)\) each of which satisfies the condition \[ \biggl| {{z^ 2 f'(z)+1} \over {\mu z^ 2 f'(z)-1 +(1+\mu) \alpha}} \biggr| <\beta \qquad (0\leq\alpha <1,\;0<\beta \leq 1,\;0\leq \mu<1) \qquad \text{ and } \] \[ \Omega^ + (p, \alpha, \beta, \gamma)= \Omega^ +(p) \cap \Omega(p, \alpha, \beta, \gamma). \] The authors show that a sufficient condition for \(f(z)\) to belong to \(\Omega (p, \alpha, \beta, \mu)\) is \[ \sum_{n=0}^ \infty (p+n) [1+ \mu\beta ]| a_{p+n} |\leq \beta(1- \alpha) (1+ \mu). \] Moreover the distortion theorem and the radius of starlikeness or convexity of \(f(z)\in \Omega^ + (p, \alpha, \beta, \gamma)\) are derived.