Certain classes of analytic functions with negative coefficients. II (Q1089456)

[For part I see the authors in ibid. 17, 1210-1223 (1986; Zbl 0608.30014).] Let T denote the class of functions \(f(z)=a_ 1z- \sum^{\infty}_{m=2}a_ mz^ m\), \(a_ m\geq 0\), \(a_ 1>0\), \(z\in E=\{| z| <1\}\). Let \(S_ n(A,B)\) be the class of functions \(f\in T\) such that \[ D^{n+1}f(z)/D^ nf(z)=(1+Aw(z))/(1+Bw(z)),-1\leq A<B\leq 1,\quad n\in N_ 0=N\cup \{0\}, \] and w(z) is a regular function with \(w(0)=0\) and \(| w(z)| <1\) for \(z\in E\). Here \(D^ nf(z)=z(z^{n-1}f(z))^{(n)}/n!\) which was termed by the reviewer [Ann. Polon. Math. 38, 87-94 (1980; Zbl 0452.30008)] as the ''nth order Ruscheweyh derivative''. Let \(K_ n(A,B)\) denote the class of functions \(f\in T\) such that \(zf'(z)\in S_ n(A,B)\). For a given \(z_ 0\), \(-1<z_ 0<1\), let \(T_ 1\) and \(T_ 2\) denote the subclasses of T satisfying the condition \(f(z_ 0)=z_ 0\) and \(f'(z_ 0)\), respectively. The authors also introduce the subclasses of T denoted by \(S_ i(z_ 0)\) and \(K_ i(z_ 0)\), \(i=1,2\) and defined as follows: \(S_ i(z_ 0)=S_ n(A,B)\cap T_ i\), \(K_ i(z_ 0)=K_ n(A,B)\cap T_ i\), \(i=1,2\). The authors obtain necessary and sufficient conditions for functions to be in \(S_ n(A,B)\), K(A,B), \(S_ i(z_ 0)\), and \(K_ i(z_ 0)\), \(i=1,2\). The sharp radius of convexity is determined for the classes \(S_ i(z_ 0)\), \(i=1,2\). Also, they establish that the necessary and sufficient condition for a subset X of the interval (0,1) to satisfy the property that \(\cup_{j\in X}S_ i(z_ j)\) and \(\cup_{j\in X}K_ i(z_ j)\), \(i=1,2\) form a convex family is that X must be a connected subset of the (0,1)-subinterval. Extreme points for these classes are also obtained. Several known results can be deduced from the author's results as special cases when the proper choices for A,B,n and \(z_ 0\) are made. Some of these results are due to \textit{H. Silverman} [Proc. Am. Math. Soc. 51, 109-116 (1975; Zbl 0311.30007)] and \textit{V. P. Gupta} and \textit{P. K. Jain} [Bull. Austral. Math. Soc. 14, 409-416 (1976; Zbl 0323.30016); ibid. 15, 467-473 (1976; Zbl 0335.30010)].