Complex linear combinations of n functions in \(S^ *(A,B)\) (Q1092196)

Let A and B, \(-1<A\leq 1\), \(-1\leq B<A\), be arbitrary real numbers. Denote by P(A,B) the class of functions \(p(z)=1+p_ 1z+..\). holomorphic in the unit disc K such that \[ p(z)=[1+A\omega (z)]/[1+B\omega (z)] \] where \(\omega\), \(\omega (0)=0\), is holomorphic in K and \(| \omega (z)| <1\). Next, let \(S^*(A,B)\) denote the class of functions f, \(f(0)=0\), \(f'(0)=1\), holomorphic in K and such that \(zf'(z)/f(z)\in P(A,B).\) In the paper the authors determine radii of starlikeness and radii of convexity of linar combinations \[ \gamma_ 1f_ 1(z)+...+\gamma_ nf_ n(z),\quad \sum^{n}_{j=1}\gamma_ j=1, \] of functions \(f_ 1,...,f_ n\) in \(S^*(A,B)\) under the condition \[ 0\leq \max [\arg (\gamma_ i/\gamma_ j)]\leq \pi,\quad 1\leq i,j\leq n. \]