On certain classes of \(p\)-valent analytic functions (Q1210460)

The author introduces the class \(P(p,\alpha)\) \((0\leq \alpha < p)\) \((p=1,2,\dots)\) consisting of functions \(f(z)\) of the form (1) \(f(z)=z^ p+\sum^ \infty_{n=1} a_{n+p} z^{n+p}\cdots\) which are analytic and satisfying \(\text{Re}\bigl[ f'(z)/z^{p-1}\bigr]>\alpha\), in the open unit disc of the complex plane. On the other hand \(f(z)\) of the form (1) is said to belong to \(P(p,\alpha,\beta)\) if \((p+1)^{-\beta} D^ \beta f\in P(p,\alpha)\) \((\beta\geq 0)\). In this paper the author obtains some inclusion theorems for the various classes \(P(p,\alpha,\beta)\) and the effect of certain integral operators on these families. For example it is proved that if \(f\in P(p,\alpha,\beta)\) then \((1+p)^ \delta I^ \delta f(z)\) also belongs to \(P(p,\alpha,\beta)\), where \[ I^ \delta f(z)={1\over \Gamma(z)} \int^ 1_ 0 \bigl(\log(1/t)\bigr)^{\delta-1} f(zt)dt\quad (\delta>0). \] {}.