On a subclass of multivalent functions with negative coefficients (Q1885187)

Let \(f(z) = z^p-\sum^\infty_{n=1} a_{n+p}z^{n+p}\), \(a_{n+p} > 0\), \(p = 1,2,\dots,\) be analytic and \(p\)-valent in \(|z| <1\) and let \[ F(z)=\frac{p+c}{z^c}\int^z_0 t^{c-1}f(t)\,dt. \] For \(0 \leq\alpha\leq 1\), \(\beta\geq 0\), \(0 <\gamma < p\), a function \(f\) is in the class \(T(\alpha,\beta,\gamma)\) if and only if \[ \sum^\infty_{n=1}\left(\frac{n+p+1}{p+1}\right)^\beta (n+\beta)(1+\gamma)a_{n+p}\leq 2p\gamma(1-\alpha). \] Theorem: (i) If \(c > -p\) and \(f\in T(\alpha,\beta,\gamma)\) then \(F\in T(\alpha,\beta,\gamma)\). (ii) If \(F\in T(\alpha,\beta,\gamma)\) and \(f(z) = \frac{z^{1-c}}{p+c}[z^c F(z)]'\), \(c = 1, 2,\dots,\) then \(f(z)\) is \(p\)-valent in \(|z|<r_p=\inf_{n\geq 1}[\frac{(1+\gamma)(p+c)}{2\gamma(1-\alpha)(n+p+c)}]^{\frac1n}\).