On certain classes of p-valent functions (Q5919226)

Let \(K^*_{n+p-1}\) be a class of p-valent functions of the form \(f(z)=z^ p-\sum^{\infty}_{k=1}a_{p+k}z^{p+k},\) \(a_{p+k}\geq 0\), for \(k=1,2,...\), and such that \(Re(D^{n+p}f(z)/D^{n+p-1}f(z))>\) for \(| z| <1\), where \[ D^{n+p-1}f(z)=n^ p(z^{n- 1}f(z))^{(n+p-1)}/(n+p-1)!=z^ p(1-z)^{-n-p}*f(z). \] In the paper some properties of the class \(K^*_{n+p-1}\) are investigated. In particular, the modified Hadamard product * is introduced \[ f*g(z)=z^ p-\sum^{\infty}_{k=1}a_{p+k}b^{p+k}_{p+k} \] and the following result is given: If \(f_ j\in K^*_{n+p-1}\), \(j=1,2,...,m\), then \(f_ 1*f_ 2*...*f_ m\in K^*_{n+p-1}\), \(n=\min \{n_ 1,n_ 2,...,n_ m\}\).