Certain subclasses of multivalent analytic functions (Q5906946)

Let \(A_p\) denote the class of functions of the form \(f(z)=z^p+ \sum^\infty_{k=1} a_{p+k}z^{p+k}\) which are analytic in the open unit disc \(E\) of the complex plane. For a positive integer \(n\) and \(p\geq 1\) define \[ \varphi_n (p,z)=\sum^\infty_{k=1} {(1-p)^n\over(k+p)^n}z^k\;(z\in E) \] and \[ D^nf(z)= f(z)*z^{p-1} \varphi_n(p,z) \] where \(f\in A_p\) and \(*\) denotes the Hadamard convolution. For \(n\) any integer and \(-1\leq B<A\leq 1\), we say \(f\in S_{n,p} (A,B)\) if \(f\in A_p\) and satisfies the condition that \[ {z\bigl(D^n f(z) \bigr)'\over D^nf(z)} \quad\text{is subordinate to}\quad p{1+Az\over 1+Bz} \quad \text{in }E. \] In this paper the authors obtain certain properties of the class \(S_{n,p}(A,B)\) using techniques of differential subordination.