Close-to-convex functions of complex order related to Ruscheweyh derivative (Q1910387)

Let \(A\) denote the class of functions \(f\) analytic in the unit disk \(E=\{z:|z|<1\}\) of the form \(f(z)=z+\sum^\infty_{k=2} a_kz^k\), \(f\) is said to be in the class \(R_n\) if \(\text{Re}\{{z(D^nf(z))'\over D^nf(z)}\}>0\), where \(D_n\) is the Ruscheweyh derivative operator defined by \[ D^nf(z)={z\over (1-z)^{n+1}}* f(z), \] denoting convolution. In this paper, a class \(K^b_n(A,B)\) of functions \(f\in A\) is introduced satisfying for \(z\in E\), \[ 1+{1\over b} \Biggl\{{z(D^nf(z))'\over D^ng(z)}-1\Biggr\}\prec {1+Az\over 1+Bz} \] for some \(g\in R_n\), \(\prec\) denoting subordination, \(b\neq 0\), any complex number and \(A\), \(B\) satisfying \(-1\leq B<A\leq 1\). Various properties of this class are studied.