On certain class of analytic functions defined by differential subordination (Q701572)

Denote by \(B_n(\lambda, \alpha,A,B)\) a class of analytic functions \(f(z)=z+ \sum^\infty_{k=n+1} a_kz^k\) in the unit disk \(U\) which satisfy the subordination relation \[ (1-\lambda) \bigl(f(z)/z)^\alpha +\lambda zf'(z)/f(z) \bigl(f(z)/z \bigr)^\alpha \prec(1+Az)/(1+Bz) \] where \(\text{Re} \alpha>0\), \(-1\leq B\leq 1\), \(A=B\), \(A\) is a real and \(\lambda\) is a complex number. For functions of this classes, the author proves subordination relations, results on inclusion, inequality properties and covering theorems. The proofs are based on the differential subordination method see \textit{S. S. Miller} and \textit{P. T. Mocanu} [Mich. Math. J. 28, 157-171 (1981; Zbl 0439.30015)]. The results are sharp and some of them generalize the related works of some authors.