On differential subordinations for a class of analytic functions defined by a linear operator (Q1777686)

Let \({\mathcal A}\) be the class of functions \(f(z)=z+\sum_{k=2}^\infty a_kz^k\) that are analytic in the unit disc \({\mathcal U}=\{z:| z| <1\}.\) In this paper the authors study the following linear operator \[ L(a,c)f(z) := z+ \sum_{n=1}^\infty \frac{(a)_n}{(c)_n}\;a_{n+1}\;z^{n+1} \] and give sufficient conditions for functions to satisfy the subordinations \[ \frac{L(a,c)f(z)}{L(a+1,c)f(z)}\prec q(z), \quad\left( \frac{L(a,c)f(z)}{L(a+1,c)f(z)}\right)^\beta \prec q(z), \] and \[ \left( \frac{L(a,c)f(z)}{z}\right)^\beta \prec q(z), \quad \frac{z}{L(a+1,c)f(z)}\prec q(z). \] Here \(f(z), q(z)\in{\mathcal A},\) \(``\prec"\) denotes the usual subordination and \((\chi)_n\) is the shifted factorial. Also, some applications of the obtained results are given and comparison with previous known results is done.