*(English)*Zbl 0924.30008

G. Sălăgean introduced in 1983 an iterate differential operator:

and used it to define the classes ${S}_{n}\left(\alpha \right)$ of functions called “$n$-starlike of order $\alpha $”. A function $f\left(z\right)=z+{a}_{2}{z}_{2}+\cdots $ is said to belong to the class ${S}_{n}\left(\alpha \right)$ if it satisfies

for some $\alpha (0\le \alpha <1)$ and $n\in {N}_{0}$.

Many authors have also used this operator to study several sets of univalent functions defined in the open unit disk.

This paper reveals a new sufficient condition so that a function should belong to ${S}_{n}\left(\alpha \right)$ which generalizes earlier results obtained by Sălăgean, Owa, Shen and Obradović. The well known Jack’s lemma is used in the proof. In the second part the authors improve a result given by Uralegaddi for the functions $n$-starlike of order $\alpha $ and are obtaining other new properties which included several known results. All results of this paper have been obtained by using the above mentioned operator.

##### MSC:

30C45 | Special classes of univalent and multivalent functions |