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Properties of the Salagean operator. (English) Zbl 0924.30008

G. Sălăgean introduced in 1983 an iterate differential operator:

${D}^{0}f\left(z\right)=f\left(z\right),\phantom{\rule{1.em}{0ex}}{D}^{\text{'}}f\left(z\right)=zf\left(z\right),{D}^{n}f\left(z\right)=D\left({D}^{n-1}f\left(z\right)\right)$

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

$\text{Re}\left\{\frac{{D}^{n+1}f\left(z\right)}{{D}^{n}f\left(z\right)}\right\}>\alpha ,\phantom{\rule{1.em}{0ex}}z\in U$

for some $\alpha \left(0\le \alpha <1\right)$ 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
##### Keywords:
Sălăgean operator; Jack’s lemma