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Properties of the Salagean operator. (English) Zbl 0924.30008
G. Sălăgean introduced in 1983 an iterate differential operator: $$D^0f(z) = f(z),\quad D'f(z) = zf(z), D^nf(z) = D(D^{n-1}f(z))$$ and used it to define the classes $S_n(\alpha)$ of functions called “$n$-starlike of order $\alpha$”. A function $f(z) = z + a_2z_2+\dots$ is said to belong to the class $S_n(\alpha)$ if it satisfies $$\text{Re} \left\{\frac{D^{n+1}f(z)}{D^nf(z)}\right\}>\alpha, \quad z\in U$$ for some $\alpha(0\leq \alpha < 1)$ 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 $S_n(\alpha)$ 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.

30C45Special classes of univalent and multivalent functions