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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.

MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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