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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(z)=f(z),D ' f(z)=zf(z),D n f(z)=D(D n-1 f(z))

and used it to define the classes S n (α) of functions called “n-starlike of order α”. A function f(z)=z+a 2 z 2 + is said to belong to the class S n (α) if it satisfies

ReD n+1 f(z) D n f(z)>α,zU

for some α(0α<1) and nN 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 (α) 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 α 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:
30C45Special classes of univalent and multivalent functions