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On integral operators. (English) Zbl 0934.30007

Let \(f_n(z)= z/(1- z)^{n+ 1}\), \(n\in N_0\), and \(f^{(-1)}_n\) be defined such that \(f_n* f^{-1}_n= {z\over 1-z}\), where \(*\) denotes convolution (Hadamard product). Let \(f\) be analytic in the unit disc \(E\). The authors introduce a new operator \(I_nf= f^{(-1)}_n* f\) which is analogous to one defined by Ruscheweyh. Using this operator, the classes \(M^*_{(n)}\) are defined. A function \(f\), analytic in \(E\), is in \(M^*_{(n)}\) if and only if \(I_nf\) is close-to-convex. The properties of \(f\in M^*_{(n)}\) are discussed in some detail. It is shown that \(M^*_{(n)}\subset M^*_{(n+ 1)}\) for \(f\in N_0\) and for \(n= 0,1\), \(M^*_{(n)}\) consists entirely of univalent functions. Closure properties of some integral operators defined on \(M^*_{(n)}\) are also given.
For these proofs of the main results, see lemmas (2,3), (3,1) and theorems (3,1), (3,5), the authors used the very known and efficient “admissible function method” which was introduced by S. S. Miller and P. T. Mocanu, and also Jack’s lemma which is a particular case of this method.

MSC:

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

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