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Angular estimations of certain integral operators. (English) Zbl 0899.30008

Let \(\mathcal A\) denote the class of functions \(f\) analytic in the unit disc \(U=\{z:| z| <1\}\). For \(\mu >0\), \(c\geq 0\) and \(f,g\in {\mathcal A}\), let \[ F(z) := J_{c,\mu}(f)=\left \{ {c+\mu\over z^c} \int_{0}^zt^{c-1}f^{\mu}(t)dt \right \}^{1/\mu} \quad \text{and} \quad G(z) := J_{c,\mu}(g). \] A function \(f\in {\mathcal A}\) is said to be in the class \(S^ *[a,b]\) (\(-1\leq b<a\leq 1\)) if \(zf'(z)/f(z)\) is subordinate to \((1+az)/(1+bz)\) for \(z\in U\). In this paper the authors determined conditions on the parameters \(\beta \in [0,1)\), \(\delta \in (0,1]\) and \(\eta\) such that \[ \left | \left ( {zf'(z)f^{\mu-1}(z) \over g^{\mu}(z)} -\beta \right) \right | < {\pi \delta \over 2} \text{ for some } g\in S^ *[a,b] \Longrightarrow \left | \left ( {zF'(z)F^{\mu-1}(z) \over G^{\mu}(z)} -\beta \right) \right | < {\pi \eta \over 2}. \]

MSC:

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