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On the Bernardi integral operator. (English) Zbl 0991.30501

Srivastava, H. M. (ed.) et al., Current topics in analytic function theory. Singapore: World Scientific. 212-219 (1992).
Summary: Let \(A\) denote the class of functions \(f\), analytic in \(D=\{z:|z|<1\}\) and sucht that \(f(0)=f'(0)-1=0\). For \(f\in A\) and \(-1<c \leq 1\), let \(F_C\) be defined by \[ F_C(z)={c+1\over z^c}\int^z_0 t^{c-1}f(t) dt. \] It is shown that if \(f\in A\) and satisfies \(\text{Re} f'(z)> -0.262\dots\), then \(F_0\) is starlike and that if \(\text{Re} f'(z)> -0.0175,\dots\), then \(F_1\) is starlike. Both results improve previous work by R. Singh and V. Singh. A similar theorem concerning \(F_C\) is obtained.
For the entire collection see [Zbl 0976.00007].

MSC:

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