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Univalence criterion for certain integral operators

Hamada H.

Demonstratio Mathematica

2005

Vol. 38, nr 1

43--46

Pescar investigated the univalence of certain integral operators. We will show that the results are obtained by the Schwarz lemma. We will also give some generalizations.

Linear invariance and integral operators of univalence functions

Dorff M., Szynal J.

Demonstratio Mathematica

2005

Vol. 38, nr 1

47--57

In this paper, we study a larger set than (1), namely the set of the minimal invariant family which contains (1), where / belongs to the linear invariant family, and thereby we obtain information about the univalence of (1). In particular, we determine the order of this minimal invariant family in the cases of univalent and convex univalent functions in D. As a result, we find the radius of close-to-convexity and the lower bound for the radius of univalence for the minimal invariant family in the case of convex univalent functions. This allows us to determine the exact region for (a, (3) where the corresponding minimal invariant family is univalent and close-to-convex. These results are sharp and generalize those which were obtained in [11].