Abstract
Let w(z) be regular in the unit disc U and let h(r, s, t) be a complex function defined in a domain of C3. The authors determine conditions on h such that |h(w(z), zw′(z), z2w″(z))| <1 implies |w(z)| <1 and such that Re h(w(z), zw′(z), z2w″(z)) >0 implies Re w(z) >0. Applications of these results to univalent function theory, differential equations and harmonic functions are given.
This work was carried out while the first author was a U.S.A. — Romania Exchange Scholar.
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Miller, S.S., Mocanu, P.T. (1979). Second order differential inequalities in the complex plane. In: Cazacu, C.A., Cornea, A., Jurchescu, M., Suciu, I. (eds) Romanian-Finnish Seminar on Complex Analysis. Lecture Notes in Mathematics, vol 743. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0079503
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DOI: https://doi.org/10.1007/BFb0079503
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