The study of Univalent function has a long history and is amongst the most fundamental questions in complex function theory. Nehari studies oscillation of second order ODE and Schwarzian derivatives. His research is initialed by the equation

d²y/dz² + p(z)y = 0.

In 1949, he gave a proof of necessary criteria and sufficient criteria of univalence, │p(z)│ ≤ 6/(1−z²)² and │p(z)│≤ 2/(1−z²)² respectively. In the same year, Hille gave an example showing that it is necessary and sufficient that p(z)(1 − z²)² is at the interior or on the boundary of the cardioid

A = −2e^i∅ − e^2i∅, −π ∅ ≤ π.

Hille’ result concides with Nehar’s only on the real axis of A-plane. Hille’s example has complex monodromy. We follow the study of univalent Schwarz map of hypergeometric function defined on unit disk with...[ Read more ]

The study of Univalent function has a long history and is amongst the most fundamental questions in complex function theory. Nehari studies oscillation of second order ODE and Schwarzian derivatives. His research is initialed by the equation

d²y/dz² + p(z)y = 0.

In 1949, he gave a proof of necessary criteria and sufficient criteria of univalence, │p(z)│ ≤ 6/(1−z²)² and │p(z)│≤ 2/(1−z²)² respectively. In the same year, Hille gave an example showing that it is necessary and sufficient that p(z)(1 − z²)² is at the interior or on the boundary of the cardioid

A = −2e^i∅ − e^2i∅, −π < ∅ ≤ π.

Hille’ result concides with Nehar’s only on the real axis of A-plane. Hille’s example has complex monodromy. We follow the study of univalent Schwarz map of hypergeometric function defined on unit disk with complex parameters. Two criteria for univalence and non-univalence are derived.

Before study of hypergeometric function, we strengthen Hille’s example that his Schwarz map gives one-to-one, finitely many-to-one, and infinitely-to-one mapping depending on the value of A relative to the cardioid curve introduced above.