THESIS
2021
1 online resource (vii, 68 pages) : illustrations
Abstract
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
d
^{2}y/dz
^{2} + p(z)y = 0.
In 1949, he gave a proof of necessary criteria and sufficient criteria of univalence, │p(z)│ ≤ 6/(1−z
^{2})
^{2} and │p(z)│≤ 2/(1−z
^{2})
^{2} respectively. In the same year, Hille gave an example showing that it is necessary and sufficient that p(z)(1 − z
^{2})
^{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
d
^{2}y/dz
^{2} + p(z)y = 0.
In 1949, he gave a proof of necessary criteria and sufficient criteria of univalence, │p(z)│ ≤ 6/(1−z
^{2})
^{2} and │p(z)│≤ 2/(1−z
^{2})
^{2} respectively. In the same year, Hille gave an example showing that it is necessary and sufficient that p(z)(1 − z
^{2})
^{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.
Post a Comment