We consider the chordal Loewner differential equation in the upper half-plane,the behavior of the driving functionλ(t)and the generated hull Kt when Kt approachesλ(0)in a fixed direction or in a sector.In the case that the hull Kt is generated by a simple curveγ(t)withγ(0)=0,we prove some sharp relations ofλ(t)/√t andγ(t)/√t as t→0 which improve the previous work.
A uniqueness theorem of a solution of a system of nonlinear equations is given. Using this result uniqueness theorems for power orthogonal polynomials, for a Gaussian quadrature formula of an extended Chebyshev system, and for a Gaussian Birkhoff quadrature formula are easily deduced.
