Perturbation of the tangential slit by conformal maps

For a tangential slit, the behavior of the driving function in the Loewner differential equation is less clear. In this paper, we investigate the tangential slit φ(Γ), where φ is a univalent real analytic function near the origin, and where Γ is a circular arc tangent at the origin. Our main aim is to give an interesting way to prove the asymptotic property of the driving function which generates the tangential slit φ(Γ).