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Abstract. For a non-tangential slit γ(t), the behavior of the driving function λ(t) near zero in the Loewner equation is well understood; for tangential slit, the situation is less clear. In this paper, we investigate the tangential slit Γ p , p > 0, where Γ is a circular arc tangent at 0; Γ p has order p+1 p near zero. Our main result is to give the exact expression of λ(t), and its Hölder exponent near 0 in terms of p, which has a natural connection with the known results. We also extend this to a general type of tangential slits, and give an estimation of the growth of λ(t) near 0.

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For the contractive iterated function system S k z = e 2 π i k / m + ρ ( z − e 2 π i k / m ) S_kz=e^{2\pi ik/m}+{\rho (z-e^{2\pi ik/m})} with 0 > ρ > 1 , k = 0 , ⋯ , m − 1 0>\rho >1, k=0,\cdots , m-1 , we let K ⊂ C K\subset \mathbb {C} be the attractor, and let μ \mu be a self-similar measure defined by μ = 1 m ∑ k = 0 m − 1 μ ∘ S k − 1 \mu =\frac 1m\sum _{k=0}^{m-1}\mu \circ S_k^{-1} . We consider the Cauchy transform F F of μ \mu . It is known that the image of F F at a small neighborhood of the boundary of K K has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate the behavior of F F away from K K ; it has nice geometry and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures.