Let $G\subset {\bf SU}(2,1)$ be a non-elementary complex hyperbolic Kleinian group. If $G$ preserves a complex line, then $G $ is $\mathbb {C}$-Fuchsian; if $ G $ preserves a Lagrangian plane, then $ G $ is $\mathbb {R}$-Fuchsian; $ G $ is Fuchsian if $ G $ is either $\mathbb {C}$-Fuchsian or $\mathbb {R}$-Fuchsian. In this paper, we prove that if the traces of all elements in $ G $ are real, then $ G $ is Fuchsian. This is an analogous result of Theorem V.G. 18 of B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988, in the setting of complex hyperbolic isometric groups. As an application of our main result, we show that $ G $ is conjugate to a subgroup of ${\bf S}(U(1)\times U(1,1))$ or ${\bf SO}(2,1)$ if each loxodromic element in $G $ is hyperbolic. Moreover, we show that the converse of our main result does not hold by giving a $\mathbb {C}$-Fuchsian group.
We consider the class H0 of sense-preserving harmonic functions f = h + \bar g defined in the unit disk |z| < 1 and normalized so that h(0) = 0 = h′(0) − 1 and g(0) = 0 = g′(0), where h and g are analytic in the unit disk. In the first part of the article we present two classes PH0(α) and GH0(β) of functions from H0 and show that if f \in PH0(α) and F \in GH0(β), then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections (partial sums) {s_{n,n}}\left( f \right)\left( z \right) = {s_n}\left( h \right)\left( z \right) + \overline {{s_n}\left( g \right)\left( z \right)} , where f = h + \bar g \in H0, sn(h) and sn(g) denote the n-th partial sums of h and g, respectively. We prove, among others, that if f = h + \bar g \in H0 is a univalent harmonic convex mapping, then sn,n(f) is univalent and close-to-convex in the disk |z| < 1/4 for n ≥ 2, and sn,n(f) is also convex in the disk |z| < 1/4 for n ≥ 2 and n ≠ 3. Moreover, we show that the section s3,3(f) of f \in CH0 is not convex in the disk |z| < 1/4 but it is convex in a smaller disk., Liulan Li, Saminathan Ponnusamy., and Obsahuje seznam literatury