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
We prove two Dyakonov type theorems which relate the modulus of continuity of a function on the unit disc with the modulus of continuity of its absolute value. The methods we use are quite elementary, they cover the case of functions which are quasiregular and harmonic, briefly hqr, in the unit disc., Miloš Arsenović, Miroslav Pavlović., and Seznam literatury