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
Our main objective was to test whether chronic orthostatic body position induces network changes in the saphenous vein superficial tributary system of the rat. Fourteen male Sprague-Dawley rats were kept in tilted tube cages (45º head-up position) for two weeks to induce chronic gravitational load to their leg veins. Ten animals housed in normal cages and four animals kept in horizontally positioned tube cages served as controls. The whole superficial network of the left saphenous vein was microprepared surgically under anesthesia, superfused with saline and observed under a videomicroscope, while normal flow and pressure were maintained in the lumen. Branching angles, lengths of venous segments and their diameters were measured offline from digitized images using special image-analyzing software. Several branching angles at the popliteal confluence were significantly reduced by 12.5-15.8 %. The in vivo diameter of the main branch (936±34 vs. 805±44 µm) and of one of the popliteal tributaries (776±38 vs. 635±36 µm) increased (p<0.05), comparing vessels from tilted animals with those from normal controls. Maintaining the animals in horizontal tube cages did not induce the above alterations. The increased diameters and reduced branching angles of the saphenous vein network observed are adaptive responses of the venous network to a long-term gravitational load., M. Lóránt, G. L. Nádasy, G. Raffai, E. Monos., and Obsahuje bibliografii