For normalized analytic functions in the unit disk, we consider subclasses of bounded turning. The geometric representation is introduced, the radii of convexity (close to convex) are calculated and some subordination relations are suggested. Moreover, the upper bound of the pre-Schwarzian norm for these functions is computed. AMS subject classifications: 30C45

We study C2-structural stability of interval maps with negative Schwarzian. It turns out that for a dense set of maps critical points either have trajectories attracted to attracting periodic orbits or are persistently recurrent. It follows that for any structurally stable unimodal map the ω-limit set of the critical point is minimal.

Higher-order degenerated versions of Fay’s trisecant identity are presented. It is shown that these lead to solutions for Schwarzian Kadomtsev–Petviashvili equations.

For every positive integer r, we solve the modular Schwarzian differential equation {h,τ}=2π2r2E4, where E4 is weight 4 Eisenstein series, by means of equivariant functions on upper half-plane. This paper supplements previous works authors [20], [21], same has been solved for infinite families rational values r. also leads to solutions y″+r2π2E4y=0 These are quasi-modular forms SL2(Z) if r even...