1. Introduction. A Riemann surface S is called symmetric if there exists an anti-conformal map 0 of S onto itself such that <p2 = identity. We say that 0 is a symmetry on S. The classical "retrospection theorem" asserts the existence of representations of closed Riemann surfaces of genus g by "Schottky groups," groups generated by Möbius transformations Ax,-,Ag such that A; maps the exterior of...

For α ≥ 0, λ > 0, we consider theM α, λ b of normalized analytic α−λ convex functions defined in the open unit disc U. In this paper, we investigate the classM α, λ b, that is, Re{ zf ′ b z /fb z 1− α α 1 − λ zf ′ b z /fb z αλ 1 zf ′′ b z /f ′ b z } > 0, with fb is Koebe type, that is, fb z : z/ 1−zn . The subordination result for the aforementioned class will be given. Further, bymaking use of...

Using Mazur’s theorem on torsions of elliptic curves, an upper bound 24 for the order finite Galois group ℋ associated with weighted walks in quarter plane ℤ + 2 is obtained. The explicit criterion to have 4 or 6 rederived by simple geometric arguments. division polynomials, a recursive 4m 4m+2 also As corollary, 8 given through method simpler than existing one.