in this paper‎, ‎we introduce a new class$t_{k}^{s,a}[a,b,alpha‎ ,‎beta ]$ of analytic functions by using a‎ ‎newly defined convolution operator‎. ‎this class contains many known classes of‎ ‎analytic and univalent functions as special cases‎. ‎we derived some‎ ‎interesting results including inclusion relationships‎, ‎a radius problem and‎ ‎sharp coefficient bound for this class‎.

The aim of this paper is to study the subclasses of Janowski functions with bounded boundary and bounded radius rotations of order a. The order of a function from the class of Janowski functions with bounded boundary rotations to be from bounded radius rotations is of major interest and some of its applications are also discussed here. 2000 Mathematics subject classification: 30C45 • 30C50

Let f be a homeomorphism of the closed annulus A that preserves the orientation, the boundary components and that has a lift f̃ to the infinite strip à which is transitive. We show that, if the rotation numbers of both boundary components of A are strictly positive, then there exists a closed nonempty unbounded set B− ⊂ à such that B− is bounded to the right, the projection of B− to A is dense, ...

Consider an analytic function f which has a Siegel disk properly contained in the domain of holomorphy. We prove that if the rotation number is of bounded type, then f has a critical point in the boundary of the Siegel disk.