THE TILTED CARATHEODORY FUNCTION CLASS AND ITS PRACTICAL APPLICATIONS

In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for the tilted Carathe<acute accent>o dory function of order beta and angle alpha (alpha is an element of (-pi/2, pi/2) and beta is an element of [0, cos alpha)), which maps the unit disk D into the region {w is an element of C : Re{e(i alpha) w} > beta}. Using these conditions, we also derive conditions for an analytic function that maps D into a sector defined by {w is an element of C : | arg(w - gamma)|< (pi/2)delta}, where gamma is an element of [0,1) and delta is an element of (0, 1]. The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in D.