Let L(f, r) denote the length of the closed curve which is the image of |z| = r < 1 under the mapping w = f(z). We establish some sufficient conditions for L(f, r) to be bounded and for f(z) to in the classes of strongly close-to-convex function of order α and to be strongly Bazilevič function of type β of order α. Moreover, we prove an inequality connected with the Fejér-Riesz's inequality.
Let ƒ be analytic in the unit disk B and normalized so that ƒ (z) = z + a2z2 + a3z3 +܁܁܁. In this paper, we give upper bounds of the Hankel determinant of second order for the classes of starlike functions of order α, Ozaki close-to-convex functions and two other classes of analytic functions. Some of the estimates are sharp.