On some length problems for univalent functions

Let A be the class of functionsf ( z ) = z + ∑ n = 2 ∞a nz n ,which are analytic in the unit disk D = { z : | z | < 1 } . Let C ( r ) be the closed curve that is the image of the circle | z |= r < 1 under the mapping w = f ( z ), L ( r ) the length of C ( r ), and let A ( r ) be the area enclosed by the curve C ( r ). In 1968 D. K. Thomas shown that if f ∈ A , f is starlike with respect to the origin, and for 0≤ r < 1, A ( r ) < A , an absolute constant, thenL ( r ) = O log 1 1 − ras r → 1 .Later, in 1969 Nunokawa has shown that if f is convex univalent, thenL ( r ) = OA ( r ) log 1 1 − r1 / 2as r → 1 .This paper is devoted to obtaining a related correspondence between f ( z ) and L ( r ) for the case when f is univalent. Copyright © 2016 John Wiley & Sons, Ltd.