A class of multivalent functions

This inequality has been proved for n = 2 and n = 3, and for special subclasses of the family of univalent functions (1.2) has been proved for all n. In particular, if all the coefficients in (1.1) are real then (1.2) holds and is sharp for all positive integers n. In fact Rogosinski [8](1), Dieudonne [2] and Szasz [9] have proved (1.2) when all the coefficients are real and f(z) satisfies a weaker condition than univalency, namely that for z zf 0 when and only when az>0. Such a function is called typically-real in the unit circle. The situation is quite different for the class V(p) of functions (1.1) which are regular and multivalent of order p (p-valent) in lzI <1. Almost without exception, the previous investigations were concerned only with the order of magnitude of the coefficients b, and the most recent result due to Biernacki [1] that