A Remark on p-Valent Functions

James A. Jenkins, Kôtaro Oikawa
1971 Journal of the Australian Mathematical Society  
Communicated by E. Strzelecki In the theory of multivalent functions there are several different levels of postulates for /^-valency. Perhaps the most well-known is the class of mean pvalent functions in the sense of Spencer [8] (we shall refer to them as areally mean />-valent functions), whose basic properties are found, e.g., in Hayman [4] . Recently Eke [1,2] extended to these functions a number of results which had been known for circumferentially mean p-valent functions. On the other
more » ... . On the other hand, Garabedian-Royden [3] and Jenkins [5] have introduced a wider class, for which they discussed the extension of Koebe's 1/4-theorem. Functions in this class are referred to as weakly mean />-valent functions by the former, and logarithmically areally mean />-valent functions by the latter. There are various other properties of areally mean />-valent functions which are satisfied by those functions also. In the present paper, we shall discuss a negative aspect of logarithmically areally mean />-valent functions. It will be shown that the above mentioned result of Eke cannot be extended to those functions. We shall also give a glance at s-dimensionally mean /7-valent functions, discussed in Spencer [8] , which lie in between areally mean />-valent functions and logarithmically areally mean j?-valent functions. Given a regular function / on the unit disc \z\ < 1, let n(w) be the number of w-points counted with multiplicity, and consider its circumferential mean p(R) = 1 r n (Re ie )d9, 2nJ 0 0 g R < oo. It is a non-negative lower-semicontinuous function and is such that p{R) > 0 if and only if there exists z satisfying R = |/(z)|.
doi:10.1017/s1446788700010260 fatcat:uhvboduxufgnnalzqsumdv4zgi