Convex family of meromorphically multivalent functions on connected sets
Abstract In this paper, by using a differential operator we have defined a new class of meromorphically multivalent functions. Also two useful subclasses of this class involving fixed points are investigated. Important properties of these classes like convex linear combination, coefficient estimates, convex family etc. are found.
Subclasses of Meromorphically p-Valent Convex Functions
In this paper two new subclasses C(n, p,A,B) and K(n, p,A,B) of meromorphically p-valent convex functions in the unit disc are introduced and certain properties of these classes such as inclusion relations and integral transforms are studied.
Coefficients of meromorphic schlicht functions
This paper presents an elementary proof of a known theorem on the coefficients of meromorphic schlicht functions: if fG Ezand bk = Ofor 1 _k 1 except for a simple pole at oo with residue 1. Let 2o be the subclass of 2 for which bo = 0. It follows from the area theorem that Ibi < 1, and Schiffer [7] obtained the sharp estimate Ib2 <?2/3. The form of the extremal functions suggested that I bn| I 2/(n+l) for all n. This has proved to be quite false, but it is true for certain subclasses of 2. For example, the following theorem goes back to Goluzin [3 ]. THEOREM. Let f C2 and suppose b= = = b * -b = 0 for some m _ 1. Then I b|I < 2/(n+l), n =m, m+l, * , 2m. The inequality I b2j <2/3 is a special case of this theorem. Jenkins [5] proved the theorem by the method of quadratic differentials. I claimed [1] to give an elementary proof based on the Grunsky inequalities. However, Professor Jenkins has pointed out to me that my proof contains an error, since the square-root transformation V(f(z2)) will introduce a branch point wherever f(z2) =0. Furthermore, Goluzin made essentially the same mistake (see the footnote in [4, p. 279]). The purpose of this paper is to correct the error in [1 ] and thus to deduce the theorem in an elementary way from the Grunsky inequalities. The main idea was suggested by Pommerenke's recent proof [6 ] that I b2| < 2/3. The nth Faber polynomial Fn(w) of a function f 2 is defined by