SUBCLASSES OF MEROMORPHICALLY P-VALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS ASSOCIATED WITH A LINEAR OPERATOR
The aim of this paper is to investigate various inclusion relations and some other properties of a certain subclasses �p,k(�,A,B,n) and � ∗(�,A,B,n) of meromorphically p-valent analytic funtions with negative coefficients which are defined here by means of a linear operator. The familier concept of neighborhoods of analytic functions is also extended and applied to the functions considered here. We also derive many interesting results for the modified Hadamard products of functions belonging to the class � ∗(�,A,B,n).
NEIGHBOURHOODS OF UNIVALENT FUNCTIONS
Abstract The main result shows that a small perturbation of a univalent function is again a univalent function, hence a univalent function has a neighbourhood consisting entirely of univalent functions. For the particular choice of a linear function in the hypothesis of the main theorem, we obtain a corollary which is equivalent to the classical Noshiro–Warschawski–Wolff univalence criterion. We also present an application of the main result in terms of Taylor series, and we show that the hypothesis of our main result is sharp.
International EMS systems: New South Wales, Australia.
Australia has an area of 7,692,024 km 2 and a population of 19,169,083. The state of New South Wales (NSW) has an area of 801,428 square kilometres and a population of 6,663,700 (June 2002). It contributes 10% of the land area of Australia and 34% of the population. The length of the coastline is 2007 km with an east-west distance of 1450 km (Fig. 1). Approximately 90% of the NSW population resides within 100 km of the coast. The population density is greatest in the Greater Metropolitan Region around Sydney, an area of 1580 square km, population of 4,154,722. Only 0.6% live in regions classified as remote or very remote (based upon distance to regional townships). The climate is generally mild, although temperature extremes range from � /20 8C (Charlotte’s Pass) to 50 8C (far Western NSW). The highest point is 2228 m at Mt. Kosciuszko. However, most of NSW is below 500 m in altitude.