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.
Multivariate Image Texture by Multivariate Variogram for Multispectral Image Classification
Traditional image texture measure usually allows a texture description of a single band of the spectrum, characterizing the spatial variability of gray-level values within the singleband image. A problem with the approach while applied to multispectral images is that it only uses the texture information from selected bands. In this paper, we propose a new multivariate texture measure based on the multivariate variogram. The multivariate texture is computed from all bands of a multispectral image, which characterizes the multivariate spatial autocorrelation among those bands. In order to evaluate the performance of the proposed texture measure, the derived multivariate texture image is combined with the spectral data in image classification. The result is compared to classifications using spectral data alone and plus traditional texture images. A machine learning classifier based on Support Vector Machines (SVMs) is used for image classification. The experimental results demonstrate that the inclusion of multivariate texture information in multispectral image classification significantly improves the overall accuracy, with 5 to 13.5 percent of improvement, compared to the classification with spectral information alone. The results also show that when incorporated in image classification as an additional band, the multivariate texture results in high overall accuracy, which is comparable with or higher than the best results from the existing single-band and two-band texture measures, such as the variogram, cross variogram and Gray-Level Co-occurrence Matrix (GLCM) based texture. Overall, the multivariate texture provides the useful spatial information for land-cover classification, which is different from the traditional single band texture. Moreover, it avoids the band selection procedure which is prerequisite to traditional texture computation and would help to achieve high accuracy in the most classification tasks.