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The theory of operators is an important subject in analytic function theory, geometric function theory and univalent function theory. It also is an important subject in applied sciences. It is still an active field of research and various types of problems, which can be solved by generalising operators. Recently, many researchers have shown great interests in the study of differential operators in the theory of univalent functions and various subclasses of analytic functions defined in the open unit disc. In this paper a generalised derivative operator will be used to derive some results concerning the subordination and superordination of analytic function in the open unit disc.

An optimized image watermarking technique based on LU factorization and entropy analysis in combination with lifting wavelet transform and discrete cosine transform is presented in this paper. At first, the original image is decomposed by a 2-level lifting wavelet transform for obtaining the coefficients of a high-frequency subband followed by discrete cosine transform. Afterward, non-overlapping blocks are obtained by dividing the coefficients of discrete cosine transform whereas LU factorization is applied to each nonoverlapping blocks based on pseudo-random sequences. Then, the watermark is embedded into the first row, the first column element of the upper triangular matrix of LU factorization. The normalized cross-correlation (NC), and the peak signal-to-noise ratio (PSNR) are used to evaluate the invisibility and robustness of the presented technique. The experimental results have indicated that the presented technique fulfills all watermarking requirements in terms of invisibility, robustness, security, and capacity. The comparison with the existing scheme has shown that the proposed watermarking technique has a superior performance in terms of invisibility than the existing scheme.

The division algorithm for multivariate polynomials over fields has been introduced not very long ago, in connection with algorithmic and computational problems in the rings since the work of Buchberger. The well-known fact is a term order must be a well-ordering and the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not leading term, but is freely chosen. In this paper, we will show the original geometric approach to the classification of all possible orders on the ring of polynomials in two variables is given. The connection between this classification and the well-known classification of Robiano is exposed in details

The theory of operators is an important subject in analytic function theory, geometric function theory and univalent function theory. It also is an important subject in applied sciences. It is still an active field of research and various types of problems, which can be solved by generalising operators. Recently, many researchers have shown great interests in the study of differential operators in the theory of univalent functions and various subclasses of analytic functions defined in the open unit disc. In this paper a generalised derivative operator will be used to derive some results concerning the subordination and superordination of analytic function in the open unit disc.