General Subclasses Research Articles

Some m-fold symmetric bi-univalent function classes and their associated Taylor-Maclaurin coefficient bounds

The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class Σm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Sigma_{m}$\\end{document} of m-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients |am+1|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\vert a_{m+1} \\vert $\\end{document} and |a2m+1|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\vert a_{2 m+1} \\vert $\\end{document} are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. Additionally, the Fekete-Szegö inequalities for these classes are investigated. The results presented could generalize and improve some recent and earlier works. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, the utilization of the Ruscheweyh derivative operator can encompass a broad spectrum of engineering applications, including the robotic manipulation control, optimizing optical systems, antenna array signal processing, image compression, and control system filter design. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications.

Coefficient bounds for the General Subclasses of Close-to-Convex Functions of Complex Order

Coefficient bounds for the General Subclasses of Close-to-Convex Functions of Complex Order

MEGH: A parametric class of general hazard models for clustered survival data.

In many applications of survival data analysis, the individuals are treated in different medical centres or belong to different clusters defined by geographical or administrative regions. The analysis of such data requires accounting for between-cluster variability. Ignoring such variability would impose unrealistic assumptions in the analysis and could affect the inference on the statistical models. We develop a novel parametric mixed-effects general hazard (MEGH) model that is particularly suitable for the analysis of clustered survival data. The proposed structure generalises the mixed-effects proportional hazards and mixed-effects accelerated failure time structures, among other structures, which are obtained as special cases of the MEGH structure. We develop a likelihood-based algorithm for parameter estimation in general subclasses of the MEGH model, which is implemented in our R package MEGH. We propose diagnostic tools for assessing the random effects and their distributional assumption in the proposed MEGH model. We investigate the performance of the MEGH model using theoretical and simulation studies, as well as a real data application on leukaemia.

Combinatorial characterization of a certain class of words and a conjectured connection with general subclasses of phylogenetic tree-child networks

The combinatorial study of phylogenetic networks has attracted much attention in recent times. In particular, one class of them, the so-called tree-child networks, are becoming the most prominent ones. However, their combinatorial properties are largely unknown. In this paper we address the problem of exactly counting them. We conjecture a relationship with the cardinality of a certain class of words. By solving the counting problem for the words, and on the basis of the conjecture, several simple recurrence formulas for general cases arise. Moreover, a precise asymptotic analysis is provided. Our results coincide with all current formulas in the literature for particular subclasses of tree-child networks, as well as with numerical results obtained for small networks. We expect that the study of the relationship between the newly defined words and the networks will lead to further combinatoric characterizations of this class of phylogenetic networks.

New subclasses of analytic and bi-univalent functions endowed with coefficient estimate problems

Inspired by the recent works of Srivastava et al. (Appl Math Lett 23(10):1188–1192, 2010), Frasin and Aouf (Appl Math Lett 24(9):1569–1573, 2011), and Çağlar et al. (Filomat 27(7):1165–1171, 2013), we introduce and investigate in the present paper two new general subclasses of the class consisting of normalized analytic and bi-univalent functions in the open unit disk $${\mathbb {U}}=\{z\in {\mathbb {C}}:\left| z\right| <1\}$$ . For functions belonging to these general subclasses introduced here, we obtain estimates on the Taylor–Maclaurin coefficients $$|a_{2}|$$ and $$|a_{3}|$$ . Several connections to some of the earlier known results are also pointed out. The results presented in this paper would generalize and improve those in related works of several earlier authors.

COEFFICIENT ESTIMATES FOR CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS ASSOCIATED WITH HORADAM POLYNOMIAL

In this present article, we studied and examined the novel general subclasses of the function class $\Sigma$ of bi-univalent function defined in the open unit disk, which are associated with the Horadam polynomial. This study locates estimates on the Taylor - Maclaurin coefficients $|a_{2}|$ {\it and} $|a_{3}|$ in functions of the class which are considered. Additionally, Fekete-Szeg\"{o} inequality of functions belonging to this subclasses are also obtained.

Comprehensive subclasses of analytic functions and coefficient bounds

In this paper, we introduce two general subclasses of analytic functions by means of the principle of subordination and investigate the coefficient bounds for functions in these classes. The well-known results are obtained as a corollary of our main results. Especially, we improve the results of Altintas and Kilic [ 1 ].

Critical Hardy inequalities

We prove a range of critical Hardy inequalities and uncertainty type principles on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. Moreover, we establish a new type of critical Hardy inequality and prove Hardy-Sobolev type inequalities. Most of the obtained estimates are new already for the case of $\mathbb R^{n}$. For example, for any $f\in C_{0}^{\infty}(\mathbb{R}^{n}\backslash\{0\})$ our results imply the range of critical Hardy inequalities of the form $$\qquad \underset{R>0}{\sup}\left\|\frac{f-f_{R}}{|x|^{\frac{n}{p}}{\log}\frac{R}{|x|}}\right\|_{L^{p}(\mathbb{R}^{n})}\leq \frac{p}{p-1}\left\| \frac{1}{|x|^{\frac{n}{p}-1}} \nabla f\right\|_{L^{p}(\mathbb{R}^{n})},\quad 1<p<\infty,$$ where $f_{R}=f(R\frac{x}{|x|})$, with sharp constant $\frac{p}{p-1}$, recovering the known cases of $p=n$ and $p=2$. Moreover, our results also imply a new type of a critical Hardy inequality of the form $$\left\|\frac{f}{|x|}\right\|_{L^{n}(\mathbb{R}^{n})}\leq n\left\|(\log|x|)\nabla f\right\|_{L^{n}\mathbb{R}^{n})}, $$ for all $f\in C_{0}^{\infty}(\mathbb{R}^{n}\backslash\{0\}),$ where the constant $n$ is sharp. However, homogeneous groups provide a perfect degree of generality to talk about such estimates without using specific properties of $\mathbb R^n$ or of the Euclidean distance.

Building multivariate Sato models with linear dependence

The increased trading in multi-name financial products has required the development of state-of-the-art multivariate models. These models should be computationally tractable and, at the same time, flexible enough to explain the stylized facts of asset log-returns and of their dependence structure. The popular class of multivariate Lévy models provides a variety of tractable models, but suffers from one major shortcoming: Lévy models can replicate single-name derivative prices for a given time-to-maturity, but not for the whole range of quoted strikes and maturities, especially during periods of market turmoil. Moreover, there is a significant discrepancy between the moment term structure of Lévy models and the one observed in the market. Sato processes on the other hand exhibit a moment term structure that is more in line with empirical evidence and allow for a better replication of single-name option price surfaces. In this paper, we propose a general framework for multivariate models characterized by independent and time-inhomogeneous increments, where the asset log-return processes at unit time are modeled as linear combinations of independent self-decomposable random variables, where at least one self-decomposable random variable is shared by all the assets. As examples, we consider two general subclasses within this new framework, where we assume a normal variance-mean mixture with a one-sided tempered stable mixing density or a difference of one-sided tempered stable laws for the distribution of the risk factors. Particular attention is given to the models' ability to explain the asset dependence structure. A numerical study reveals the advantages of these new types of models.

Lp-Caffarelli–Kohn–Nirenberg type inequalities on homogeneous groups

We prove $L^{p}$-Caffarelli-Kohn-Nirenberg type inequalities on homogeneous groups, which is one of most general subclasses of nilpotent Lie groups, all with sharp constants. We also discuss some of their consequences. Already in the abelian case of $\mathbb{R}^{n}$ our results provide new insights in view of the arbitrariness of the choice of the not necessarily Euclidean quasi-norm.

A Note on Hardy Inequalities on Homogeneous Groups

We provide the necessary and sufficient characterizations on a pair of positive radial functions so that the two-weight Hardy inequalities hold true on homogeneous groups, one of most general subclasses of nilpotent Lie groups. We also present a simple condition on a couple of positive radial weight functions for the Lp −Hardy type inequalities to be valid.

Coefficient estimates for a new subclass of bi-univalent functions defined by convolution

In this paper we introduce general subclasses of bi-univalent functions by using convolution. Bounds for the first two coefficients |a2| and |a3| for bi-univalent functions in these classes are obtained. The obtained results generalize the results which are given in [Murugusundaramoorthy, G., Magesh, M., Prameela, V., Coefficient bounds for certain subclasses of bi-univalent function, Abstr. Appl. Anal., (2013), Art. ID 573017, 3 pp.] and [Brannan, D. A. and Taha, T. S., On some classes of bi-univalent functions, Studia Univ. Babes¸ Bolyai Math., 31 (1986), No. 2, 70–77].

The General Subclasses of the Analytic Functions and Their Various Properties

The object of the present paper is to introduce and investigate two new general subclasses ${{S}^{*}}C(\alpha ,\beta ;\gamma )$ and $T{{S}^{*}}C(\alpha ,\beta ;\gamma )~~(\alpha, \beta \in [0,1),~\gamma \in [0,1])$ of the analytic functions. Here, we give sufficient conditions as well as necessary and sufficient conditions for the functions belonging to the classes.

Coefficients Bounds for a General Subclasses of m-Fold Symmetric Bi-Univalent Functions

In this paper, we introduce and investigate a new general subclasses and of consisting of analytic and m-fold symmetric bi-univalent functions in the open unit disk We obtain estimates on the Taylor-Maclaurin coefficients and Also, we obtain new special cases for our results.

Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups

We give sharp remainder terms of Lp and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy–Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.

Coefficient estimates for some general subclasses of analytic and bi-univalent functions

In the present investigation, we consider two new general subclasses \(\mathcal {H}_{\Sigma }(\tau , \mu , \lambda , \gamma ; \alpha )\) and \(\mathcal {H}_{\Sigma }(\tau , \mu , \lambda , \gamma ; \beta )\) of the class \(\Sigma \) consisting of analytic and bi-univalent functions in the open unit disk \(\mathbb {U}\). For functions belonging to the two classes introduced here, we find estimates on the Taylor–Maclaurin coeffcients \(|a_{2}|\) and \(|a_{3}|\). Several connections to some of the earlier known results are also pointed out.

Initial coefficient estimates for some subclasses of M-Fold symmetric BI-Univalent functions

In the present investigation, we consider two new general subclasses B∑m(τ,λ;α) and B∑m*(τ,λ;β) of ∑m consisting of analytic and m-fold symmetric bi-univalent functions in the open unit disk U. For functions belonging to the two classes introduced here, we derive non-sharp estimates on the initial coefficients |am+1| and |a2m+1|. Several connections to some of the earlier known results are also pointed out.

Coefficient estimates for general subclasses of $m$-foldsymmetric analytic bi-univalent functions

In this work, we introduce and investigate two new subclasses of the bi-univalent functions in which both $f$ and $f^{-1}$ are $ m$-fold symmetric analytic functions. For functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for $ \left\vert a_{m+1}\right\vert $ and $\left\vert a_{2m+1}\right\vert .$

Using the Structure of Subfields in the Construction of Goppa Codes and Extended Goppa Codes

It is shown that for some location sets and some integral multiple power Galois fields that some new general subclasses of Goppa codes may be defined which have improved lower bounds to code dimension and minimum distance compared with ordinary Goppa codes. Some previously published results are shown to be particular cases of these general subclasses of codes. A new subclass of reversible Goppa codes is also presented. Examples of code construction for these subclasses, demonstrating the improved code parameters, are presented for both non binary and binary codes.

Some extensions of coefficient problems for bi-univalent Ma-Minda starlike and convex functions

Motivated by the works of H.M. Srivastava et al. [7], we introduce and investigate two new general subclasses HT (?,?,?), h,pHT (?) of bi-starlike and bi-convex of Ma-Minda type functions. Bounds on the first two coefficients |a2| and |a3| for functions in HT (?,?,?), h,pHT (?) are given. The results here generalize and improve the corresponding earlier works done by Ali et al.[1] and Brannan et al.[2].