Geometric Function Research Articles

Coefficient Bounds of Bi-Univalent Function Involving Pseudo-Starlikeness Associatedwith Sigmoid Function Defined by Salagean Operator via Chebyshev Polynomial

Some special classes of univalent functions play an important role in geometric function theory because of their geometric properties. Many of such classes have been introduced and studied; some became well known, for example, the classes of convex, starlike, close to convex, strongly convex and strongly starlike functions. Previous studies by Awolere and Oladipupo (2018) now served as motivation and background to investigate certain classes of analytic, univalent and bi-univalent functions in terms of their coefficient bounds involving salagean and sigmoid functions via Chebyshev poynomial. The classes 퐻푛(휆,훽,훾(푠),휙(푧,푡))are newly established classes for which coefficient bounds will be determined. The aim of the present work is to investigate coefficient bound for class 퐻푛(휆,훽,훾(푠),휙(푧,푡))of pseudo-starlikeness associated with sigmoid functions defined by Salagean operator via Chebyshev polynomial, Fekete-szego problem will also be established and the Hankel of the function will be determined

Triplet Upconversion under Ambient Conditions Enables Digital Light Processing 3D Printing.

The rapid photochemical conversion of materials from liquid to solid (i.e., curing) has enabled the fabrication of modern plastics used in microelectronics, dentistry, and medicine. However, industrialized photocurables remain restricted to unimolecular bond homolysis reactions (Type I photoinitiations) that are driven by high-energy UV light. This narrow mechanistic scope both challenges the production of high-resolution objects and restricts the materials that can be produced using emergent manufacturing technologies (e.g., 3D printing). Herein we develop a photosystem based on triplet-triplet annihilation upconversion (TTA-UC) that efficiently drives a Type I photocuring process using green light at low power density (<10 mW/cm2) and in the presence of ambient oxygen. This system also exhibits a superlinear dependence of its cure depth on the light exposure intensity, which enhances spatial resolution. This enables for the first-time integration of TTA-UC in an inexpensive, rapid, and high-resolution manufacturing process, digital light processing (DLP) 3D printing. Moreover, relative to traditional Type I and Type II (photoredox) strategies, the present TTA-UC photoinitiation method results in improved cure depth confinement and resin shelf stability. This report provides a user-friendly avenue to utilize TTA-UC in ambient photochemical processes and paves the way toward fabrication of next-generation plastics with improved geometric precision and functionality.

Introduction in third-order fuzzy differential subordination

In light of the well-established and widely-used theory of differential subordination, recent works incorporating fuzzy elements into Geometric Function Theory have given rise to the concept of fuzzy differential subordination. Second-order fuzzy differential subordinations were taken into consideration for studies up until this point. The research described in this paper aims to expand the concept of fuzzy differential subordination to third-order fuzzy differential subordination, building on an idea first put forth in 2011 by José A. Antonino and Sanford S. Miller and still being investigated by scholars today. The key concepts and preliminary findings required for the development of this branch of fuzzy differential subordination are introduced. The class of admissible functions is specified, the fundamental theorems are established and the fundamental concepts of the third-order fuzzy subordination approach are presented. The example given demonstrates the applicability of the new findings.

Inequalities for meromorphic functions with prescribed poles

The extremal problems for functions of complex variables, as well as approaches for obtaining classical inequalities on the base of various methods of the geometric function theory, are known for various norms and for many classes of functions such as rational functions with various constraints and for various domains in the complex plane. It is important to mention that different types of Bernstein-type inequalities appeared in the literature in more generalized forms in which the underlying polynomial was replaced by a more general class of functions. One such generalization is the passage from polynomials to rational functions. In this paper, we prove some inequalities for meromorphic functions with prescribed poles and restricted zeros. These results not only generalize some Bernstein-type inequalities for rational functions, but also improve and generalize some known polynomial inequalities. These inequalities have their own importance in the approximation theory.

Applications of fuzzy differential subordination theory on analytic $ p $ -valent functions connected with $ \mathfrak{q} $-calculus operator

In recent years, the concept of fuzzy set has been incorporated into the field of geometric function theory, leading to the evolution of the classical concept of differential subordination into that of fuzzy differential subordination. In this study, certain generalized classes of $ p $ -valent analytic functions are defined in the context of fuzzy subordination. It is highlighted that for particular functions used in the definitions of those classes, the classes of fuzzy $ p $-valent convex and starlike functions are obtained, respectively. The new classes are introduced by using a $ \mathfrak{q} $-calculus operator defined in this investigation using the concept of convolution. Some inclusion results are discussed concerning the newly introduced classes based on the means given by the fuzzy differential subordination theory. Furthermore, connections are shown between the important results of this investigation and earlier ones. The second part of the investigation concerns a new generalized $ \mathfrak{q} $-calculus operator, defined here and having the $ (p, \mathfrak{q)} $-Bernardi operator as particular case, applied to the functions belonging to the new classes introduced in this study. Connections between the classes are established through this operator.

Inclusion properties for analytic functions of $ q $-analogue multiplier-Ruscheweyh operator

&lt;abstract&gt;&lt;p&gt;The results of this work have a connection with the geometric function theory and they were obtained using methods based on subordination along with information on $ \mathfrak{q} $-calculus operators. We defined the $ \mathfrak{q} $-analogue of multiplier- Ruscheweyh operator of a certain family of linear operators $ I_{\mathfrak{q}, \mu }^{s}(\lambda, \ell) \mathfrak{f}(\varsigma) \; (s\in \mathbb{N}_{0} = \mathbb{N}\cup \{0\}, \mathbb{ N} = \left\{ 1, 2, 3, ..\right\}; \ell, \lambda, \mu \geq 0, 0 &amp;lt; \mathfrak{q} &amp;lt; 1) $. Our major goal was to build some analytic function subclasses using $ I_{ \mathfrak{q}, \mu }^{s}(\lambda, \ell)\mathfrak{f}(\varsigma) $ and to look into various inclusion relationships that have integral preservation features.&lt;/p&gt;&lt;/abstract&gt;

Analytic Studies of a Class of Langevin Differential Equations Dominated by a Class of Julia Fractal Functions

In this investigation, we study a class of analytic functions of type Carathéodory style in the open unit disk connected with some fractal domains. This class of analytic functions is formulated based on a kind of Langevin differential equations (LDEs). We aim to study the analytic solvability of LDEs in the advantage of geometric function theory consuming the geometric properties of the Julia fractal (JF) and other fractal connected with the logarithmic function. The analytic solutions of the LDEs are obtainable by employing the subordination theory.

Subordinations and superordinations studies using $ q $-difference operator

&lt;abstract&gt;&lt;p&gt;The results of this work belong to the field of geometric function theory, being based on differential subordination methods. Using the idea of the $ \mathfrak{q} $-calculus operators, we define the $ \mathfrak{q} $-analogue of the multiplier- Ruscheweyh operator of a specific family of linear operators, $ I_{\mathfrak{q}, \mu }^{s}(\lambda, \ell). $ Our major goal is to build and investigate some analytic function subclasses using $ I_{\mathfrak{q}, \mu }^{s}(\lambda, \ell) $. Also, some differential subordination and superordination results are obtained. Moreover, based on the new theoretical results, several examples are constructed. For every differential superordination under investigation, the best subordinant is provided.&lt;/p&gt;&lt;/abstract&gt;

Lower large deviations for geometric functionals in sparse, critical and dense regimes

Lower large deviations for geometric functionals in sparse, critical and dense regimes

Sharp inequalities for $ q $-starlike functions associated with differential subordination and $ q $-calculus

&lt;p&gt;This paper employs differential subordination and quantum calculus to investigate a new class of $ q $-starlike functions associated with an eight-like image domain. Our study laid a foundational understanding of the behavior of these $ q $-starlike functions. We derived the results in first-order differential subordination. We established sharp inequalities for the initial Taylor coefficients and provided optimal estimates for solving the Fekete-Szegö problem and a second-order Hankel determinant applicable to all $ q $-starlike functions in this class. Furthermore, we presented a series of corollaries that demonstrate the broader implications of our findings in geometric function theory.&lt;/p&gt;

Certain geometric properties of the fractional integral of the Bessel function of the first kind

&lt;abstract&gt;&lt;p&gt;This paper revealed new fractional calculus applications of special functions in the geometric function theory. The aim of the study presented here was to introduce and begin the investigations on a new fractional calculus integral operator defined as the fractional integral of order $ \lambda $ for the Bessel function of the first kind. The focus of this research was on obtaining certain geometric properties that give necessary and sufficient univalence conditions for the new fractional calculus operator using the methods associated to differential subordination theory, also referred to as admissible functions theory, developed by Sanford S. Miller and Petru T. Mocanu. The paper discussed, in the proved theorems and corollaries, conditions that the fractional integral of the Bessel function of the first kind must comply in order to be a part of the sets of starlike functions, positive and negative order starlike functions, convex functions, positive and negative order convex functions, and close-to-convex functions, respectively. The geometric properties proved for the fractional integral of the Bessel function of the first kind recommend this function as a useful tool for future developments, both in geometric function theory in general, as well as in differential subordination and superordination theories in particular.&lt;/p&gt;&lt;/abstract&gt;

Q-Hermite-Hadamard Integral Inequality for the Coordinated Convex Functions

The calculus without the notion of limits is quantum calculus. Its study dates back to L. Euler in the middle of the eighteenth century whereas the systematic initiation on it was done by F.H. Jackson in the beginning of the twentieth century. The rapid growth on q-calculus is due to its applications in various branches of mathematical and physical sciences. Of them, one of the most basic and important functions in the theory of geometric function is convexity having its wider applications in pure and applied mathematics. As it still lacks the intensive study on quantum estimates on the various types of integral inequalities, we focus our study on quantum estimates of Hermite-Hadamard type integral inequality especially on coordinated convex functions. In this paper, we have extended Hermite-Hadamard type integral inequality for coordinated convex function in terms of quantum framework.

Applications of Fuzzy Differential Subordination to the Subclass of Analytic Functions Involving Riemann–Liouville Fractional Integral Operator

In this research, we combine ideas from geometric function theory and fuzzy set theory. We define a new operator Dτ−λLα,ζm:A→A of analytic functions in the open unit disc Δ with the help of the Riemann–Liouville fractional integral operator, the linear combination of the Noor integral operator, and the generalized Sălăgean differential operator. Further, we use this newly defined operator Dτ−λLα,ζm together with a fuzzy set, and we next define a new class of analytic functions denoted by Rϝζ(m,α,δ). Several innovative results are found using the concept of fuzzy differential subordination for the functions belonging to this newly defined class, Rϝζ(m,α,δ). The study includes examples that demonstrate the application of the fundamental theorems and corollaries.

Analytic Functions Related to a Balloon-Shaped Domain

One of the fundamental parts of Geometric Function Theory is the study of analytic functions in different domains with critical geometrical interpretations. This article defines a new generalized domain obtained based on the quotient of two analytic functions. We derive various properties of the new class of normalized analytic functions X defined in the new domain, including the sharp estimates for the coefficients a2,a3, and a4, and for three second-order and third-order Hankel determinants, H2,1X,H2,2X, and H3,1X. The optimality of each obtained estimate is given as well.

Nonlinear dynamic analysis of a geometrically imperfect sandwich beam with functionally graded material facets and auxetic honeycomb core in thermal environment

In this study, a model for the nonlinear dynamic analysis of the geometrically imperfect sandwich beam with functionally graded material (FGM) facets and auxetic honeycomb core is proposed. The auxetic feature of the honeycomb material is realized by a homogenization method, which gives the expressions of the effective material coefficients in terms of material property and cellular geometric parameters. The nonlinear motion equation of the geometrically imperfect beam is derived based on the Reddy's higher shear deformation theory, von Kármán nonlinear theory, as well as general geometric imperfection function. The thermal effect on the material properties and structural rigidity is also taken into account in this model. The Newmark method and Newton–Raphson iterative method are employed to solve the nonlinear motion equation of the sandwich beam. In numerical examples, the dynamic response of the sandwich beam under two types of load, i.e. moving load and impulse load, are discussed in details. The influences of various factors (such as material parameter, geometric imperfection, and thermal condition) on the nonlinear dynamic behaviors of the sandwich beam are elaborately investigated. The numerical results show those factors can play significant roles in the bearing ability, structural stiffness, or nonlinear dynamic behaviors of the sandwich beam in certain conditions. Some key conclusions on the influences are drawn from the numerical results, which can be a useful reference for future investigations.

On the Study of Starlike Functions Associated with the Generalized Sine Hyperbolic Function

Geometric function theory, a subfield of complex analysis that examines the geometrical characteristics of analytic functions, has seen a sharp increase in research in recent years. In particular, by employing subordination notions, the contributions of different subclasses of analytic functions associated with innovative image domains are of significant interest and are extensively investigated. Since ℜ(1+sinh(z))≯0, it implies that the class Ssinh* introduced in reference third by Kumar et al. is not a subclass of starlike functions. Now, we have introduced a parameter λ with the restriction 0≤λ≤ln(1+2), and by doing that, ℜ(1+sinh(λz))&gt;0. The present research intends to provide a novel subclass of starlike functions in the open unit disk U, denoted as Ssinhλ*, and investigate its geometric nature. For this newly defined subclass, we obtain sharp upper bounds of the coefficients an for n=2,3,4,5. Then, we prove a lemma, in which the largest disk contained in the image domain of q0(z)=1+sinh(λz) and the smallest disk containing q0(U) are investigated. This lemma has a central role in proving our radius problems. We discuss radius problems of various known classes, including S*(β) and K(β) of starlike functions of order β and convex functions of order β. Investigating Ssinhλ* radii for several geometrically known classes and some classes of functions defined as ratios of functions are also part of the present research. The methodology used for finding Ssinhλ* radii of different subclasses is the calculation of that value of the radius r&lt;1 for which the image domain of any function belonging to a specified class is contained in the largest disk of this lemma. A new representation of functions in this class, but for a more restricted range of λ, is also obtained.

An Operator Defined on Hadamard Product Pertaining to Generalized Hurwitz-Lerch Zeta Function with Conical Section

The author’s goal is to highlight the most recent developments in the research on the study of complex-valued functions as seen through an understanding of geometric function theory. Contributions will be required for any aspect of the Hadamard product associated to the generalized Hurwitz-Lerch Zeta function convoluted with the theory of functions that are univalent. In the present investigation, the author focused into the inclusion relations of a few subclasses of the -starlike functions, -uniformly convex functions, and -quasi-convex functions which together make up the generalized Hurwitz-Lerch Zeta function.

Phase field-based cohesive zone approach to model delamination in fiber-reinforced polymer composites

In the present work, a unified phase field-based cohesive zone model (PF-CZM) has been utilized to predict bulk fracture and most importantly interfacial delamination in carbon fiber-reinforced polymer composites (CFRP). The formulation is based on the classical phase-field approach of energy minimization, with a rational degradation function and a polynomial-type crack geometric function. Incorporation of those provides the PF theory to integrate various traction–separation behaviors by choosing optimal constitutive functions and makes the theory capable of predicting the quasi-brittle type of fracture. Since the interfacial delamination in composites occurs either due to adhesive failure or decohesion between stacked lamina, the PF-CZM model is used to predict such failure by exploiting the cohesive zone flavor of the model. Further, the model inherently incorporates the strength of the material without any additional calibration. Various standard simulations are performed considering CFRP geometry where delaminations due to adhesive failure or interface decohesion occur in a mode-I and mode-II fashion. The model is observed to predict mode-I and mode-II separation accurately. To test the accuracy of the model an experimental validation is also performed. In addition to the delamination, open hole tension (OHT) simulations of CFRP lamina for different fiber orientations are considered to validate the anisotropic crack growth predictions, and the results are compared with experiments.

Monocular depth estimation using self-supervised learning with more effective geometric constraints

Self-supervised learning-based depth estimation from monocular videos is a challenging yet promising way for 3D environment perception. Existing methods that use photometric consistency as supervision are often fragile in the case of textureless environments or illumination variations. In this paper, we adopt geometric constraints to improve the reliability and accuracy of monocular depth estimation. We introduce an equation derived from the two-view imaging geometry, from which we also develop a novel geometric loss function that is shown to be more effective than the epipolar constraint in self-supervised learning. In addition, we design a depth reconstruction loss by explicitly aligning the depth scale of the consecutive estimated depth maps to keep the depth scale consistent. Finally, a new self-supervised learning framework is formulated by incorporating the proposed geometric constraints into depth, pose, and optical flow estimation models. Extensive experiments on KITTI, Make3D, and NYU Depth v2 datasets show that, by adding our geometric constraints, the depth estimation model compares favorably with state-of-the-art self-supervised learning methods, and each sub-task has a significant improvement.

Fuzzy Differential Subordination Associated with a General Linear Transformation

In this study, we investigate a possible relationship between fuzzy differential subordination and the theory of geometric functions. First, using the Al-Oboudi differential operator and the Babalola convolution operator, we establish the new operator BSα,λm,t:An→An in the open unit disc U. The second step is to develop fuzzy differential subordination for the operator BSα,λm,t. By considering linear transformations of the operator BSα,λm,t, we define a new fuzzy class of analytic functions in U which we denote by Tϝλ,t(m,α,δ). Several innovative results are found using the concept of fuzzy differential subordination and the operator BSα,λm,t for the function f in the class Tϝλ,t(m,α,δ). In addition, we explore a number of examples and corollaries to illustrate the implications of our key findings. Finally, we highlight several established results to demonstrate the connections between our work and existing studies.