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In this particular work, we define a modified q-sigmoid function and use the mean function to study the convex and concave properties. We further apply the modified q-sigmoid function to geometric function theory defined in a complex plane with examples. [ABSTRACT FROM AUTHOR]

The objective of this paper is to obtain an upper bound to the third Hankel determinant for certain subclass of p-valent functions, using Toeplitz determinants. [ABSTRACT FROM AUTHOR]

Analytic functions are characterized by the geometry of their image domains. That's why, geometry of image domain is of substantial importance to have a comprehensive study of analytic functions. To introduce and study new geometrical structures as image domain and to define their subsequent analytic functions is an ongoing part of research in geometric function theory. We introduced a new domain named as cardioid domain and defined the corresponding analytic function, see [14]. Here we further study the cardioid domain, to define and study starlike functions associated with cardioid domain. [ABSTRACT FROM AUTHOR]

In this work, bi-univalent functions, which are one of the most important research areas of Geometric Function Theory and which are still very popular in recent years, have been investigated. For the studies of functions, it is customary to determine the bound. In this direction, firstly, we introduce a new subclass of bi-univalent functions in the open unit disk D = {z ∈ C: /z/ < 1}. For this purpose, the Komatu integral operator developed for complex functions and subordination princible are used. Afterwards, the relation between the Fibonacci number sequence and the functions with the positive real part is given. This relation is a fundamental role for Results section. The upper bounds of the first two Taylor-Maclaurin coefficients are investigated. Finally, we derive Fekete-Szegö inequalities for functions belonging to this newly-defined class. The obtained results are compared with studies in the literature. [ABSTRACT FROM AUTHOR]

Non extreme points of compact, convex integral families of analytic functions are investigated. Knowledge about extreme points provides a valuable tool in the optimization of linear extremal problems. The functions studied are determined by a 2-parameter collection of kernel functions integrated against measures on the torus. Families from classical geometric function theory such as the closed convex hull of the derivatives of normalized close-to-convex functions, the ratio of starlike functions of different orders, as well as many others are included. However for these families of analytic functions, identifying "all" the extreme points remains a difficult challenge except in some special cases. Aharonov and Friedland [1] identified a band of points on the unit circle which corresponds to the set of extreme points for these 2-parameter collections of kernel functions. Later this band of extreme points was further extended by introducing a new technique by Dow and Wilken [3]. On the other hand, a technique to identify a non extreme point was not investigated much in the past probably because identifying non extreme points does not directly help solving the optimization of linear extremal problems. So far only one point on the unit circle has been identified which corresponds to a non extreme point for a 2-parameter collections of kernel functions. This leaves a big gap between the band of extreme points and one non extreme point. The author believes it is worth developing some techniques, and identifying non extreme points will shed a new light in the exact determination of the extreme points. The ultimate goal is to identify the point on the unit circle that separates the band of extreme points from non extreme points. The main result introduces a new class of non extreme points. [ABSTRACT FROM AUTHOR]

A new approach to modeling the geometric structure of real materials based on a new type of functions – porous functions – is introduced. The general concept, basic operations, operations of calculus, visualization and interpretations, and potential applications of porous functions are outlined. [ABSTRACT FROM AUTHOR]

By this research notes, the well-known Tremblay operator and certain core knowledge therewith are firstly introduced and an extensive result containing numerous (analytic and) geometric properties (of possible applications of the related operator) along with a number of special implications are then constituted. As method for proving, the well-known assertion proposed by [8] is also considered there. [ABSTRACT FROM AUTHOR]

In this paper, we investigated the Schur geometric convexity of related function for Holders Inequality by using majorization inequality theory, giving a complete critical condition of Schur Geometrically convex function for Holders Inequality related function and some applications are established. [ABSTRACT FROM AUTHOR]

For analytic function f, q-difference operator Dq is defined as Dqf(z) = f(z) - f(qz)/(1 - q)z; q ∊ (0; 1); and z ∊ E = { z: |z| < 1: Using convolution (*), it follows that Dqf(z) = 1/z { f(z) * z/(1 - qz)(1 - z)}, lim q→1- f(z) = f'(z) and Dqf(0) = f'(0): Using this concept, several classes of analytic functions are defined and their properties such as coefficient results, radius problems and inclusion relations are discussed. Some special cases are also pointed out. [ABSTRACT FROM AUTHOR]

In this paper, we derive some bounds for the real parts and arguments of the functionals given by ... where Js;b is the widely-investigated Srivastava-Attiya operator defined on the class of normalized analytic functions f in the open unit disk ... with suitable real parameters s and b. These results reduce upon specialization to some well-known inclusion relationships for several classes of functions with given geometric properties. We also make a comparison between one of the results obtained here and an already known result for some specific cases. [ABSTRACT FROM AUTHOR]

We consider the well-known problem of the geometric theory of functions of a complex variable on non-overlapping domains with free poles on radial systems. The main results of the present work strengthen and generalize several known results for this problem. [ABSTRACT FROM AUTHOR]

The present work is an attempt to give partial proofs of certain conjectures on the fifth coefficient of certain normalized analytic functions. Further, bounds on the sixth and seventh coefficients for the starlike functions related to a lune are also investigated. The non-sharp bound on third and fourth Hankel determinants are also obtained. [ABSTRACT FROM AUTHOR]

Quantum calculus (the calculus without limit) appeared for the first time in fluid mechanics, noncommutative geometry and combinatorics studies. Recently, it has been included into the field of geometric function theory to extend differential operators, integral operators, and classes of analytic functions, especially the classes that are generated by convolution product (Hadamard product). In this effort, we aim to introduce a quantum symmetric conformable differential operator (Q-SCDO). This operator generalized some well-know differential operators such as Sàlàgean differential operator. By employing the Q-SCDO, we present subclasses of analytic functions to study some of its geometric solutions of q-Painlevé differential equation (type III). [ABSTRACT FROM AUTHOR]

Grunsky operators play an important role in classical geometric function theory and in the study of Teichmüller spaces. The Grunsky map is known to be holomorphic on the universal Teichmüller space. In this paper the authors deal with the compactness of a Grunsky differential operator. They will give upper and lower estimates of the essential norm of a Grunsky differential operator and discuss when a Grunsky differential operator is a p-Schatten class operator. [ABSTRACT FROM AUTHOR]

The concept of hyperelastic deformations of bi-conformal energy is developed as an extension of quasiconformality. These deformations are homeomorphisms h : X → onto Y between domains X , Y ⊂ R n of the Sobolev class W loc 1 , n (X , Y) whose inverse f = = def h - 1 : Y → onto X also belongs to W loc 1 , n (Y , X) . Thus the paper opens new topics in Geometric Function Theory (GFT) with connections to mathematical models of Nonlinear Elasticity (NE). In seeking differences and similarities with quasiconformal mappings we examine closely the modulus of continuity of deformations of bi-conformal energy. This leads us to a new characterization of quasiconformality. Specifically, it is observed that quasiconformal mappings behave locally at every point like radial stretchings; if a quasiconformal map h admits a function ϕ as its optimal modulus of continuity at a point x ∘ , then f = h - 1 admits the inverse function ψ = ϕ - 1 as its modulus of continuity at y ∘ = h (x ∘) . That is to say, a poor (possibly harmful) continuity of h at a given point x ∘ is always compensated by a better continuity of f at y ∘ , and vice versa. Such a gain/loss property, seemingly overlooked by many authors, is actually characteristic of quasiconformal mappings. It turns out that the elastic deformations of bi-conformal energy are very different in this respect. Unexpectedly, such a map may have the same optimal modulus of continuity as its inverse deformation. In line with Hooke's Law, when trying to restore the original shape of the body (by the inverse transformation), the modulus of continuity may neither be improved nor become worse. However, examples to confirm this phenomenon are far from being obvious; indeed, elaborate computations are on the way. We eventually hope that our examples will gain an interest in the materials science, particularly in mathematical models of hyperelasticity. [ABSTRACT FROM AUTHOR]

Inequality study is a magnificent field for investigating the geometric behaviors of analytic functions in the open unit disk calling the subordination and superordination. In this work, we aim to formulate a generalized differential-difference operator. We introduce a new class of analytic functions having the generalized operator. Some subordination results are included in the sequel. [ABSTRACT FROM AUTHOR]

In this paper, we present the best possible parameters α1, α2, α3 and β1, β2, β3 such that the double inequalities Cα1 (a; b)G1-α1 (a, b) < T(a, b) < Cβ1 (a, b)G1-β1 (a, b); α2C(a, b) + (1 - α2)G(a, b) < T(a, b) < β2C(a, b) + (1 - β2)G(a, b), α3/G(a, b)+1 - α3/C(a, b) < 1/T(a, b) < β3/G(a, b) + 1 - β3/C(a, b) hold for all a, b > 0 with a 6 = b, where G(a, b) = √ab, C(a, b) = (a² + b²) / (a+b) and T(a, b) = 2 ∫ π/20 √a2 cos2(t) + b² sin²(t)dt/π are the geometric, contraharmonic and Toader means of a and b, respectively. [ABSTRACT FROM AUTHOR]

We consider an open extremal problem in geometric function theory of complex variables on the maximum of the functional [ABSTRACT FROM AUTHOR]

Although much research (f. e. [1], [3], [5], [7-15]) has been devoted to the extremal problems of a geometric function theory associated with estimates of functionals defined on systems of non-overlapping domains, however, in the general case the problems remain unsolved. The paper describes the problem of finding the maximum of a functional. This problem is to find a maximum of the product of inner radii of mutually non-overlapping symmetric domains with respect to a unit circle and the inner radius in some positive certain degree of the domain with respect to zero and description of extreme configurations. The topic of the paper is devoted to the study of the problem of the classical direction of the geometric theory of complex variable functions, namely, the extremal problems for non-overlapping domains. [ABSTRACT FROM AUTHOR]

In the first part of this work we present several new geometric properties of analytic functions by applying the differential subordination. In addition, several results in the geometric functions theory pointed out. In the second part we find upper bounds for coefficients of functions in class B∑q,µ (β, λ, h) which is defined by fractional q-calculus operators. [ABSTRACT FROM AUTHOR]

In the present paper, by using the Lp,q,n(x) functions, our methodology intertwine to yield the Theory of Geometric Functions and that of Special Functions, which are usually considered as very different fields. Thus, we aim at introducing a new class of biunivalent functions defined through the (p, qJ-Lucas polynomials. Furthermore, we derive coefficient inequalities and obtain Fekete-Szego problem for this new function class. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the q-analogue of a certain family of linear operators in geometric function theory. Our main purpose is to define some subclasses of analytic functions by means of the q-analogue of linear operators and investigate various inclusion relationships with integral preserving properties. [ABSTRACT FROM AUTHOR]

The class B¯ B(α) of non-Bazilevič functions was introduced by Obradović. Later, estimates on the second coefficient and Fekete-Szegö functional for normalized analytic functions in the class B¯ B(α) were investigated by Tuneski and Darus. In the present work, sharp estimate on third to eighth coefficients for normalized analytic functions f(z) = z + a2z2 + a3z3 + … ∊ B¯(α) are investigated. Further sharp estimate on the functional ®a2a3 - a4® is also obtained. [ABSTRACT FROM AUTHOR]

A new approach to modeling the geometric structure of real materials based on a new type of functions – porous functions – is introduced. The general concept, basic operations, operations of calculus, visualization and interpretations, and potential applications of porous functions are outlined. [ABSTRACT FROM AUTHOR]

Geometric Function Theory is a remarkable area in complex analysis and it plays a significant role and contributes new ideas and results even today. The study of univalent and multivalent functions is one of the main branches of geometric function theory and one of the most fascinating aspects of the theory of analytical functions of a complex variable. The purpose of this paper is to introduce new classes of uniformly convex functions Rp(γ, α, β, q) and KRp(γ, α, β, q) in U. Also obtain some sufficient and necessary conditions for the function belongs to these classes. As well as, we investigate some geometri [ABSTRACT FROM AUTHOR]

Let 𝓐 denote the family of analytic functions f with f(0) = f′(0) – 1 = 0, in the open unit disk Δ. We consider a class S c s ∗ (α) := f ∈ A : z f ′ (z) f (z) − 1 ≺ z 1 + α − 1 z − α z 2 , z ∈ Δ , $$\begin{array}{} \displaystyle \mathcal{S}^{\ast}_{cs}(\alpha):=\left\{f\in\mathcal{A} : \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z}{1+\left(\alpha-1\right) z-\alpha z^2}, \,\, z\in \Delta\right\}, \end{array}$$ where 0 ≤ α ≤ 1/2, and ≺ is the subordination relation. The methods and techniques of geometric function theory are used to get characteristics of the functions in this class. Further, the sharp inequality for the logarithmic coefficients γn of f ∈ S c s ∗ $\begin{array}{} \mathcal{S}^{\ast}_{cs} \end{array}$ (α): ∑ n = 1 ∞ γ n 2 ≤ 1 4 1 + α 2 π 2 6 − 2 L i 2 − α + L i 2 α 2 , $$\begin{array}{} \displaystyle \sum_{n=1}^{\infty}\left|\gamma_n\right|^2 \leq \frac{1}{4\left(1+\alpha\right)^2}\left(\frac{\pi^2}{6}-2 \mathrm{Li}_2\left(-\alpha\right)+ \mathrm{Li}_2\left(\alpha^2\right)\right), \end{array}$$ where Li2 denotes the dilogarithm function are investigated. [ABSTRACT FROM AUTHOR]

We formulate the equations for point vortex dynamics on a closed two-dimensional Riemannian manifold in the language of affine and other kinds of connections. This can be viewed as a relaxation of standard approaches, using the Riemannian metric directly, to an approach based more on local coordinates provided with a minimal amount of extra structure. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin function and the Levi-Civita connection associated with the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevantHamiltonian function consists of two main terms. One of the terms is the wellknown quadratic form based on a matrix whose entries are Green and Robin functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green functions. One main issue of the paper is a detailed analysis of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian. This analysis confirms the mentioned dynamical equations formulated in terms of connections. [ABSTRACT FROM AUTHOR]

Some extremal problems of the geometric theory of functions of a complex variable related to the estimates of functionals defined on systems of non-overlapping domains are considered. Till now, many such problems have not been solved, though some partial solutions are available. In the paper, the improved method is proposed for solving the problems on extremal decomposition of the complex plane. The main results generalize and strengthen some known results in the theory of non-overlapping domains with free poles to the case of an arbitrary arrangement of systems of points on the complex plane. [ABSTRACT FROM AUTHOR]

We review the main results of the geometric theory of ordinary differential equations obtained by the prominent Russian geometer N. V. Stepanov (1926–1991). Some results obtained by Stepanov are illustrated by examples of third- and five-order equations. [ABSTRACT FROM AUTHOR]

In the present study, by using the Horadam Polnomials and q-analogue of Noor integral oprerator, we target to construct an interesting connection between the geometric function theory and that of special functions. Also, by defining a new class of bi-univalent analytic functions, we investigate coefficient estimates and famous Fekete-Szegö inequality for functions belonging to this interesting class. [ABSTRACT FROM AUTHOR]

Recently, Lucas polynomials and other special polynomials gained importance in the field of geometric function theory. In this study, by connecting these polynomials, subordination, and the Al-Oboudi differential operator, we introduce a new class of bi-univalent functions and obtain coefficient estimates and Fekete-Szegö inequalities for this new class. [ABSTRACT FROM AUTHOR]

The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin's paper in the journal "Russian Mathematical Surveys" in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1; 2:57] and generalized this result to the case of multidimensional complex space. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of analytic functions associated with Mittag-Leffler fuction in the open unit disk. Several properties of functions belonging to this class are derived. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of the continuous Newton method for solving nonlinear equations with holomorphic mappings in complex Banach spaces. Our contribution is based on recent progress in the geometric theory of spiral-like functions. We prove convergence theorems and illustrate them by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the concept of fuzzy differential subordination is used in the fiield of Geometric Function Theory. Certain integral operators are studied with reference to this newly introduced notion. Some interesting results with special cases are discussed. [ABSTRACT FROM AUTHOR]

We introduce the compactness locus of a geometric functor between rigidly‐compactly generated tensor‐triangulated categories, and describe it for several examples arising in equivariant homotopy theory and algebraic geometry. It is a subset of the tensor‐triangular spectrum of the target category which, crudely speaking, measures the failure of the functor to satisfy Grothendieck–Neeman duality (or equivalently, to admit a left adjoint). We prove that any geometric functor — even one which does not admit a left adjoint — gives rise to a Wirthmüller isomorphism once one passes to a colocalization of the target category determined by the compactness locus. When applied to the inflation functor in equivariant stable homotopy theory, this produces the Adams isomorphism. [ABSTRACT FROM AUTHOR]

Dorff et al. ['Convolutions of harmonic convex mappings', Complex Var. Elliptic Equ. 57 (5) (2012), 489–503] formulated a question concerning the convolution of two right half-plane mappings, where the normalisation of the functions was considered incorrectly. In this paper, we reformulate the problem correctly and provide a solution to it in a more general form. We also obtain two new theorems which correct and improve related results. [ABSTRACT FROM AUTHOR]

We aim to present a new linear integral operator and a certain subclass of regular functions with bounded turning property in correlation with the extended generalised Mittag-Leffler functions. Several results concerning an application of differential subordination are also investigated. [ABSTRACT FROM AUTHOR]

We construct counterexamples to classical calculus facts such as the inverse and implicit function theorems in scale calculus--a generalization of multivariable calculus to infinite-dimensional vector spaces, in which the reparameterization maps relevant to symplectic geometry are smooth. Scale calculus is a corner stone of polyfold theory, which was introduced by Hofer, Wysocki, and Zehnder as a broadly applicable tool for regularizing moduli spaces of pseudoholomorphic curves. We show how the novel nonlinear scale-Fredholm notion in polyfold theory overcomes the lack of implicit function theorems, by formally establishing an often implicitly used fact: The differentials of basic germs--the local models for scale-Fredholm maps--vary continuously in the space of bounded operators when the base point changes. We moreover demonstrate that this continuity holds only in specific coordinates, by constructing an example of a scale-diffeomorphism and scale-Fredholm map with discontinuous differentials. This justifies the high technical complexity in the foundations of polyfold theory. [ABSTRACT FROM AUTHOR]

In this paper, by using the (P, Q) -Lucas polynomials and the q-analogue of the Noor integral operator, we aim to build a bridge between the theory of geometric functions and that of special functions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate q- analogues of Bieberbach- de Branges theorems and Fekete-Szegö inequalities for certain families of q- convex and q- close-to-convex functions. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to introduce a class Dκ,α λ,δ C0 (ß) of bi-concave functions defined by a differential operator. We find estimates on the Taylor-Maclaurin coefficients |a2 | and |a3 | for functions in this class. Several consequences of these results are also pointed out in the form of corollaries. [ABSTRACT FROM AUTHOR]

Abstract Suppose H is a finite dimensional reproducing kernel Hilbert space of functions on X. If H has the complete Pick property then there is an isometric map, Φ, from X , with the metric induced by H , into complex hyperbolic space, CH n , with its pseudohyperbolic metric. We investigate the relationships between the geometry of Φ (X) and the function theory of H and its multiplier algebra. [ABSTRACT FROM AUTHOR]

Abstract Group encryption (GE) is the natural encryption analogue of group signatures in that it allows verifiably encrypting messages for some anonymous member of a group while providing evidence that the receiver is a properly certified group member. Should the need arise, an opening authority is capable of identifying the receiver of any ciphertext. As introduced by Kiayias, Tsiounis and Yung (Asiacrypt'07), GE is motivated by applications in the context of oblivious retriever storage systems, anonymous third parties and hierarchical group signatures. This paper provides the first realization of group encryption under lattice assumptions. Our construction is proved secure in the standard model (assuming interaction in the proving phase) under the Learning-With-Errors (LWE) and Short-Integer-Solution (SIS) assumptions. As a crucial component of our system, we describe a new zero-knowledge argument system allowing to demonstrate that a given ciphertext is a valid encryption under some hidden but certified public key, which incurs to prove quadratic statements about LWE relations. Specifically, our protocol allows arguing knowledge of witnesses consisting of X ∈ Z q m × n , s ∈ Z q n and a small-norm e ∈ Z m which underlie a public vector b = X ⋅ s + e ∈ Z q m while simultaneously proving that the matrix X ∈ Z q m × n has been correctly certified. We believe our proof system to be useful in other applications involving zero-knowledge proofs in the lattice setting. [ABSTRACT FROM AUTHOR]