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Let S 1 , S 2 , S 3 {S_{1},S_{2},S_{3}} be oriented non-flat minimal surfaces immersed in ℝ n + 1 {{\mathbb{R}}^{n+1}} ( n ≥ 2 ) {(n\geq 2)} with the Gauss maps G 1 , G 2 , G 3 {G_{1},G_{2},G_{3}} into ℙ n ⁢ ( ℂ ) {{\mathbb{P}}^{n}({\mathbb{C}})} , respectively. Assume that there are conformal diffeomorphisms Φ 2 , Φ 3 {\Phi_{2},\Phi_{3}} of S 1 {S_{1}} onto S 2 , S 3 {S_{2},S_{3}} respectively and let f 1 = G 1 {f^{1}=G_{1}} , f 2 = G 2 ∘ Φ 2 {f^{2}=G_{2}\circ\Phi_{2}} , f 3 = G 3 ∘ Φ 3 {f^{3}=G_{3}\circ\Phi_{3}} . In this paper, we will show that f 1 , f 2 , f 3 {f^{1},f^{2},f^{3}} are algebraic dependence, i.e., f 1 ∧ f 2 ∧ f 3 ≡ 0 {f^{1}\wedge f^{2}\wedge f^{3}\equiv 0} , if they have the same inverse images for a few hyperplanes of ℙ n ⁢ ( ℂ ) {{\mathbb{P}}^{n}({\mathbb{C}})} in general position with some certain conditions. [ABSTRACT FROM AUTHOR]

We present a condition under which the thermal quasi-geostrophic (TQG) model possesses a solution that is holomorphic in time with values in the Gevrey space of complex analytic functions. This can be seen as the complex extension of the work by Levermore and Oliver (1997) for the generalised Euler equation but applied to the TQG model. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a refinement of the coefficient inequality for subordinate functions on the unit disc U in ℂ . Next, as applications of this inequality, we will obtain some refinements of the Fekete and Szegő inequalities for a class of holomorphic mappings associated with starlike mappings and quasi-convex mappings of type B on the unit ball B of a complex Banach space. The results presented here would generalize and improve some recent works of several authors [12, 20, 23]. [ABSTRACT FROM AUTHOR]

The structural complexity of high-tech industries is often compromised by a combination of thermal, mechanical, and geometric weaknesses. New generation materials and engineering the structure of materials are among the techniques that engineers employ to eliminate these effects. In this study, a comprehensive analysis solution is derived using Lekhnitskii's complex variable approach with the use of general mapping functions, the concept of functionally graded materials (FGMs), and holomorphic functions in the form of Laurent series. This general solution is used for the thermoelastic analysis of perforated functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates with polygonal hole. A refined-calibrated rule of mixtures is used to approximate the material property of FG-CNTRC plates according to gradational changes in direction of thickness and available molecular dynamics simulations results. After validation of present analytical solution results with finite element analysis results and available mechanical analysis of composite plates results, stress and moment resultants due to remoting heat flux-mechanical loading is studied. The effect of FG-CNTRC material properties, heat flux condition, and four parameters affecting the shape of the polygonal holes has been investigated. During the present parametric analysis, the results clearly show that the parameters related to the FG-CNTRC material properties, flux conditions, and hole geometry each provide a reliable tool for designers to influence the stress and moment resultants to minimize undesirable stresses. This general formulation is able to calculate thermoelastic parameters (thermal and mechanical parameters, separately) for the generalized problems of the FGM plate or composite laminates with a polygonal hole. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give expressions for modular linear differential operators (MLDOs) of any order. In particular, we show that they can all be described in terms of Rankin-Cohen brackets and a modified Rankin-Cohen bracket found by Kaneko and Koike. We also give more uniform descriptions of MLDOs in terms of canonically defined higher Serre derivatives and an extension of Rankin-Cohen brackets, as well as in terms of quasimodular forms and almost holomorphic modular forms. The last of these descriptions involves the holomorphic projection map. The paper also includes some general results on the theory of quasimodular forms on both cocompact and non-cocompact subgroups of SL_2(\mathbb {R}), as well as a slight sharpening of a theorem of Martin and Royer on Rankin-Cohen brackets of quasimodular forms. [ABSTRACT FROM AUTHOR]

We investigate harmonic maps on almost contact metric manifolds which are locally conformal to almost cosymplectic manifolds. We obtain the necessary and sufficient conditions for the holomorphy to imply harmonicity and then we find obstructions to the existence of non-constant pluriharmonic maps. We also establish some results on the stability of the identity map on a locally conformal almost cosymplectic manifold of pointwise constant φ-holomorphic sectional curvature. [ABSTRACT FROM AUTHOR]

Let Hp = Hp(Bd) denote the Hardy space in the open unit ball Bd of C d , d ≥ 1. We characterize the reverse Carleson measures for Hp, 1 < p < ∞, that is, we describe all finite positive Borel measures μ defined on the closed ball B ¯ d and such that ‖ f ‖ H p ≤ c ‖ f ‖ L p B ¯ d , μ for all f ∈ Hp(Bd) ∩ C( B ¯ d ) and a universal constant c > 0. Given a non-inner holomorphic function b : Bd → B1, we obtain properties of the reverse Carleson measures for the de Branges–Rovnyak space H b . [ABSTRACT FROM AUTHOR]

Herglotz's representation of holomorphic functions with positive real part and Carathéodory's theorem on approximation by inner functions are two well‐known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi‐connected domain Ω$\Omega$, a version of Heglotz's representation is known. Carathéodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Korányi and Pukánszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carathéodory's approximation theorem while proving this new form of Herglotz's representation. Carathéodory's approximation theorem is also proved for operator‐valued functions on a multi‐connected domain. [ABSTRACT FROM AUTHOR]

Let \mathrm {L}^2_a(\mathbb {D}) be the classical Bergman space and let M_h denote the operator of multiplication by a bounded holomorphic function h. Let B be a finite Blaschke product of order n. An open question proposed by R. G. Douglas is whether the operators M_B on \mathrm {L}^2_a(\mathbb {D}) similar to \oplus _1^n M_z on \oplus _1^n \mathrm {L}^2_a(\mathbb {D})? The question was answered in the affirmative, not only for Bergman space but also for many other Hilbert spaces with reproducing kernel. Since the operator M_z^* is in Cowen-Douglas class B_1(\mathbb {D}) in many cases, Douglas question can be reformulated for operators in B_1(\mathbb {D}), and the answer is affirmative for many operators in B_1(\mathbb {D}). A natural question occurs for operators in Cowen-Douglas class B_n(\mathbb {D}) (n>1). In this paper, we investigate a family of operators, which are in a norm dense subclass of Cowen-Douglas class B_2(\mathbb {D}), and give a negative answer. [ABSTRACT FROM AUTHOR]

In this paper, we define a new class R t , δ , υ m , n , σ (A , B) of holomorphic functions in the open unit disk defined connected with the combination binomial series and Babalola operator using the differential subordination with Janowski-type functions. Using the well-known Carathéodory's inequality for function with real positive parts and the Keogh-Merkes and Ma-Minda's in equalities, we determined the upper bound for the first two initial coefficients of the Taylor-Maclaurin power series expansion. Then, we found an upper bound for the Fekete-Szegö functional for the functions in this family. Further, a similar result for the first two coefficients and for the Fekete-Szegő inequality have been done the function G − 1 when G ∈ R t , δ , υ m , n , σ (A , B) . Next, for the functions of these newly defined family we determine coefficient estimates, distortion bounds, radius problems, and the radius of starlikeness and close-to-convexity. The novelty of the results is that we were able to investigate basic properties of these new classes of functions using simple methods and these classes are connected with the new convolution operator and the Janowski functions. [ABSTRACT FROM AUTHOR]

We propose a second-order method for unconditional minimization of functions f(z) of complex arguments. We call it the mixed Newton method due to the use of the mixed Wirtinger derivative ∂ 2 f ∂ z ¯ ∂ z for computation of the search direction, as opposed to the full Hessian ∂ 2 f ∂ (z , z ¯) 2 in the classical Newton method. The method has been developed for specific applications in wireless network communications, but its global convergence properties are shown to be superior on a more general class of functions f, namely sums of squares of absolute values of holomorphic functions. In particular, for such objective functions minima are surrounded by attraction basins, while the iterates are repelled from other types of critical points. We provide formulas for the asymptotic convergence rate and show that in the scalar case the method reduces to the well-known complex Newton method for the search of zeros of holomorphic functions. In this case, it exhibits generically fractal global convergence patterns. [ABSTRACT FROM AUTHOR]

This paper addresses the classification problem of integrable deformations of solutions of “degenerate” Riemann–Hilbert–Birkhoff (RHB) problems. These consist of those RHB problems whose initial datum has diagonal pole part with coalescing eigenvalues. On the one hand, according to theorems of Malgrange, Jimbo, Miwa, and Ueno, in the non-degenerate case, there exists a universal integrable deformation inducing (via a unique map) all other deformations [M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformations of linear ordinary differential equations with rational coefficients I, Physica D 2 (1981) 306–352; B. Malgrange, Déformations de systèmes différentiels et microdifférentiels, in Séminaire E.N.S. Mathématique et Physique, eds. L. Boutet de Monvel, A. Douady and J.-L. Verdier, Progress in Mathematics, Vol. 37 (Birkhäuser, Basel, 1983), pp. 351–379; B.Malgrange, Sur les déformations isomonodromiques, II, in Séminaire E.N.S. Mathématique et Physique, eds. L. Boutet de Monvel, A. Douady and J.-L. Verdier, Progress in Mathematics, Vol. 37 (Birkhäuser, Basel, 1983), pp. 427–438; B. Malgrange, Deformations of differential systems, II, J. Ramanujan Math. Soc. 1 (1986) 3–15]. On the other hand, in the degenerate case, Sabbah proved, under sharp conditions, the existence of an integrable deformation of solutions, sharing many properties of the one constructed by Malgrange–Jimbo–Miwa–Ueno [C.Sabbah, Integrable deformations and degenerations of some irregular singularities, Publ. RIMS Kyoto Univ. 57(3–4) (2021) 755–794; arXiv:1711.08514v3]. Albeit the integrable deformation constructed by Sabbah is not, stricto sensu, universal, we prove that it satisfies a relative universal property. We show the existence and uniqueness of a maximal class of integrable deformations all induced (via a unique map) by Sabbah’s integrable deformation. Furthermore, we show that such a class is large enough to include all generic integrable deformations whose pole and deformation parts are locally holomorphically diagonalizable. In itinere, we also obtain a characterization of holomorphic matrix-valued maps which are locally holomorphically Jordanizable. This extends, to the case of several complex variables, already known results independently obtained by Thijsse and Wasow [Ph. G. A.Thijsse, Global holomorphic similarity to a Jordan form, Results Math. 8 (1985) 78–87; W.Wasow, Linear Turning Point Theory (Springer-Verlag, New York, 1985)]. [ABSTRACT FROM AUTHOR]

In this paper, we first derive a Landau-type theorem and a Bloch-type theorem for the class of meromorphic harmonic mappings. Then we establish two new versions of Bloch-type theorems for certain K-quasiregular meromorphic harmonic mappings in the unit disk. [ABSTRACT FROM AUTHOR]

We study the stable dynamics of non-polynomial automorphisms of ℂ2, more precisely we construct a special class of transcendental Hénon maps of the form F(z,w) = (e−zm + δe2π m iw,z) with m ≥ 2 a natural number and ℝ ∋ δ > 2.One of the points of greatest interest is that we get a combinatorial dynamic behavior, indeed there are cycles of escaping Fatou components where the dynamics vary depending on whether m is even or odd. Moreover, we have two distinct limit functions on each cycle, both of which have generic rank 1. Additionally, each Fatou component in each cycle has two disjoint and hyperbolic limit sets on the line at infinity, with the exception of the Fatou components belonging to the short cycle of period m (that occurs if m is odd) which have the same hyperbolic limit set. [ABSTRACT FROM AUTHOR]

This survey considers continuous functions defined on the boundary of a bounded domain D in C n , n > 1, and possessing the Morera boundary property. We study the question of the existence of a holomorphic extension of such functions into the domain D for some sufficient sets Γ of complex lines. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we investigate the approximation property for the predual of the weighted space of holomorphic mappings, where is a countable family of weights defined on a balanced open subset U of a Banach space E. We introduce a locally convex topology on and show that is topologically isomorphic to for a suitably restricted family . As a consequence, certain characterizations of the approximation property for the space are proved. [ABSTRACT FROM AUTHOR]

Considering the bundle of 2-jets as a realization of the holomorphic manifold over 3-dimensional nilpotent algebra, the authors introduce a new class of lifts of connections in the bundle of 2-jets which is a generalization of the complete lifts. [ABSTRACT FROM AUTHOR]

We suggest a general construction of continuous Banach bundles of holomorphic function algebras on subvarieties of the closed noncommutative ball. These algebras are of the form , where is the noncommutative disc algebra defined by G. Popescu, and is the closure in of a graded ideal in the algebra of noncommutative polynomials, depending continuously on a point of a topological space . Moreover, we construct bundles of Fréchet algebras of holomorphic functions on subvarieties of the open noncommutative ball. The algebra of free holomorphic functions on the unit ball was also introduced by G. Popescu, and stands for the closure in of a graded ideal in the algebra of noncommutative polynomials, depending continuously on a point . [ABSTRACT FROM AUTHOR]

Biunivalent holomorphic functions form an interesting class in geometric function theory and are connected with special functions and solutions of complex differential equations. This class has been investigated by many authors, mainly to find the coefficient estimates. The assumption of biunivalence is rigid; this rigidity means that, for example, only the initial Taylor coefficients have been estimated. The aim of this paper is to develop a variational technique for biunivalent functions, which provides a power tool for solving the general extremal problems on the classes of such functions. It involves quasiconformal analysis. [ABSTRACT FROM AUTHOR]

The space \mathbf {H}^{4,2} of vectors of norm -1 in \mathbb {R}^{4,3} has a natural pseudo-Riemannian metric and a compatible almost complex structure. The group of automorphisms of both of these structures is the split real form \mathsf {G}_2'. In this paper we consider a class of holomorphic curves in \mathbf {H}^{4,2} which we call alternating. We show that such curves admit a so called Frenet framing. Using this framing, we show that the space of alternating holomorphic curves which are equivariant with respect to a surface group is naturally parameterized by certain \mathsf {G}_2'-Higgs bundles. This leads to a holomorphic description of the moduli space as a fibration over Teichmüller space with a holomorphic action of the mapping class group. Using a generalization of Labourie's cyclic surfaces, we then show that equivariant alternating holomorphic curves are infinitesimally rigid. [ABSTRACT FROM AUTHOR]

Let $H^\infty $ be the algebra of bounded holomorphic functions on the open unit disk, and let $\mathfrak M$ be its maximal ideal space. Let $\mathfrak M_a$ be the union of nontrivial Gleason parts (analytic disks) of $\mathfrak M$. In this paper, we study the problem of extensions of bounded Banach-valued holomorphic functions and holomorphic maps with values in Oka manifolds from Gleason parts of $\mathfrak M_a\setminus \mathbb {D}$. The resulting extensions satisfy the uniform boundedness principle in the sense that their norms are bounded by constants that do not depend on the choice of the Gleason part. The results extend fundamental results of D. Suárez on the characterization of the algebra of restrictions of Gelfand transforms of functions in $H^\infty $ to Gleason parts of $ \mathfrak M_a\setminus \mathbb {D}$. The proofs utilize our recent advances on $\bar \partial $ -equations on quasi-interpolating sets and Runge-type approximations. [ABSTRACT FROM AUTHOR]

Computation of the exact norm of the composition operator acting on the spaces of holomorphic functions has shown to be difficult to perform. In this paper, we provide estimates of the norm of this operator acting on the mixed norm space Hp,q, α . For some values of parameters p,q and α, the proposed upper bound of the norm generalizes the upper bound of the norm of the composition operator acting on the weighted Bergman space. [ABSTRACT FROM AUTHOR]

We investigate a widely used application of compactness of bounded linear operators T : X(B) →Y, where X(B) is a Banach space of holomorphic functions on the open unit ball B ⊂ CN and Y is a Banach space. In particular, we show that compactness of the operator when X(B) is not reflexive, is not a sufficient condition for the property that every bounded sequence (fn)n∈N in X(B) such that fn→0 with respect to the compact open topology as n→∞, implies that T(fn)→0 with respect to the norm of Y as n→∞. [ABSTRACT FROM AUTHOR]

In this paper, we define a new class $ \mathcal{R}_{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right) $ of holomorphic functions in the open unit disk defined connected with the combination binomial series and Babalola operator using the differential subordination with Janowski-type functions. Using the well-known Carathéodory's inequality for function with real positive parts and the Keogh-Merkes and Ma-Minda's in equalities, we determined the upper bound for the first two initial coefficients of the Taylor-Maclaurin power series expansion. Then, we found an upper bound for the Fekete-Szegö functional for the functions in this family. Further, a similar result for the first two coefficients and for the Fekete-Szegő inequality have been done the function $ \mathcal{G}^{-1} $ when $ \mathcal{G}\in \mathcal{R} _{t, \delta, \upsilon }^{m, n, \sigma }\left(\mathcal{A}, \mathcal{B}\right) $. Next, for the functions of these newly defined family we determine coefficient estimates, distortion bounds, radius problems, and the radius of starlikeness and close-to-convexity. The novelty of the results is that we were able to investigate basic properties of these new classes of functions using simple methods and these classes are connected with the new convolution operator and the Janowski functions.

Let D = {{ C : ||| < 1} be the open unit disk in the complex plane C and let H(D) be the space of all holomorphic functions on D. For a non-negative integer n and a function f H(D), the nthh order differentiation operator is defined as Dnf = f(n). The weighted composition operator together with nthh order differentiation operator give rise to a new operator generally termed as generalized weighted composition operator denoted by Wn, and is defined by Wn, f(() = (()f(n)(((()), f H(D); D, where H(D) and is a holomorphic self-map of D. This operator is basically the combination of multiplication operator MM, composition operator CC and nthh order differentiation operator Dn. We study the boundedness and compactness of this operator between Dirichlet-type spaces and Bloch-type spaces. [ABSTRACT FROM AUTHOR]

It has been conjectured that every stable manifold arising from a holomorphic automorphism, that acts hyperbolically on a compact invariant set, is biholomorphic to complex Euclidean space. Such stable manifolds are known to be biholomorphic to the basin of a uniformly attracting family of holomorphic maps. It is shown that the basin of a uniformly attracting family of holomorphic maps is biholomorphic to complex Euclidean space and this resolves the conjecture on the biholomorphism type of such stable manifolds affirmatively. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new generalized squeezing function defined on bounded domains in $ \mathbb {C}^{n} $ C n and we describe its properties. We compare our generalized squeezing function with the squeezing function corresponding to the polydisk and we show that fundamental properties such as invariance under biholomorphic transformations, existence of extremal maps and continuity remain valid in our frame. Finally, we examine the relationship between the limit of squeezing functions of a sequence of domains and the squeezing function of the limit domain. [ABSTRACT FROM AUTHOR]

In the recent years, the study of the Hankel determinant problems have been widely investigated by many researchers. We were essentially motivated by the recent research going on with the Hankel determinant and other coefficient bounds problems. In this research article, we first considered the subclass of analytic starlike functions connected with the domain of the tangent function. We then derived the initial four sharp coefficient bounds, the sharp Fekete-Szegö inequality, and the sharp second and third order Hankel determinant for the defined class. Also, we derived sharp estimates like sharp coefficient bounds, Fekete-Szegö estimate, and sharp second order Hankel determinant for the functions having logarithmic coefficient and for the inverse coefficient, respectively, for the defined functions class.

Padé approximations are approximations of holomorphic functions by rational functions. The application of Padé approximations to Diophantine approximations has a long history dating back to Hermite. In this paper, we use the Maier–Chudnovsky construction of Padé-type approximation to study irrationality properties about values of functions with the form f (x) = ∑ k = 0 ∞ x k k ! (b k + s) (b k + s + 1) ⋯ (b k + t) , where b , t , s are positive integers and obtain upper bounds for irrationality measures of their values at nonzero rational points. Important examples includes exponential integral, Gauss error function and Kummer's confluent hypergeometric functions. [ABSTRACT FROM AUTHOR]

The aim of this research is to investigate certain problems of differential subordination and superordination of analytic univalent functions in an open unit disk. Furthermore, the universal by‐product operator and geometric properties such as coefficient inequalities are investigated. Remarkable results on the differential subordination and superordination of analytic univalent functions and results using higher derivative operator for differential subordination are obtained. Some interesting applications of subordination in the unit disk are obtained using convolution and convexity. [ABSTRACT FROM AUTHOR]

The given text discusses the concept of representative points in a graph and how they can be used to define a function. It explains that by choosing three rational points, one inside each region of the graph, a function can be created. The text aims to show that this function is injective, meaning that it is one-to-one. It provides a diagram and references a related problem about the possibility of having uncountably many disjoint figure 8s in the plane. The text concludes by mentioning that the result discussed is due to R. L. Moore. [Extracted from the article]

This paper addresses the weighted composition operators C ϕ ψ from the (α , k) -Bloch spaces to the A (β , k) spaces of bounded holomorphic functions on W, where W is a generalized Hua domain of the fourth kind. Additionally, we obtain some necessary and sufficient conditions for the boundedness and compactness of these operators. [ABSTRACT FROM AUTHOR]

In this paper, we scrutinize a collection of meromorphic functions known as normal families, prove the theorem that normal families share a holomorphic function, and present several illustrative counterexamples. [ABSTRACT FROM AUTHOR]

First we survey and explain the strategy of some recent results that construct holomorphic sl (2 , ℂ) -differential systems over some Riemann surfaces Σ g of genus g ≥ 2 , satisfying the condition that the image of the associated monodromy homomorphism is (real) Fuchsian [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, Fuchsian sl (2 , ℂ) -systems of compact Riemann surfaces [with an appendix by Takuro Mochizuki], preprint, arXiv:org/abs/2104.04818] or some cocompact Kleinian subgroup Γ ⊂ SL (2 , ℂ) as in [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of SL (2 , ℂ) , preprint, arXiv:org/abs/2112.03131]. As a consequence, there exist holomorphic maps from Σ g to the quotient space SL (2 , ℂ) / Γ , where Γ ⊂ SL (2 , ℂ) is a cocompact lattice, that do not factor through any elliptic curve [I. Biswas, S. Dumitrescu, L. Heller and S. Heller, On the existence of holomorphic curves in compact quotients of SL (2 , ℂ) , preprint, arXiv:org/abs/2112.03131]. This answers positively a question of Ghys in [E. Ghys, Déformations des structures complexes sur les espaces homogènes de SL (2 , ℂ) , J. Reine Angew. Math. 468 (1995) 113–138]; the question was also raised by Huckleberry and Winkelmann in [A. H. Huckleberry and J. Winkelmann, Subvarieties of parallelizable manifolds, Math. Ann. 295 (1993) 469–483]. Then we prove that when M is a Riemann surface, a Torelli-type theorem holds for the affine group scheme over ℂ obtained from the category of holomorphic connections on étale trivial holomorphic bundles. After that, we explain how to compute in a simple way the holonomy of a holomorphic connection on a free vector bundle. Finally, for a compact Kähler manifold M , we investigate the neutral Tannakian category given by the holomorphic connections on étale trivial holomorphic bundles over M. If ϖ (respectively, Θ) stands for the affine group scheme over ℂ obtained from the category of connections (respectively, connections on free (trivial) vector bundles), then the natural inclusion produces a morphism v : (Θ) → (ϖ) of Hopf algebras. We present a description of the transpose of v in terms of the iterated integrals. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some new truncated second main theorems by using the Jackson p -Casorati determinant, where the truncated counting functions are assigned varying weights. We specifically concentrate on the Jackson difference operator applied to zero-order holomorphic mappings intersecting a finite set of slowly moving targets in P n (C) . As an application, we prove a uniqueness theorem of meromorphic functions sharing some small functions with the Jackson-type counting functions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of singular directions, singular radii and indirect singular points of holomorphic curves in P n ⁢ ( ℂ ) {P^{n}(\mathbb{C})} . [ABSTRACT FROM AUTHOR]

For weighted -spaces of holomorphic functions in the unit bidisc, the following problem is investigated: what type of function spaces in the unit matrix disc are obtained when the mentioned spaces are extended to the domain . [ABSTRACT FROM AUTHOR]

We extend classical results of Perego and Rapagnetta on moduli spaces of sheaves of type OG10 to moduli spaces of Bridgeland semistable objects on the Kuznetsov component of a cubic fourfold. In particular, we determine the period of this class of varieties and use it to understand when they become birational to moduli spaces of sheaves on a K3 surface. [ABSTRACT FROM AUTHOR]

In the recent years, the study of the Hankel determinant problems have been widely investigated by many researchers. We were essentially motivated by the recent research going on with the Hankel determinant and other coefficient bounds problems. In this research article, we first considered the subclass of analytic starlike functions connected with the domain of the tangent function. We then derived the initial four sharp coeffcient bounds, the sharp Fekete-Szegö inequality, and the sharp second and third order Hankel determinant for the defined class. Also, we derived sharp estimates like sharp coefficient bounds, Fekete-Szegö estimate, and sharp second order Hankel determinant for the functions having logarithmic coefficient and for the inverse coefficient, respectively, for the defined functions class. [ABSTRACT FROM AUTHOR]

We consider functions on the unit disc whose images are contained in a strip of the complex plane. Under an additional condition, such functions are constants. We also consider appropriate operator valued versions. Applications are found to the theory of semigroups. [ABSTRACT FROM AUTHOR]

In the current paper, we define some subclasses of holomorphic functions in the open disk u that are defined by the differential operator. These subclasses are defined in the open disk u. We make some estimates for the bounds of the Fekete-Szego inequality as well as the coefficients | a2 | and | a3 | for functions that belong to these subclasses. Also, some of the results of these subclasses have been generalized, particularly those that involve the majorization and quasi-subordination concepts. [ABSTRACT FROM AUTHOR]

We obtain different results about the mean ergodicity of weighted composition operators when acting on the spaces H(B)$H(B)$, Hb(B)$H_b(B)$, and H∞(B)$H^\infty (B)$, where B$B$ is the open unit ball of a Banach space, as well as about the compactness and the mean ergodicity of the multiplication operator. This study relates these properties of the operators with properties of the symbol and the weight defining such operators. [ABSTRACT FROM AUTHOR]

In the theory of bicomplex holomorphic functions, there is not concept of isolated singularities; that is, such functions do not have singularities just at a point like holomorphic functions in one complex variable. However, there are other type of singularities that behave similarly to the isolated singularities in one complex variable. In this work, we describe how they can be classified in such a way that it resembles the classification made for the complex analysis case. It turns out that to singularities there corresponds their orders which are hyperbolic numbers with integer components, not real integers. We give also the Residue Theorem in the bicomplex analysis setting. [ABSTRACT FROM AUTHOR]

In a previous work, the hyperbolic conformality for bicomplex functions was introduced, and it was proved that, with the adequate hypothesis, a bicomplex holomorphic function is hyperbolic conformal. The aim of this paper is to extend this idea to 픻n, with 픻 the set of hyperbolic numbers. Thus, the fundaments of the analysis in 픻n are presented here, as well as the generalization of some geometric hyperbolic objects that were defined in the context of bicomplex analysis. [ABSTRACT FROM AUTHOR]

The concepts of various kinds of asymptotically Toeplitz operators have been originally defined and studied for the Hardy–Hilbert space. The aim of this article is to extend these definitions to the case of operators acting on Banach spaces of holomorphic functions, including Hardy spaces, Hardy–Lorentz spaces, and Hardy–Orlicz spaces. We give a function-theoretic characterization of uniformly asymptotically Toeplitzness and mean weakly asymptotically Toeplitzness for weighted composition operators. We also investigate strong and weak asymptotic Toeplitzness of weighted composition operators. [ABSTRACT FROM AUTHOR]

Leibniz algebras are non-antisymmetric generalizations of Lie algebras. In this paper, we investigate the properties of complete Leibniz algebras under certain conditions on their extensions. Additionally, we explore the properties of derivations and direct sums of Leibniz algebras, proving several results analogous to those in Lie algebras. [ABSTRACT FROM AUTHOR]

The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine the geometric series, logarithm, and Hurwitz–Lerch zeta functions to formulate a new special function, namely, the logarithm-Hurwitz–Lerch zeta function (LHL-Z function). This investigation then contributes to the study of the LHL-Z function in terms of the geometric theory of holomorphic functions, based on the differential subordination methodology, to discuss and determine the univalence and convexity conditions of the LHL-Z function. Moreover, there are other subordination and superordination connections that may be visually represented using geometric methods. Functions often exhibit symmetry when subjected to conformal mappings. The investigation of the symmetries of these mappings may provide a clearer understanding of how subordination and superordination with the Hurwitz–Lerch zeta function behave under different transformations. [ABSTRACT FROM AUTHOR]

Carleman’s formulas solve the problem of recovery of a function from such a class of its values on the set of uniqueness M ⊂ ∂D for this class that does not contain the Shilov boundary ∂D. In this paper, by using the matrix upper half-plane and the biholomorphic equivalence of the matrix unit disc, the Carleman formula for the Cartesian product of matrix upper half-planes is proved. [ABSTRACT FROM AUTHOR]

We study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first‐level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire transcendental functions in holomorphic dynamics. [ABSTRACT FROM AUTHOR]

There is considered an interpolation problem f(λn) = bn in the class of holomorphic in the unit disk U(0; 1) = {z ε C: |z| < 1} functions of finite η-type, i.e such that... where η: [1;+∞) → [0;+∞) is an increasing convex function with respect to ln t and ln t = o (η(t)) (t → +∞). There were received sufficient conditions of the interpolation problem solvability in terms of the counting functions.... Earlier, in 2004, necessary conditions were obtained (Ukr. Math. J., 56 (2004), №3) in these terms. For the moderate growth of f (when the majorant η = ψ satisfies the condition ψ (2x) = O (ψ (x)), x → +∞) that problem was solved in J. Math. Anal. Appl., 414 (2014), №1. In this paper, we remove any restrictions on the growth of η and construct an interpolation function f such that... [ABSTRACT FROM AUTHOR]