Search

We present a result concerning the essential norm of linear operators mapping some general weighted‐type spaces of holomorphic functions on the open unit ball in ℂn$$ {\mathrm{\mathbb{C}}}^n $$ to Banach spaces. As some applications, we compare essential norms of a product‐type operator of integral type and a weighted composition operator between some of the general weighted‐type spaces. [ABSTRACT FROM AUTHOR]

Let X$X$ and Y$Y$ be complex Banach spaces with BX$B_X$ denoting the open unit ball of X$X$. This paper studies various aspects of the holomorphic Lipschitz space HL0(BX,Y)$\mathcal {H}L_0(B_X,Y)$, endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets Lip0(BX,Y)$\operatorname{Lip}_0(B_X,Y)$ of Lipschitz mappings and H∞(BX,Y)$\mathcal {H}^\infty (B_X,Y)$ of bounded holomorphic mappings, from BX$B_X$ to Y$Y$. Thanks to the Dixmier–Ng theorem, HL0(BX,C)$\mathcal {H}L_0(B_X, \mathbb {C})$ is indeed a dual space, whose predual G0(BX)$\mathcal {G}_0(B_X)$ shares linearization properties with both the Lipschitz‐free space and Dineen–Mujica predual of H∞(BX)$\mathcal {H}^\infty (B_X)$. We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that G0(BX)$\mathcal {G}_0(B_X)$ contains a 1‐complemented subspace isometric to X$X$ and that G0(X)$\mathcal {G}_0(X)$ has the (metric) approximation property whenever X$X$ has it. We also analyze when G0(BX)$\mathcal {G}_0(B_X)$ is a subspace of G0(BY)$\mathcal {G}_0(B_Y)$, and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context. [ABSTRACT FROM AUTHOR]

The aim of this work is to derive a symbol calculus on L 2 (R n) for multidimensional Hausdorff operators. Two aspects of this activity result in two almost independent parts. While throughout the perturbation matrices are supposed to be self-adjoint and form a commuting family, in the second part they are additionally assumed to be positive definite. What relates these two parts is the powerful method of diagonalization of a normal Hausdorff operator elaborated earlier by the second named author. [ABSTRACT FROM AUTHOR]

We determine the sharp Bohr inequality for holomorphic functions defined on a balanced domain G of a complex Banach space X and takes values from the unit disk. Also, we investigate the Bohr-type inequalities for harmonic functions $ f=h+\bar {g} $ f = h + g ¯ defined on a balanced domain $ G\subset X $ G ⊂ X. [ABSTRACT FROM AUTHOR]

The properties of the unique nontrivial analytic solution, defined implicitly by a functional equation, are pointed out. This work provides local estimations and global inequalities for the involved solution. The corresponding operatorial equation is studied as well. The second part of the paper is devoted to the full classical moment problem, which is an inverse problem. Two constraints are imposed on the solution. One of them requires the solution to be dominated by a concrete convex operator defined on the positive cone of the domain space. A one-dimensional operator is valued, and a multidimensional scalar moment problem is solved. In both cases, the existence and the uniqueness of the solution are proved. The general idea of the paper is to provide detailed information on solutions which are not expressible in terms of elementary functions. [ABSTRACT FROM AUTHOR]

We study the pointwise limit of a sequence of holomorphic functions, both of one and several complex variables. We analyze the set on which the limit is holomorphic, and the set on which it is not. [ABSTRACT FROM AUTHOR]

The study presented in this paper follows a line of research familiar for Geometric Function Theory, which consists in defining new integral operators and conducting studies for revealing certain geometric properties of those integral operators such as univalence, starlikness, or convexity. The present research focuses on the Bessel function of the first kind and order ν unveiling the conditions for this function to be univalent and further using its univalent form in order to define a new integral operator on the space of holomorphic functions. For particular values of the parameters implicated in the definition of the new integral operator involving the Bessel function of the first kind, the well-known Alexander, Libera, and Bernardi integral operators can be obtained. In the first part of the study, necessary and sufficient conditions are obtained for the Bessel function of the first kind and order ν to be a starlike function or starlike of order α ∈ [ 0 , 1) . The renowned prolific method of differential subordination due to Sanford S. Miller and Petru T. Mocanu is employed in the reasoning. In the second part of the study, the outcome of the first part is applied in order to introduce the new integral operator involving the form of the Bessel function of the first kind, which is starlike. Further investigations disclose the necessary and sufficient conditions for this new integral operator to be starlike or starlike of order 1 2 . [ABSTRACT FROM AUTHOR]

Based on Cauchy’s integral formula and conformal maps, this paper presents a new method for constructing barycentric rational interpolation formulae for complex functions, which may contain singularities such as poles, branch cuts, or essential singularities. The resulting interpolations are pole-free, exponentially convergent, and numerically stable, requiring only O (N) operations. Inspired by the logarithm equilibrium potential, we introduce a Möbius transform to concentrate nodes to the vicinity of singularity to get a spectacular improvement on approximation quality. A thorough convergence analysis is provided, alongside numerous numerical examples that illustrate the theoretical results and demonstrate the accuracy and efficiency of the methodology. Meanwhile, the paper also discusses some applications of the method including the numerical solutions of boundary value problems and the zero locations of holomorphic functions. [ABSTRACT FROM AUTHOR]

It is shown that if K⊂E$K\subset E$ and L⊂F$L\subset F$ are balanced compact determining subsets of separable Banach spaces with the approximation property, then the algebras of holomorphic germs H(K)$\mathcal {H}(K)$ and H(L)$\mathcal {H}(L)$ are topologically algebra isomorphic if, and only if, K̂P(E)$\hat{K}_{\mathcal {P}(E)}$ and L̂P(F)$\hat{L}_{\mathcal {P}(F)}$ (the polynomial convex hulls of K and L) are biholomorphically equivalent. [ABSTRACT FROM AUTHOR]

In the complex field, special functions are closely related to geometric holomorphic functions. Koebe function is a notable contribution to the study of the geometric function theory (GFT), which is a univalent function. This sequel introduces a new class that includes a more general Koebe function which is holomorphic in a complex domain. The purpose of this work is to present a new operator correlated with GFT. A new generalized Koebe operator is proposed in terms of the convolution principle. This Koebe operator refers to the generality of a prominent differential operator, namely the Ruscheweyh operator. Theoretical investigations in this effort lead to a number of implementations in the subordination function theory. The tight upper and lower bounds are discussed in the sense of subordinate structure. Consequently, the subordinate sandwich is acquired. Moreover, certain relevant specific cases are examined. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a special holomorphic function in U = { z ∈ C : | z | < 1 } which is associated with new generalized multiplier transformations. We investigate several properties of the defined function using the concept of subordination, then highlight a number of cases with interesting results. [ABSTRACT FROM AUTHOR]

Given an open subset U of C n , a weight v on U and a complex Banach space F, let H v (U , F) denote the Banach space of all weighted holomorphic mappings f : U → F , under the weighted supremum norm f v : = sup v (z) f (z) : z ∈ U. We prove that the set of all mappings f ∈ H v (U , F) that attain their weighted supremum norms is norm dense in H v (U , F) , provided that the closed unit ball of the little weighted holomorphic space H v 0 (U , F) is compact-open dense in the closed unit ball of H v (U , F). We also prove a similar result for mappings f ∈ H v (U , F) such that vf has a relatively compact range. [ABSTRACT FROM AUTHOR]

In this short note, by using a general three circles theorem, we show the rigidity of the sharp Bezout estimate first found by Gang Liu on nonnegatively curved Riemann surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we first obtain a refined version of the Bohr inequality of norm-type for holomorphic mappings with lacunary series on the polydisk in C n under some restricted conditions. Next, we determine the refined version of the Bohr inequality for holomorphic functions defined on a balanced domain G of a complex Banach space X and take values in the unit disk D . Furthermore, as a consequence of one of these results, we obtain a refined version of the Bohr-type inequality for harmonic functions f = h + g ¯ defined on a balanced domain G ⊂ X . All the results are proved to be sharp. [ABSTRACT FROM AUTHOR]

The main goal of this investigation is to obtain sharp upper bounds for Fekete-Szegö functional and the third Hankel determinant for a certain subclass SL ∗ u , v , α of holomorphic functions defined by the Carlson-Shaffer operator in the unit disk. Finally, for some special values of parameters, several corollaries were presented. [ABSTRACT FROM AUTHOR]

We determine the sharp Bohr inequality for holomorphic functions defined on a balanced domain G of a complex Banach space X and takes values from the unit disk. Also, we investigate the Bohr-type inequalities for harmonic functions f = h + g ¯ defined on a balanced domain G ⊂ X . [ABSTRACT FROM AUTHOR]

We have previously shown that a -fold holomorphic covering of a domain in the complex plane by another domain is extremal in the majorization principle for -valent functions and quadratic forms associated with Green's functions of these domains. In this paper, dual majorization principles involving both Green's and Neumann functions are obtained, in which -fold coverings are also extremal. The results are exemplified by applications of these principles to geometric function theory. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define and explore a certain class of analytic functions involving the (p , q) -Wanas operator related to the Janowski functions. We discuss geometric properties, growth and distortion bounds, necessary and sufficient conditions, the Fekete–Szegö problem, partial sums, and convex combinations for the newly defined class. We solve the Fekete–Szegö problem related to the convolution product and discuss applications to probability distribution. [ABSTRACT FROM AUTHOR]

In this paper, we define certain families S E * (ϑ) and C E (ϑ) of holomorphic and bi-univalent functions which are defined in the open unit disk U. We establish upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szegö type inequalities for functions in these families. [ABSTRACT FROM AUTHOR]

Dosi and, quite recently, the author showed that, on the character space of a nilpotent Lie algebra, there exists a sheaf of Fréchet–Arens–Michael algebras (of noncommutative holomorphic functions in the complex case and of noncommutative smooth functions in the real case). We construct similar sheaves (both versions, holomorphic and smooth) on a special space of representations for the Lie algebra of the group of affine transformations of the real line (which is the simplest nonnilpotent solvable Lie algebra). [ABSTRACT FROM AUTHOR]

In the present paper, making use of Gegenbauer polynomials, we initiate and explore a new family J Σ (λ , γ , s , t , q ; h) of holomorphic and bi-univalent functions which were defined in the unit disk D associated with the q-Srivastava–Attiya operator. We establish the bounds for | a 2 | and | a 3 | , where a 2 , a 3 are the initial Taylor–Maclaurin coefficients. For the new family of functions J Σ (λ , γ , s , t , q ; h) we investigate the Fekete-Szegö inequality, special cases and consequences. [ABSTRACT FROM AUTHOR]

We introduce a new subfamily of holomorphic functions and attempt to estimate an upper bound for the Hankel determinant of the second and third kind for the normalized regular mapping f, a member of this class. [ABSTRACT FROM AUTHOR]

We study the homogeneous Schwarz problem for Douglis analytic functions. We consider two-dimensional matrices J with a multiple eigenvalue and a eigenvector, which is not proportional to a real vector. We obtain a sufficient condition for the matrix J under which there exist two linearly independent solutions of the problem defined in a certain domain D. We present an example. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem on the existence of the upper (lower) envelope of a convex cone or, more generally, a convex set for functions on the projective limit of vector lattices with values in the completion of the Kantorovich space or on the extended real line. We propose vectorial, ordinal, and topological dual interpretations of the existence conditions for such envelopes and the method of constructing it. Applications to the problem on the existence of a nontrivial (pluri)subharmonic and/or (pluri)harmonic minorant for functions in domains of a finite-dimensional real or complex space are considered. We also propose general approaches to problems on the nontriviality of weight classes of holomorphic functions, describing zero (sub)sets for such classes of holomorphic functions, and to the problem of representing a meromorphic function as a ratio of holomorphic function from a given weight class. [ABSTRACT FROM AUTHOR]

Let H(𝔹n) be the set of all holomorphic functions on the open unit ball 𝔹n in ℂn, φ a holomorphic self‐map of 𝔹n, u∈H(𝔹n), and ℜm the mth iterated radial derivative operator on H(𝔹n). We characterize the metrical boundedness and metrical compactness of the weighted iterated radial composition operator ℜu,φm from the weighted Bergman–Orlicz space to the weighted‐type space. [ABSTRACT FROM AUTHOR]

A boundary version of the Schwarz lemma for meromorphic functions is investigated. For the function Inf(z) = 1/z +∑∞ k=2 knck-2zk-2, belonging to the class of W, we estimate from below the modulus of the angular derivative of the function on the boundary point of the unit disc. [ABSTRACT FROM AUTHOR]

We prove an analogue of one of Cesàro's results and use it to study the problem of interpolation by generalized exponential series, namely, by series whose terms themselves are Dirichlet series. [ABSTRACT FROM AUTHOR]

We consider a boundary version of the Schwarz Lemma on a certain class of functions which is denoted by N. For the function f(z) = z + a2z² + a3z³ + ... which is defined in the unit disc D such that the function f(z) belongs to the class N, we estimate from below the modulus of the angular derivative of the function f"(z)/f(z) at the boundary point c with f0(c) = 0. The sharpness of these inequalities is also proved. [ABSTRACT FROM AUTHOR]

In the present article, we introduce and study two certain classes of holomorphic prestarlike and bi-univalent functions associated with Bazilevič function. We determinate upper bounds for the Taylor-Maclaurin coefficients |a2| and |a3| for functions belonging to these classes. Further we point out certain special cases for our results. [ABSTRACT FROM AUTHOR]

Holomorphic embedding (HE) methods have been proposed to solve different power flow problems using tailored embedded systems; however, a solid theoretical foundation to ensure the effectiveness of HE methods is still missing. In addition, several analytical and numerical issues associated with HE methods remain to be resolved. The present study is devoted to the theoretical foundation of HE methods and develops existence theorems and sufficient conditions to deal with two key issues: how to ensure the holomorphic and conjugate properties of solution functions for a general embedded system for the power flow problem. Useful implications and guidelines are derived from these theoretical results. Examples and simulation results are conducted to demonstrate the developed theory. [ABSTRACT FROM AUTHOR]

The study of the Cauchy problem for solutions of the heat equation in a cylindrical domain with data on the lateral surface by the Fourier method raises the problem of calculating the inverse Laplace transform of the entire function . This problem has no solution in the standard theory of the Laplace transform. We give an explicit formula for the inverse Laplace transform of using the theory of analytic functionals. This solution suits well to efficiently develop the regularization of solutions to Cauchy problems for parabolic equations with data on noncharacteristic surfaces. [ABSTRACT FROM AUTHOR]

We show that a comparison of the capacities of suitable condensers gives an inequality for the Schwarzian derivatives of a holomorphic p-valent function defined on the unit disk and ranging in a given domain of the complex plane. If that domain is a disk as well and the function is univalent, then this inequality essentially coincides with the classical Nehari inequality. The cases of equality in the resulting relation are discussed. [ABSTRACT FROM AUTHOR]

In this note, we prove that the uni‐modular mono‐components from Blaschke products in the unit disc are eigenvectors of the Sturm‐Liouville operators and investigate a kind of wave equations with some initial conditions, which give rise to mono‐components. [ABSTRACT FROM AUTHOR]

In this paper, a criterion for a sequence of composition operators defined on the space of holomorphic functions in a complex domain to be frequently hypercyclic is provided. Such a criterion improves some already known special cases, and, in addition, it is also valid to provide dense vector subspaces as well as large closed ones consisting entirely, except for zero, of functions that are frequently hypercyclic. [ABSTRACT FROM AUTHOR]

Highlights • Brief historical overview of the discovery of complex numbers. • Comparison of real and complex arithmetic for (non)linear vibration problems. • Definition of complex differentiability and derivation of Cauchy-Riemann equations. • Analytic solutions of SDOF nonlinear oscillators which are not complex differentiable. Abstract In many technical applications, non-linearities play an important role and must be taken into account as early as the design phase. In the field of dynamics and vibration, nonlinear effects can dramatically change the forced response behavior. For example, several vibration states can be possible for one excitation frequency. The Harmonic Balance Method (HBM) is widely used to solve differential equations resulting from nonlinear dynamic problems. Periodic stationary oscillations are efficiently approximated by a truncated Fourier series, where the Fourier coefficients are to be calculated from a nonlinear algebraic system of equations. As in linear theory, the harmonic signals can be defined with real or complex arithmetic and so HBM can be applied in real or complex form. However, since the resulting equation system for the Fourier coefficients is usually solved numerically with Newton's method, one encounters a problem when using the complex representation. For the complex residual function, the complex derivative simply does not exist, which is why it must be separated into real and imaginary parts. The present paper deals with the problem of non-differentiable complex functions by first briefly discussing the history of complex numbers. Complex differentiability is explained using simple mathematical examples. For a Duffing oscillator and for a Friction oscillator it is shown in detail that no complex derivative exists for the associated HBM residual function. Furthermore, general amplitude-dependent nonlinear terms are shown to end up in complex non-differentiability with the need of splitting the problem into real and imaginary parts. [ABSTRACT FROM AUTHOR]

In this note we establish an advanced version of the inverse function theorem and study some local geometrical properties like starlikeness and hyperbolic convexity of the inverse function under natural restrictions on the numerical range of the underlying mapping. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of families of so-called nonlinear resolvents. Namely, we construct polynomial transformations which map the closed unit polydisks onto the coefficient bodies for the resolvent families. As immediate applications of our results we present a covering theorem and a sharp estimate for the Schwarzian derivative at zero on the class of resolvents. [ABSTRACT FROM AUTHOR]

In the present investigation, by making use of strong differential subordinations and superordinations, we introduce and study two new classes of holomorphic functions containing generalized differential oper- ator. Also we determine important properties for functions belongs to these classes. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Liouville theorem for holomorphic functions on a class of complete Gauduchon manifolds. This generalizes a result of Yau for complete Kähler manifolds to the complete non-Kähler case. [ABSTRACT FROM AUTHOR]

Abstract In this note, the universality of a sequence of operators associated to the partial sums of the Taylor series of a holomorphic function is investigated. The emphasis is put on the fact that the Taylor series are evaluated at a prescribed point and the variable is the center of the expansion. The dynamics of the sequence of operators linked to the partial sums of a power series that is not generated by an entire function is also studied. [ABSTRACT FROM AUTHOR]

Let M be a subharmonic function in a domain D ⊂ ℂn with Riesz measure νM, and let Z ⊂ D. As was shown in the first of the preceding papers, if there exists a holomorphic function f ≠ 0 in D such that f(Z) = 0 and |f| ⩽ exp M on D, then one has a scale of integral uniform upper bounds for the distribution of the set Z via νM. The present paper shows that for n = 1 this result "almost has a converse." Namely, it follows from such a scale of estimates for the distribution of points of the sequence Z ≔ {zk}k=1,2,... ⊂ D ⊂ ℂ via νM that there exists a nonzero holomorphic function f in D such that f(Z) = 0 and |f| ⩽ exp M↑r on D, where the function M↑r ⩾ M on D is constructed from the averages of M over circles rapidly narrowing when approaching the boundary of D with a possible additive logarithmic term associated with the rate of narrowing of these circles. [ABSTRACT FROM AUTHOR]

We compute the Bass stable rank of the ring г(X,OX) of global sections of the structure sheaf OX on a finite-dimensional Stein space (X,OX) and then apply this result to the problem of the factorization of invertible holomorphic matrices on X. [ABSTRACT FROM AUTHOR]

This brief communication contains estimates for the zero divisors of holomorphic functions from weight classes in terms of subharmonic real test functions. [ABSTRACT FROM AUTHOR]

We investigate a boundary version of the Schwarz lemma for classes H(α). Also, we estimate a modulus of the angular derivative of f(z) zfunction at the boundary point b with f(b) = b/β√2, 0 < β ≤ 1. The sharpness of these inequalities is also proved. [ABSTRACT FROM AUTHOR]

Let U be a balanced open subset of a Hausdorff complex locally convex space E . In this paper we represent the predual G ( U ) of the space of holomorphic functions as the projective limit of the preduals of spaces of holomorphic functions that are bounded on certain subsets of U . As an application we prove that if E is a Banach space, then it has the approximation property if and only if G ( U ) has the approximation property. We also give a new proof of a result due to Aron and Schottenloher stating that if E is a complex Banach space with the approximation property then the space of holomorphic functions H ( U ) , with the compact-open topology, has the approximation property. [ABSTRACT FROM AUTHOR]

Let M be a subharmonic function with Riesz measure νM in a domain D in the n-dimensional complex Euclidean space ℂn, and let f be a nonzero function that is holomorphic in D, vanishes on a set Z ⊂ D, and satisfies |f| ⩽ expM on D. Then restrictions on the growth of νM near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework. [ABSTRACT FROM AUTHOR]

In this paper, the results of the Carathèodory inequality have been generalized. C. T. Rajagopal further strengthened the inequality (1.8) by considering the zeros of the function f (z) . We will obtain more general results for the inequalities (1.8) and (1.9) by considering both the zeros and the poles of the function f (z) . [ABSTRACT FROM AUTHOR]