The purpose of the present paper is to extend the study of Reinhardt domains from ℂ^n to toric varieties. The main result clarifies pseudoconvexity and envelopes of holomorphy for subdomains of simplicial affine toric varieties. In conclusion, suggestions and open questions on the general case and related extension problems on toric varieties are discussed. The article is reasonably self-contained in order to make the material accessible to complex analysts.
The well-known concept of subordination between two analytic functions is referred to as weak subordination in the harmonic case, due to certain limitations. Let 𝔻 denote the open unit disk centered at the origin in the complex plane ℂ . In this article, a family of convex and univalent harmonic mappings, ℱ_β:𝔻→ℂ,β∈ℕ, is constructed using harmonic Alexander operator, and mapping properties of members of this family are discussed. It is established that the sequence {ℱ_β :β∈ℕ} forms a weak subordination chain.
We give the first main theorem and the second main theorem of Nevanlinna theory for meromorphic functions on complete Kähler manifolds which admit Green’s functions regular at infinity. Our method is applicable also for the cases when the domain manifolds of meromorphic functions are strictly pseudo-convex domains and bounded regular domains in complex Euclidean spaces.
Growth properties of entire function solutions of certain homogeneous partial differential equations have been discussed by various authors. The growth estimates of entire/analytic function solutions of non-homogenous partial differential equations have not been studied so far. Therefore, in this paper, we tried to bridge this gap. Moreover, the classical order, lower order, and type of entire function solutions of non-homogeneous partial differential equations of generalized axially symmetric potential theory in terms of coefficients occurring in its power series expansion in two complex variables have been obtained.
In this paper, we find sharp upper bounds for the higher-order Schwarzian functionals σ _n(f)(0) , for n=3,4 , in two subclasses of univalent functions defined using the exponential function. These subclasses are the exponential starlike class 𝒮_e^*={ f∈𝒮:zf'(z)/f(z)≺e^z} and the exponential convex class 𝒞_e={ f∈𝒮:1+zf”(z)/f'(z)≺e^z} . For functions in 𝒞_e , we prove that |σ _3(f)(0)|≤ 1, |σ _4(f)(0)|≤ 2. For functions in 𝒮_e^* , we show that |σ _3(f)(0)|≤ 3, |σ _4(f)(0)|≤ 16√(311). As application, the sharp Schwarzian bounds induce corresponding sharp estimates for the initial Grunsky coefficients b_n,m , for exponentially subordinate functions.
Consider the set 𝒱_p(λ ) , which is defined as the collection of those functions f on the open unit disc 𝔻 in the plane of complex numbers which have a simple pole at z=p , provided p is in the interval (0, 1). They are analytic in 𝔻 except at z=p . Additionally, they satisfy the normalizations f(0)=0, f'(0)-1=0 , and for all z in 𝔻 and λ in the interval (0, 1], the inequality | (z/f(z))^2 f'(z)-1| < λ is satisfied. Every function f∈𝒱_p(λ ) can be expressed as a Taylor series expansion: f(z)=z+∑ _n=2^∞a_n z^n, |z|
The purpose of this paper is to establish a set of sufficient conditions under which the normalized form of the generalized Hurwitz–Lerch Zeta functions possesses certain geometric properties, including close-to-convexity, univalency, convexity, and starlikeness in the open unit disk. The relevance of the obtained results with some known results in the literature is also discussed.
Let p(z)=c∏ _j=1^n(z-z_j) with c 0 be a polynomial of degree n and p^γ (z) be its generalized derivative, where 0γ =(γ _1,γ _2,… ,γ _n) is an n-tuples of non-negative real numbers in the Euclidean space ℝ^n . In this paper, we consider a class of polynomials having all zeros in |z|≤ρ , ρ≤ 1 and investigate the dependence of max _|z|=1 |zp^γ (z)+Λβ/1+ρp(z) | on max _|z|=1 |p(z)|, where β is any complex number with |β |≤ 1 and Λ =∑ _j=1^nγ _j. Our results not only generalize several established polynomial inequalities, but also enable the derivation of a variety of interesting outcomes through a consistent approach. In addition to it, we also present some generalizations of Turán’s inequality.
In this paper, interpretations of wedge products of the geometric Segre, respectively, geometric Chern, forms of a holomorphic vector bundle E (with hermitian metric | | ) over a complex space Y are given by showing that: (a) the geometric Segre forms of E coincide with the Segre forms constructed by means of the Chern–Weil theory, in Sect. 6; (b) as current the Segre wedge product s_1(E_∞;| |)^β _1∧⋯∧s_p+1(E_∞;| |)^β _p+1, β _j ∈ℤ[0,∞ ), p∈ℕ is extendible to a generalized Schubert cycle on Y, provided E is semi-globally spanned, in Sect. 7; (c) the cup product (-1)^w(β )s_1(E)^β _1∪⋯∪s_p+1(E)^β _p+1, if non-vanishing, is equal to the fundamental class of an analytic intersection cycle supported by a Schubert type analytic set, provided E is globally spanned, in Sect. 8; and (d) similar results hold for the geometric Chern forms and the (analogously defined) Chern wedge products. As prerequisites multi-symbol Schubert type analytic sets and the (corresponding) Chern–Cowen forms are first introduced for a semi-globally spanned vector bundle E→ Y.
Schröder’s theorem studied the convergence of Newton’s method for the complex quadratic function, showing that on either side of the perpendicular bisector of the roots, Newton’s method converges to the root on that particular side. This paper rigorously proves that Schröder’s theorem also holds for a new variant of Newton’s method—named Backtracking New Q-Newton’s method (BNQN). However, the orbit behavior of the BNQN method differs from Newton’s method on the perpendicular bisector of the roots. The motivation for our rigorous proof comes from a remarkable experimental discovery that the basins of attraction of the BNQN method of polynomials of any degree seem to have piecewise smooth boundaries, while for Newton’s method of polynomials of degrees larger than two are usually fractal.
The fundamental aim of this paper is to apply a novel q-differential operator in 𝕌 and to make use of starlike functions to define a new subfamily of analytic functions in 𝕌 . The goals of the study are to establish the characterization properties of this new class of bi-univalent functions, and to determine their covering, distortion, and growth characteristics. In addition, certain closure, inclusion, subordination characteristics, and radius problems are investigated for functions belonging to this new class of bi-univalent functions. The proposed class consists of analytic functions in 𝕌 , and the extremal functions corresponding to this family are also obtained.
This paper investigates the relationship between the shift of an entire function with hyper-order less than 1 and its kth order derivative, focusing on cases where both functions share a doubleton set of small functions. To the best of the author’s knowledge, the most significant existing result in this context is due to Huang and Fang [Comput Methods Funct Theory, 21:523–532, 2021), who studied two-value (IM) sharing between the shift of an entire function f and its first derivative f^' . However, when it comes to the sharing properties between the shift of an entire function and its kth order derivative, particularly in both the value sharing and set sharing contexts with a minimal cardinality of two, no significant work has been done. Therefore, the main result of this paper provides a significant contribution to this area of research. Additionally, this finding refines a result from Banerjee and Roy (J Anal 32:1265–1280, 2024) by reducing the cardinality of the shared set.
We explore the problem of estimating the steady-state temperature in a two-dimensional domain at a point knowing the temperature to high order at another point. We find connections to the Bergman kernel of the domain, Runge’s theorem, and approximate null quadrature identities.
In this paper, we study upper bounds for the 2nd- and 3rd-order Hankel determinants, with a particular focus on estimating the third-order Hankel functional ℋ_3(1) for a newly defined class of bi-univalent functions denoted by 𝒪𝒮_Σ ^*(λ , β ) . This class is associated with the function 2√(1 + ζ)/1 + e^-ζ and is defined in balloon domain. Using analytic techniques and coefficient estimates, we derive meaningful upper bounds for ℋ_3(1) . To obtain these estimates, we also examine upper bounds for the individual Taylor coefficients involved. The geometry of the balloon domain introduces new aspects in the analysis, enriching the structural study of these functions. Several special cases are discussed to illustrate the effectiveness and generality of the results obtained.
This study primarily aims to establish pathway fractional integral representations of the extended Gauss and Kummer hypergeometric logarithmic functions. By selecting particular parameter values, various established special functions can be generated as special cases of these functions, demonstrating the validity of our proposed results. The proposed findings are novel and have the potential for significant applications in a variety of applied science and mathematics disciplines. These results not only extend existing fractional integral formulas but also provide a flexible analytical tool for unifying and generalizing a broad class of special functions.
This paper is concerned with the existence of solutions to the quasilinear subelliptic Dirichlet problem - _p,Xu=|u|^p_Q^*-2u+g(x,u) in Ω , u≥ 0, u∈ W_X,0^1,p(Ω ), where Ω is a bounded open domain of ℝ^n , _p,X denotes the p-sub-Laplacian associated with smooth Baouendi–Grushin-type vector fields, p_Q^*=pQ/Q-p is the critical Sobolev exponent, and g(x, u) is a subcritical perturbation. By applying variational methods, we prove the existence of a nontrivial non-negative solution. Furthermore, for the particular case g(x,u)=λ |u|^q-2u , we establish the existence of a positive solution.
In this paper, we study several coefficient problems for functions belonging to the class 𝒮^*_car . Sharp bounds are obtained for the logarithmic and logarithmic inverse coefficients, as well as for their differences. Furthermore, we derive precise estimates of Toeplitz determinants associated with these coefficients and establish a generalized Fekete–Szegö functional for the same class.
Inspired by the celebrated work of Rothschild and Stein, we investigate explicit liftings of some smooth vector fields satisfying the bracket-generating condition with step depending on the point, to Lie algebras. The examples we focus on are certain decoupled domains introduced by Nagel and Stein, whose boundaries provide a natural setting for such constructions. We present explicit models for the lifted Lie algebras associated with these boundaries and analyze their structural properties.
This paper follows the one (Derridj in Pure Appl Funct Anal (to appear), 2020) in which we studied the analytic-Gevrey wave front set with respect to iterates of hypoelliptic second-order operators, introduced by Hörmander, but of first kind (more precisely of degenerate elliptic kind). In the present work, considering this last hypothesis not satisfied, the situation seems less easy to handle and so, we introduce a subclass of Hörmander’s operators, which we call of Kolmogorov kind, for which we can study the singularity of WF_s(u;P) (see definitions in next sections). We define the index of parabolic degeneracy at a point (x_0,ξ _0) of such P and establish results as in the case of elliptic degeneracy, but with results which seem not optimally related to microlocal index k of parabolic degeneracy. The subclass we introduce contains the Kolmogorov operator P=∂ _x^2+x∂ _y+∂ _t mentioned by Hörmander in Baouendi and Metivier (Am J Math 104(2):287–319, 1982) as an example, for which k=1 . But in the last section of this paper, we study more closely the case of one-order parabolic degeneracy at (x_0,ξ _0) and establish an optimal result, for operators with k=1 . In particular, for the Kolmogorov operator, one obtains an optimal result in a conic neighborhood of any point (x_0,ξ _0)∈ω× W (elliptic, degenerate elliptic, parabolic or degenerate parabolic).
We study the L^p -mean distortion functionals, ℰ_p[f] = ∫ _𝕐 K^p_f(z) dz, for Sobolev homeomorphisms where 𝕏 and 𝕐 are bounded simply connected domains, and f coincides with a given boundary map f_0 :∂𝕐→∂𝕏 . Here, K_f(z) denotes the pointwise distortion function of f. It is conjectured that for every 1< p < ∞ , the functional ℰ_p admits a minimizer that is a diffeomorphism. We prove that if such a diffeomorphic minimizer exists, then it is unique within the class of diffeomorphisms f:𝕐𝕏 with f|_∂𝕐=f_0 and L^p -mean distortion that minimizes ℰ_p .
