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In the paper 'Composition operators on weighted Bergman spaces of Dirichlet series. J Math Anal Appl. 2015;426:340–363', Bailleul completely characterized the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case of symbols with $ c_0\ge 1 $ c 0 ≥ 1. But the sufficient conditions for the other case $ c_0=0 $ c 0 = 0 were unsolved. In this paper, we follow this line and study the boundedness of composition operators on weighted Bergman spaces of Dirichlet series for the case $ c_0=0 $ c 0 = 0. Moreover, we also obtain the compact characterizations of composition operators with $ c_0\geq 1 $ c 0 ≥ 1. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of multiple solutions of the generalized quasilinear Schrödinger equation of Kirchhoff-type ... where b>0 is a parameter, g C¹(R,R+). In preceding approaches, the order of the nonlinear term f (x, t) about t at infinity is usually supposed to be higher than |t|³. But in this paper, this condition is not necessary. Under weaker condition t0 f (x, s) ds = µ|G(t)|r - |G(t)|r-1, where G(t) is the primitive function of g(t), 2 < r < 6, we get the existence of infinitely many solutions of the equation. Our work can be regarded as the extension of those such as [J. Zhang et al. Existence of multiple solutions of Kirchhoff type equation with sign-changing potential. Appl Math Comput. 2014;242:491-499] and related works. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of at least three weak solutions for the fourth-order problem with indefinite weights involving the Leray–Lions operator with nonstandard growth conditions. We generalize the result obtained by Kefi et al. [On the fourth-order Leray–Lions problem with indefinite weight and nonstandard growth conditions. Bull Math Sci. 2021;DOI:10.1142/S1664360721500089]. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano [On the structure of the critical set of nondifferentiable functions with a weak compactness condition. Appl Anal. 2010;89:1–10]. [ABSTRACT FROM AUTHOR]

In this paper, we study a maximizing problem associated with the critical Sobolev inequality under inhomogeneous constraints. The problem of this type was previously studied by Ishiwata and Wadade [On the effect of equivalent constrain on a maximizing problem associated with the Sobolev type embedding in R N . Math Ann. 2016;364:1043–1068; On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934], Ishiwata [On variational problems associated with Sobolev type inequalities and related topics. Available from: ] and Nguyen [Maximizers for the variational problems associated with Sobolev type inequalities under constraints. Math Ann. 2018;372(1–2):229–255]. Our results give a complete picture of the effect of the constraints on the attainability and non-attainability of the problem. The sharp Sobolev inequality plays a crucial role in our argument. Our method also provides a new and simple proof for the recent results of Ishiwata and Wadade [On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934] concerning the sub-critical Sobolev-type inequalities (or Gagliardo–Nirenberg inequalities). [ABSTRACT FROM AUTHOR]

If $ P(z)=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a n ∏ v = 1 n (z − z v) is a polynomial of degree n having all its zeros in $ |z|\le k, k\ge 1 $ | z | ≤ k , k ≥ 1 then Aziz [Inequalities for the derivative of a polynomial. Proc Am Math Soc. 1983;89(2):259–266] proved that \[ \max_{|z|=1}|P'(z)|\ge \frac{2}{1+k^n}\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \] max | z | = 1 | P ′ (z) | ≥ 2 1 + k n ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. Recently, Kumar [On the inequalities concerning polynomials. Complex Anal Oper Theory. 2020;14(6):1–11 (Article ID 65)] established a generalization of this inequality and proved under the same hypothesis for a polynomial $ P(z)=a_0+a_1z+a_2z^2+\cdots +a_nz^n=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n = a n ∏ v = 1 n (z − z v) , that $$\begin{align*} & \max_{|z|=1}|P'(z)| \\ & \ge \left(\frac{2}{1+k^n}+\frac{(|a_n|k^n-|a_0|)(k-1)}{(1+k^n)(|a_n|k^n+k|a_0|)}\right)\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \end{align*} $$ max | z | = 1 | P ′ (z) | ≥ (2 1 + k n + (| a n | k n − | a 0 |) (k − 1) (1 + k n) (| a n | k n + k | a 0 |)) ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. In this paper, we sharpen the above inequalities and further extend the obtained results to the polar derivative of a polynomial. As a consequence, our results also sharpens considerably some results of Dewan and Upadhye [Inequalities for the polar derivative of a polynomial. J Ineq Pure Appl Math. 2008;9:1–9 (Article ID 119)]. [ABSTRACT FROM AUTHOR]

Let $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) be the familiar class of almost spirallike functions of type β and order α in the unit disk (see Definition 1.1). In this paper, first, we prove that for a function $ f(z)=z+\sum _{n=2}^\infty a_nz^n $ f (z) = z + ∑ n = 2 ∞ a n z n in the class $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) , then $$\begin{align*} & |a_{3}-\lambda a_{2}^2|\\ & \quad \leq (1-\alpha)\cos{\beta}\max\{1,|1-4(1-\alpha)(1-\lambda)\cos{\beta}\,{\rm e}^{{\rm i}\beta}|\},\quad \lambda\in \mathbb{C}. \end{align*} $$ | a 3 − λ a 2 2 | ≤ (1 − α) cos ⁡ β max { 1 , | 1 − 4 (1 − α) (1 − λ) cos ⁡ β e i β | } , λ ∈ C. The above estimation is sharp. Second, we extend this result to the bounded starlike circular domain in $ \mathbb {C}^{n} $ C n and obtain the sharp estimates. The results presented here would provide extensions of those given by Xu et al. [The Fekete and Szegö problem on the bounded starlike circular domain in $ \mathbb {C}^n $ C n . Pure Appl Math Q. 2016;12:621–638] and Xiong [Sharp coefficients bounds for class of almost starlike mappings of order α in $ \mathbb {C}^n $ C n . J Math Inequalities. 2020;14:853–865]. Finally, a certain conjecture is also formulated. [ABSTRACT FROM AUTHOR]

Let $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) be the familiar class of almost spirallike functions of type β and order α in the unit disk (see Definition 1.1). In this paper, first, we prove that for a function $ f(z)=z+\sum _{n=2}^\infty a_nz^n $ f (z) = z + ∑ n = 2 ∞ a n z n in the class $ \hat {\mathcal {S}}_{(\alpha,\beta)} $ S ^ (α , β) , then $$\begin{align*} & |a_{3}-\lambda a_{2}^2|\\ & \quad \leq (1-\alpha)\cos{\beta}\max\{1,|1-4(1-\alpha)(1-\lambda)\cos{\beta}\,{\rm e}^{{\rm i}\beta}|\},\quad \lambda\in \mathbb{C}. \end{align*} $$ | a 3 − λ a 2 2 | ≤ (1 − α) cos ⁡ β max { 1 , | 1 − 4 (1 − α) (1 − λ) cos ⁡ β e i β | } , λ ∈ C. The above estimation is sharp. Second, we extend this result to the bounded starlike circular domain in $ \mathbb {C}^{n} $ C n and obtain the sharp estimates. The results presented here would provide extensions of those given by Xu et al. [The Fekete and Szegö problem on the bounded starlike circular domain in $ \mathbb {C}^n $ C n . Pure Appl Math Q. 2016;12:621–638] and Xiong [Sharp coefficients bounds for class of almost starlike mappings of order α in $ \mathbb {C}^n $ C n . J Math Inequalities. 2020;14:853–865]. Finally, a certain conjecture is also formulated. [ABSTRACT FROM AUTHOR]

Two Kähler manifolds are called relatives if they admit a common Kähler submanifold with the same induced metrics. In this paper, we show that a Hartogs domain over an irreducible bounded symmetric domain equipped with the Bergman metric is not a relative to the complex Euclidean space. This generalizes the results in Cheng and Hao [On the non-existence of common submanifolds of Kähler manifolds and complex space forms. Ann Glob Anal Geom. 2021;60(1):167–180] and Cheng X, Niu Y. Submanifolds of Cartan–Hartogs domains and complex Euclidean spaces. J Math Anal Appl 2017;452(2):1262–1268.] and the novelty here is that the Bergman kernel of the Hartogs domain is not necessarily Nash algebraic. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Choquard equation: − Δ u + u = (I α ∗ F (u)) F ′ (u) i n R N where N ⩾ 3 , α ∈ (0 , N) , I α is the Riesz potential, and F (u) := 1 p | u | p + 1 q | u | q , where p = N + α N and q = N + α N − 2 are lower and upper critical exponents in sense of the Hardy–Littlewood–Sobolev inequality. Based on perturbation method and the invariant sets of descending flow, we prove that the above equation possesses infinitely many sign-changing solutions. Our results extend the results in Seok [Nonlinear Choquard equations: doubly critical case. Appl Math Lett. 2018;76:148–156] and Su [New result for nonlinear Choquard equations: doubly critical case. Appl Math Lett. 2020;102(106092):0–7]. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Hénon–Hardy type systems on R n : (− Δ) α 2 u (x) = | x | a v p (x) , u (x) ≥ 0 , x ∈ R n , (− Δ) α 2 v (x) = | x | b u q (x) , v (x) ≥ 0 , x ∈ R n , where n ≥ 2 , n > α , 0 < α ≤ 2 or α = 2 m. We prove Liouville theorems (i.e. non-existence of nontrivial nonnegative solutions) for the above Hénon–Hardy systems. The arguments used in our proof is the method of scaling spheres developed in [Dai W, Qin GLiouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752.]. Our results generalize the Liouville theorems for single Hénon–Hardy equation on R n in Bidaut-Véron and Pohozaev [Nonexistence results and estimates for some nonlinear elliptic problems. J Anal Math. 2001;84:1.49], Chen et al. [Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy–Hénon equations in R N . preprint, submitted, arXiv: 1808.06609], Dai et al. [Liouville type theorems, a priori estimates and existence of solutions for non-critical higher-order Lane–Emden–Hardy equations. preprint, submitted for publication, arXiv: 1808–10771], Dai and Qin [Liouville type theorems for Hardy–Hénon equations with concave nonlinearities. Math Nachrichten. 2020;293(6):1084–1093. ; Liouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752], Guo and Liu [Liouville-type theorems for polyharmonic equations in R N and in Liouville-type theorems for. Proc Roy Soc Edinburgh Sect A. 2008;138(2):339–359], and Phan and Souplet [Liouville-type theorems and bounds of solutions of Hardy–Hénon equations. J Diff Equ. 2012;252:2544–2562] to systems. [ABSTRACT FROM AUTHOR]

In 2011, Li [A uniqueness theorem for Dirichlet series satisfying a Riemann type functional equation. Adv Math. 2011;226(5):4198–4211] proved that if two L-functions L 1 and L 2 satisfy the same functional equation with a (1) = 1 and L 1 − 1 (c j) = L 2 − 1 (c j) for two distinct finite complex numbers c 1 and c 2 , then L 1 = L 2. We prove that if two L-functions L 1 and L 2 have positive degree and satisfy the same functional equation with a (1) = 1 and E L 1 (S) = E L 2 (S) for a finite set S = { c 1 , c 2 , c 3 } , where c 1 , c 2 and c 3 are three distinct finite complex values, then L 1 = L 2. The main results of this paper are concerning some questions posed by Gross [Factorization of meromorphic functions and some open problems. Complex Analysis, Kentucky 1976 (Proc.Conf.); Berlin: Springer-Verlag; 1977. p. 51–69. (Lecture Notes in Mathematics; 599)]. [ABSTRACT FROM AUTHOR]

This paper presents the global Borel-Pompieu- and the global Cauchy-type integral formulas for the quaternionic slice regular functions using the relationship between this function space and a non-constant coefficient differential operator given by G := ∥ x → ∥ 2 ∂ 0 + x → ∑ k = 1 3 x k ∂ k , according to [González-Cervantes JO. On cauchy integral theorem for quaternionic slice regular functions. Complex Anal Oper Theory. 2019;13(6):2527–2539; Colombo F, González-Cervantes JO, Sabadini I. A non-constant coefficients differential operator associated to slice monogenic functions. Trans Am Math Soc. 2013;365:303–318]. This association allows to show a behavior of the theory of slice regular functions similar to the well known theories of the hypercomplex analysis. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Hénon–Hardy type systems on R n : (− Δ) α 2 u (x) = | x | a v p (x) , u (x) ≥ 0 , x ∈ R n , (− Δ) α 2 v (x) = | x | b u q (x) , v (x) ≥ 0 , x ∈ R n , where n ≥ 2 , n > α , 0 < α ≤ 2 or α = 2 m. We prove Liouville theorems (i.e. non-existence of nontrivial nonnegative solutions) for the above Hénon–Hardy systems. The arguments used in our proof is the method of scaling spheres developed in [Dai W, Qin GLiouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752.]. Our results generalize the Liouville theorems for single Hénon–Hardy equation on R n in Bidaut-Véron and Pohozaev [Nonexistence results and estimates for some nonlinear elliptic problems. J Anal Math. 2001;84:1.49], Chen et al. [Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy–Hénon equations in R N . preprint, submitted, arXiv: 1808.06609], Dai et al. [Liouville type theorems, a priori estimates and existence of solutions for non-critical higher-order Lane–Emden–Hardy equations. preprint, submitted for publication, arXiv: 1808–10771], Dai and Qin [Liouville type theorems for Hardy–Hénon equations with concave nonlinearities. Math Nachrichten. 2020;293(6):1084–1093. ; Liouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752], Guo and Liu [Liouville-type theorems for polyharmonic equations in R N and in Liouville-type theorems for. Proc Roy Soc Edinburgh Sect A. 2008;138(2):339–359], and Phan and Souplet [Liouville-type theorems and bounds of solutions of Hardy–Hénon equations. J Diff Equ. 2012;252:2544–2562] to systems. [ABSTRACT FROM AUTHOR]

This paper deals with extension of analytic covers. We prove topological extension theorems for analytic covers. The main result is an extension theorem which only uses the extension of the ramification divisor. We give also a Thullen-type and a Hartogs-type extensions theorems. [ABSTRACT FROM AUTHOR]

In this paper, we prove Huang et al.’s conjecture stated that iffis a holomorphic function onwith-smooth extension up tosuch thatfmapsinto a cone, for some positive numberC, andfvanishes to infinite order at 0, thenfvanishes identically. In addition, some regularity properties of the Riemann mapping functions on the boundary and an example concerning Huang et al.’s conjecture are also given. [ABSTRACT FROM AUTHOR]

Let X be a connected K-complete normal complex space. If for every closed discrete set A in X there exists a family F of holomorphic functions on X such that N (F) = A , then the K-envelope H (X) of holomorphy of X in the sense of Kerner (Math Ann. 1959;138:316–328) coincides with X. [ABSTRACT FROM AUTHOR]

The aim of this note is to survey recent results contained in Nguyen H-M, Squassina M. [On anisotropic Sobolev spaces. Commun Contemp Math, to appear. DOI:]; Nguyen H-M, Pinamonti A, Squassina M, et al. [New characterizations of magnetic Sobolev spaces. Adv Nonlinear Anal. 2018;7(2):227–245]; Pinamonti A, Squassina M, Vecchi E. [Magnetic BV functions and the Bourgain-Brezis-Mironescu formula. Adv Calc Var, to appear. DOI:]; Pinamonti A, Squassina M, Vecchi E. [The Maz'ya-Shaposhnikova limit in the magnetic setting. J Math Anal Appl. 2017;449:1152–1159] and Squassina M, Volzone B. [Bourgain-Brezis-Mironescu formula for magnetic operators. C R Math Acad Sci Paris. 2016;354:825–831], where the authors extended to the magnetic setting several characterizations of Sobolev and BV functions. [ABSTRACT FROM AUTHOR]