The purpose of this paper is to find coefficient bounds for the class Gϑκ(L) of the parametrically-augmented Bazilevič-type normalized analytic functions in the open unit disk, which are associated with a lung-shaped domain. For functions belonging to this general class, we derive estimates for the initial Taylor-Maclaurin coefficients a2 and a3. We also find some realistic upper bound for the Fekete-Szegö functional for f∈Gϑκ(L). Similar outcomes are determined for the function log(f(z)z) and the inverse function f−1(z). Moreover, we evaluate the second Vandermonde determinants whose entries involve both Taylor-Maclaurin initial coefficients and the logarithmic coefficients.
Suppose 0 max{1,2−α} with lp+α−β>0. And we can show that the range of α<2 is the best possible if l≥2 and α=β. Our results generalize those of Bao and Wulan when α=β and l=2.
In this article, we investigate an integral Brezis–Nirenberg type problem on bounded domains in the Poincaré ball model, arising from the Hardy–Littlewood–Sobolev inequality on hyperbolic space. The main difficulty is the lack of translation invariance in the hyperbolic setting, which makes several Euclidean arguments inapplicable. To overcome this obstruction, we develop techniques adapted to the geometry of the Poincaré ball. First, by employing hyperbolic dilations, we analyze the sharp constants and derive the nonexistence of solutions in the supercritical case. Second, we employ a new family of bump functions acting on the integral kernels, which play the role of mollification while avoiding domain variations caused by translations. This enables us to establish compact embedding for the associated integral operator and to obtain an existence result in the subcritical case. Finally, in the critical case, we establish the existence result through a subcritical approximation scheme combined with compact Sobolev embedding and the Brezis–Lieb lemma which allows us to recover strong convergence and avoid the delicate blow-up analysis involved in the Euclidean subcritical approximation method. This compactness-based argument for the critical case can also be adapted to general compact Riemannian manifolds with boundary.
The present paper proposes a series of extensions to abstract Korovkin-type approximation theorems, incorporating nonlinearities through the use of statistical convergence. Building on the recent nonlinear versions, we derive statistical Korovkintype theorems for sequences of monotone and sublinear operators acting on various function spaces. Furthermore, the paper introduces modified multivariate Bernstein Chlodovsky Kantorovich Choquet (BCKC) polynomial operators through regular matrix methods and establishes their approximation behavior under statistical convergence. These results significantly expand the scope of Korovkin-type theory, thereby demonstrating both the theoretical depth and practical flexibility of nonlinear operators in approximation theory. We also calculate the rate of convergence in terms of modulus of continuity. Finally, we establish for the first time a Voronovskayatype approximation theorem for Bernstein Kantorovich Choquet (BKC) operators, thereby extending the scope of classical linear results to a genuinely nonlinear framework. (c) 2026 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Alexandroff paratopological groups provide a natural setting in which order-theoretic and algebraic properties interact. In this short note we prove obstruction results when considering continuous homomorphisms from Alexandroff paratopological groups into topological groups. Particularly and, as a consequence of these results, we provide an alternative proof of the fact that the only connected Alexandroff topological group is the one that carries the trivial topology.
