In this paper, we prove the anisotropic Shannon inequality for the Rényi entropy with the best constant on Folland-Stein homogeneous Lie groups. As a consequence, we also prove the optimal Shannon inequality in the same setting. Using a logarithmic Sobolev inequality in the setting of stratified groups, we prove a Heisenberg-type uncertainty principle in the latter setting.

The notion of warped product plays an important role in Riemannian geometry moreover in geodesic metric spaces. The warped product was first introduced by Bishop and O'Neill to study Riemannian manifolds of negative curvature.Warped products have been mainly used to construct new examples of Riemannian manifolds with prescribed curvature conditions. This construction can be extended for Finslerian metrics with some minor restrictions. This is motivated by Asanov's papers, where some models of relativity theory are described through the warped product of Finsler metrics. These metrics are in the form of (α,β)-metrics, which are the generalization of the Randers metrics; which are being asymmetric Finsler metrics in four-dimensional space-time. The product was later extended to the warped product case of Finsler manifolds by the work of Kozma, Peter and Verge.

In this paper, we first prove the weighted Levin-Cochran-Lee type inequalities on homogeneous Lie groups for arbitrary weights, quasi-norms, and L^p-and L^q-norms. Then, we derive a sharp weighted inequality involving specific weights given in the form of quasi-balls in homogeneous Lie groups. Finally, we also calculate the sharp constants for the aforementioned inequalities.

In this note, we announce the two-weight Hardy inequalities on metric measure spaces possessing a polar decomposition for the case p=1 and 1 ≤ q <∞ obtained in the paper Ruzhansky et al. (Hardy inequalities on metric measure spaces, IV: The case p = 1 . Forum Mathematicum, 2024, to appear. https://doi.org/10.1515/forum-2023-0319 .). We refer to Ruzhansky et al. (Hardy inequalities on metric measure spaces, IV: The case p = 1 . Forum Mathematicum, 2024, to appear. https://doi.org/10.1515/forum-2023-0319 .) for the complete proofs and more details.

In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case p=1 and 1 <= q1 . As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan-Hadamard manifolds for the case p=1 and 1 <= q

In this paper, we discuss the Hardy inequality with bilinear operators on general metric measure spaces. We give the characterization of weights for the bilinear Hardy inequality to hold on general metric measure spaces having polar decompositions. We also provide several examples of the results, finding conditions on the weights for integral Hardy inequalities on homogeneous Lie groups, as well as on hyperbolic spaces and more generally on Cartan-Hadamard manifolds.

The main aim of this note is to prove sharp weighted integral Hardy inequality and conjugate integral Hardy inequality on homogeneous Lie groups with any quasi-norm for the range 1

In this paper we discuss the Busemann-Hausdorff volume form and Holmes-Thompson volume form for the warped product Finsler metrics. With the help of these volume forms we obtain the E-curvature and the S-curvature for this class of metrics. Further, we show that the notion of isotropic E-curvature and isotropic S-curvature are equivalent for this class of metrics.

In this paper, we prove that every weakly Douglas warped product Finsler metric is a Douglas metric. Then we classify the weakly Douglas Landsberg metric. With the help of these results, we also study the unicorn problem and show that every weakly Douglas warped product Finsler metric is Berwald if and only if it is Landsberg.

In this paper, we study Finsler warped product metrics recently introduced by P. Marcal and Z. Shen and find characteristics differential equations for this metric to vanish $E$-curvature. We also prove that if this warped product Finsler metric is projectively flat, then it becomes a Riemannian metric.