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In this paper, we introduce operator geodesically convex and operator convex-log functions and characterize some properties of them. Then apply these classes of functions to present several operator Azc\'{e}l and Minkowski type inequalities…
We call a complement of a union of at least three disjoint (round) open balls in the unit sphere S^n a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the…
In this paper, two new classes of convex functions as a generalization of convexity which is called (h-s)_{1,2}-convex functions are given. We also prove some Hadamard-type inequalities and applications to the special means are given.
The Gauss-Minkowski correspondence in $\mathbb{R}^2$ states the existence of a homeomorphism between the probability measures $\mu$ on $[0,2\pi]$ such that $\int_0^{2\pi} e^{ix}d\mu(x)=0$ and the compact convex sets (CCS) of the plane with…
The intention of this survey to collect in one paper many recent results and advances related with Bergman type projection acting in various spaces of analytic functions in several complex variables in the unit ball, tubular domains over…
Pseudo-cones are a class of unbounded closed convex sets, not containing the origin. They admit a kind of polarity, called copolarity. With this, they can be considered as a counterpart to convex bodies containing the origin in the…
In this article we discuss classical theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph $K_n$, any two edges share at most one point: either a common vertex…
We give a thoroughful explanation of the general properties of different, general scales, corresponding to different (all possible) mathematical functions f(x), we mention and analyse many examples. These observations and statements might…
We investigate properties of some spherical fonctions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson-Sullivan measure class. We prove sharp decay estimates for spherical…
In contemporary convex geometry, the rapidly developing L_p-Brunn Minkowski theory is a modern analogue of the classical Brunn Minkowski theory. A cornerstone of this theory is the L_p-affine surface area for convex bodies. Here, we…
We define a generalization of convex functions, which we call $\delta$-convex functions, and show they must satisfy interior H\"older and $W^{1,p}$ estimates. As an application, we consider solutions of a certain class of fully nonlinear…
The central purpose of the present paper is to study boundary behavior of squeezing functions on bounded domains. We prove that the squeezing function of a strongly pseudoconvex domain tends to 1 near the boundary. In fact, such an estimate…
The main aspect of this paper is to introduce a new generalisation of nano open sets namely, nano h-open sets. These newly generalised sets serve as the foundation for the definition of nano h-continuous functions and some results involving…
A method of proving local continuity of concave functions on convex set possessing the $\mu$-compactness property is presented. This method is based on a special approximation of these functions. The class of $\mu$-compact sets can be…
In this article, we prove an inner product inequality for Hilbert space operators. This inequality, then, is utilized to present a general numerical radius inequality using convex functions. Applications of the new results include obtaining…
We present a new characterization of higher-order Sobolev spaces on the sphere. Building on the approach of Barcel\'o et al. (2020), we refine the square function they introduced for this purpose. In particular, we provide a detailed…
For a continuous map $f$ on a compact metric space we study the geometry and entropy of the generalized rotation set $\R(\Phi)$. Here $\Phi=(\phi_1,...,\phi_m)$ is a $m$-dimensional continuous potential and $\R(\Phi)$ is the set of all…
In this article we continue the study of properties of squeezing functions and geometry of bounded domains. The limit of squeezing functions of a sequence of bounded domains is studied. We give comparisons of intrinsic positive forms and…
A classification of SL$(n)$ contravariant Minkowski valuations on convex functions and a characterization of the projection body operator are established. The associated LYZ measure is characterized. In addition, a new SL$(n)$ covariant…
This short but self-contained survey presents a number of elegant matrix/operator inequalities for general convex or concave functions, obtained with a unitary orbit technique. Jensen, sub or super-additivity type inequalities are…