It is well-known that for Hilbert space linear operators $0 \leq A$ and $0 \leq C$, inequality $C \leq A$ does not imply $C^2 \leq A^2.$ We introduce a strong order relation $0 \leq B \lll A$, which guarantees that $C^2 \leq B^{1/2}AB^{1/2}$ text for all $0 \leq C \leq B,$ and that $C^2 \leq A^2$ when $B$ commutes with $A$. Connections of this approach with the arithmetic-geometric mean inequality of Bhatia-Kittaneh as well as the Kantorovich constant of $A$ are mentioned.

We consider linear operators $T: X \to X$ on a normed space $X$ which reverse orthogonality, i.e., satisfy the condition $$x \bot y \quad \Longrightarrow \quad Ty \bot Tx, \qquad x,y \in X,$$ where $\bot$ stands for Birkhoff orthogonality.

In this paper, we survey some recent trace inequalities for operators in Hilbert spaces that are connected to Schwarz's, Buzano's and Kato's inequalities and the reverses of Schwarz inequality known in the literature as Cassels' inequality and Shisha-Mond's inequality. Applications for some functionals that are naturally associated to some of these inequalities and for functions of operators defined by power series are given. Examples for fundamental functions such as the power, logarithmic, resolvent and exponential functions are provided as well.

In this paper, we discuss and improve some recent results about contractive and cyclic mappings established in the framework of $C^{*}$-algebra-valued $b$-metric spaces. Our proofs are much shorter than the ones in existing literature. Also, we give two examples that support our approach.

In this paper, we give converses of the Jensen and Edmundson-Lah-Ribarič inequalities which are more accurate than the existing ones. These converses are given in a difference form and they rely on the recent refinement of the Jensen inequality obtained via linear interpolation of a convex function. As an application, we also derive improved converse relations for generalized means, for the Hölder and Hermite-Hadamard inequalities as well as for the inequalities of Giaccardi and Petrović.

We consider a kernel based harmonic analysis of "boundary," and boundary representations. Our setting is general: certain classes of positive definite kernels. Our theorems extend (and are motivatedby) results and notions from classical harmonic analysis on the disk. Our positive definite kernels include those defined on infinite discrete sets, for example sets of vertices in electrical networks, or discrete sets which arise from sampling operations performed on positive definite kernels in a continuous setting.

Below we give a summary of main conclusions in the paper: Starting with a given positive definite kernel $K$ we make precise generalized boundaries for $K$. They are measure theoretic "boundaries." Using the theory of Gaussian processes, we show that there is always such a generalized boundary for any positive definite kernel.

In the present article, we propose the $(p,q)$-variant of beta function of second kind and establish a relation between the generalized beta and gamma functions using some identities of the post-quantum calculus. As an application, we also propose the $(p,q)$-Baskakov-Durrmeyer operators, estimate moments and establish some direct results.