Some Properties of Cone Inner Product Spaces over Banach Algebra

The aim of this paper is to introduce a concept of a cone inner product space over Banach algebras. This is done by replacing the co-domain of the classical inner product space by an ordered Banach algebra. Some properties such as Cauchy-Schwarz inequality, parallelogram identity and Pythagoras theorem are established in this setting. Similarly, the notion of cone normed algebra was introduced. Some illustrative examples are given to support our findings.

Tur´an type inequalities for the supertrigonometric functions

This paper is devoted to the study of Tur\’{an} type inequalities for some well-known special functions such as supersine and supercosine which are derived by using a new form of the Cauchy-Bunyakovsky-Schwarz inequality.

Note on Reliability Analysis of Cartesian Product of Networks for Components

Connectivity is a critical parameter which can measure the reliability of networks. Let [Formula: see text] be a vertex set of [Formula: see text]. If [Formula: see text] has at least [Formula: see text] components, then [Formula: see text] is a [Formula: see text]-component cut of [Formula: see text]. The [Formula: see text]-component connectivity [Formula: see text] of [Formula: see text] is the vertex number of a smallest [Formula: see text]-component cut. Cartesian product of graphs is a useful method to construct a large network. We will use Cauchy–Schwarz inequality to determine the component connectivity of Cartesian product of some graphs.

Creating new problems on proving inequalities, finding maximum and minimum values based on the critical properties and tangent inequalities of convex and concave functions

In this paper, we present some ideas and methods to create new problems of proving inequalities, problems of finding maximum and minimum values. Based on the maximum and minimum properties and tangent inequalities of convex and concave functions, we propose some ideas and methods to create new problems. We make all ideas and methods to be real via many specific functions. Especially, we combine the ideas and methods with equivalent transforms, Cauchy-Schwarz inequality, and inequality of arithmetic and geometric means to create new hard problems. New proposed examples, they have showed that our ideas and methods are important and efficient to lecturers at high schools and universities in giving questions in examinations, especially in examinations of selecting good students at levels, in Olympic examinations for high school and university students.

Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0<α≤1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven.

About the Cauchy–Bunyakovsky–Schwarz Inequality for Hilbert Space Operators

The symmetric shape of some inequalities between two sequences of real numbers generates inequalities of the same shape in operator theory. In this paper, we study a new refinement of the Cauchy–Bunyakovsky–Schwarz inequality for Euclidean spaces and several inequalities for two bounded linear operators on a Hilbert space, where we mention Bohr’s inequality and Bergström’s inequality for operators. We present an inequality of the Cauchy–Bunyakovsky–Schwarz type for bounded linear operators, by the technique of the monotony of a sequence. We also prove a refinement of the Aczél inequality for bounded linear operators on a Hilbert space. Finally, we present several applications of some identities for Hermitian operators.

Extended Cauchy–Schwarz Inequality and Its Application for the Two-Class Fisher Discriminant Analysis

We present the sufficient condition for a classical two-class problem from Fisher discriminant analysis has a solution. Actually, the solution was presented up to our knowledge with a necessary condition only. We use an extended Cauchy–Schwarz inequality as a tool.

Ulam stability for Thunsdorff and Cauchy-Schwarz equations

We consider the equality case in Thunsdorff inequality and Cauchy-Schwarz inequality. For these two equations we prove Ulam stability.