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This paper presents a brief survey of the most important and the most remarkable inequalities involving the basic arithmetic functions.

We generalize Romanoff's theorem. Also, we obtain a result on sums related to Euler's totient function.

In this paper the double-sided Talor's approximations are used to obtain generalisations and improvements of some trigonometric inequalities.

A method of proving Hardy's type inequality for orthogonal expansions is presented in a rather general setting. Then sharp multi-dimensional Hardy's inequality associated with the Laguerre functions of convolution type is proved for type…

The aim of this paper is to analyze the weighted KyFan inequality proposed in [11]. A number of numerical simulations involving the exponential weighted function is given. We show that in several cases and types of examples one can imply an…

Several general mixed affine surface areas are introduced. We prove some important properties, such as, affine invariance, for these general mixed affine surface areas. We also establish new Alexandrov-Fenchel type inequalities,…

In this paper, we introduce the notion of generalized squeezing function and study the basic properties of generalized squeezing functions and Fridman invariants. We also study the comparison of these two invariants, in terms of the…

WWe give a rational closed form expression for the higher derivatives of the inverse tangent function and discuss its relation to Chebyshev polynomials, trigonometric expansions and Appell sequences of polynomials.

We establish Hanner's inequality for arbitrarily many functions in the setting where the Rademacher distribution is replaced with higher dimensional random vectors uniform on Euclidean spheres.

A crucial extension of quaternionic function theory to octonions is the concept of slice regular functions, introduced to handle holomorphic-like properties in a non-associative setting. In this paper, first we present a generalization of…

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 show that recent multivariate generalizations of the Araki-Lieb-Thirring inequality and the Golden-Thompson inequality [Sutter, Berta, and Tomamichel, Comm. Math. Phys. (2016)] for Schatten norms hold more generally for all unitarily…

In this paper, several Bohr-type inequalities are generalized to the form with two parameters for the bounded analytic function. Most of the results are sharp.

Matrix inequalities that extend certain scalar ones have been at the center of numerous researchers' attention. In this article, we explore the celebrated subadditive inequality for matrices via concave functions and present a reversed…

We extend the inequality of Audenaert et al to general von Neumann algebras.

We consider the $p$-generalized arithmetic-geometric mean inequality for vectors chosen randomly from the $\ell_p^n$-ball in $\mathbb{R}^n$. In this setting the inequality can be improved or reversed up to a respective scalar constant with…

We prove inequality (1) for the modified Steiner functional A(M), which extends the notion of the integral of mean curvature for convex surfaces.We also establish an exression for A(M) in terms of an integral over all hyperplanes…

In this short note, we improve the famous Reid Inequality related to linear operators.

For convex univalent functions we give instances where the sharp bound for various coefficient functionals are identical to those for the corresponding bound for the inverse function. We give instances where the sharp bounds differ and also…