In this note, we mainly study the arithmetic-geometric mean and Cauchy-Schwarz matrix norm inequalities. By using the generalized Holder inequality for symmetric gauge functions, we obtain a more general version of a norm inequality for positive real numbers alpha, beta, gamma and r satisfying 1/alpha + 1/beta= 1/gamma and r > max{ 1/alpha, 1/beta }. The obtained result generalizes the inequalities of Audenaert [K. M. R. Audenaert, Oper. Matrices, 9 (2015), 475-479], Zou [L. Zou, Linear Algebra Appl., 562 (2019), 154-162] and Zou and Jiang [L. Zou, Y. Jiang, J. Math. Inequal., 10 (2016), 1119-1122].

In this work, we mainly study refinements and generalizations of the Young’s and its reverse inequalities. First, utilizing the famous weighted arithmetic and geometric mean inequality, we provide an alternative proof for the power version of the Young’s inequality given by Al-Manasrah and Kittaneh. In addition, some generalizations of the Al-Manasrah–Kittaneh’s inequalities are also given. Further, we also establish refinement of the Alzer–Fonseca–Kova $$\check{c}$$ ec’s version of the Young’s inequality as follows. Let a, b, $$\mu $$ and v be positive real numbers with $$0\le \mu < v\le 1$$ and m be a positive integer. Then $$\begin{aligned} \big [\mu a+(1-\mu )b\big ]^{m}&-\big (a^{\mu }b^{1-\mu }\big )^{m} +r\Big [a^{\frac{m}{2}}-\big (a^{v}b^{1-v}\big )^{\frac{m}{2}}\Big ]^{2}\\&\le \Big (\frac{1-\mu }{1-v}\Big )^{m}\Big [\big (va+(1-v)b\big )^{m}-\big (a^{v}b^{1-v}\big )^{m}\Big ] \end{aligned}$$ and $$\begin{aligned} r_{1}\Big [a^{\frac{m}{2}}-\big (a^{v}b^{1-v}\big )^{\frac{m}{2}}\Big ]^{2}&+\Big (\frac{\mu }{v}\Big )^{m}\Big [\big (va+(1-v)b\big )^{m}-\big (a^{v}b^{1-v}\big )^{m}\Big ]\nonumber \\&\le \big [\mu a+(1-\mu )b\big ]^{m}-\big (a^{\mu }b^{1-\mu }\big )^{m}, \end{aligned}$$ where r is a constant with $$0<r\le \Big (\frac{1-\mu }{1-v}\Big )^{m}v^{m}-\mu ^{m}$$ and $$r_{1}$$ is a constant with $$0<r_{1}\le \min \Big \{\Big (\frac{\mu }{v}\Big )^{m},\mu ^{m}-\Big (\frac{\mu }{v}\Big )^{m}v^{m}\Big \}$$ . As an application, we present corresponding operator and matrix inequalities following from the established scalar inequalities.

In this note, some singular value inequalities on majorisation for positive semi-definite matrices are given. First, the weak log-majorisation of the quantity f(A)g(B)± f(B)g(A) for certain functions is obtained, where A, B≥ 0 . As a consequence, the weak log-majorisation of the quantity A^pB^q± B^pA^q for p, q>0 is also obtained. Finally, different estimates for the singular values of the quantity A^pB^q± B^pA^q are deduced for A, B>0 and 0

In this work, we investigate inequalities of singular values and unitarily invariant norms for sums and products of matrices. First, we give an another more concise and clear proof to inequality obtained by Chen and Zhang [6, Theorem 5]. Then, we establish an inequality for singular values. In addition, we also give a singular values inequality for sums and products of matrices. As applications of this inequality, we present some unitarily invariant norms inequalities.

The theory of inequality in mathematics is an extensive and interesting area of research. The classical Young inequality has a great importance in several fields of mathematics, statistics and other disciplines. Many results were obtained on refinements, generalizations and converses of the Young inequality. In this note, we mainly investigate the improved Young and its reverse inequalities. First, we present a new version of a generalization of reverse Young inequality. In addition, we also get the Young and its reverse inequalities for scalars with the classical Kantorovich constant. Finally, we establish the Young and its reverse inequalities for operators on infinite and finite dimensional Hilbert spaces.

Let A(i), B-i is an element of M-n be positive semidefinite matrices with A(i)B(i) = B(i)A(i) (i = 1,2, ..., m). Then sigma((Sigma(m )(i=1)(A(i)B(i))(1/2 ))(r)) <(wlog) sigma(Sigma(m)(i=1) A(i))(r/4) (Sigma(m)(i=1) B-i)(r/2) (Sigma(m)(i=1) A(i))(r/4)), where r >= 1. This result is a refinement of M. Hayajneh, S. Hayajneh and F. Kittaneh's result.

In this note, some operator inequalities for operator means and positive linear maps are investigated. The conclusion based on operator means is presented as follows: Let ? : B(H) ? B(K) be a strictly positive unital linear map and h-1 IH ? A ? h1IH and h-12 IH ? B ? h2IH for positive real numbers h1, h2 ? 1. Then for p > 0 and an arbitrary operator mean ?, (?(A)??(B))p ? ?p?p(A?*B), where ?p = max {?2(h1,h2)/4)p, 1/16?2p(h1,h2)}, ?(h1h2) = (h1 + h-1 1)?(h2 + h-12). Likewise, a p-th (p ? 2) power of the Diaz-Metcalf type inequality is also established.

The aim of this note is to investigate some inequalities related to Kwong functions. Refinements of Kaur's and Zhan's inequality are presented by the Hadamard product. In addition, variations of the Heinz-Heron type means on Kwong functions are also given.

In this study, we present some matrix inequalities for unitarily invariant norms.Firstly, we present an inequality for unitarily invariant norms.As a consequence of this result, Huang-Peng-Zou's result follows immediately.Furthermore, we also establish inequalities for weak log-majorizations and unitarily invariant norms related to question of Bourin's.

In this short note, we present new proofs for some singular value inequalities for matrices obtained by I. Garg and J. Aulja.

We proved some inequalities for concave functions. Those inequalities complemented a theorem obtained by Lee. Finally, we partially solved an open problem proposed by Zhang P.

This note aims to present some operator inequalities for unitarily invariant norms. First, a Zhan-type inequality for unitarily invariant norms is given. Moreover, some operator inequalities for the Cauchy Schwarz type are also established.

The aim of this work is to present some Hölder-type inequalities for sums and products of operators related to unitarily invariant norms. These results generalize some known Hölder inequalities for operators.

We present some refinements of the Cauchy-Schwarz and Heinz inequalities for operators by utilizing a refinement of the Hermite-Hadamard inequality.

Предложены новые методы локализации собственных значений и новые оценки сверху размаха комплексных матриц. Эти методы и оценки точнее, чем известные ранее. Библиография: 18 наименований.

We show new versions of reverse Young inequalities by virtue of the Kantorovich constant, and utilizing the new reverse Young inequalities we give the reverses of the weighted arithmetic-geometric and geometric-harmonic mean inequalities for two positive operators. Also, new versions of reverse Young and Heinz mean inequalities for unitarily invariant norms are established.

This note aims to present some operator inequalities and matrices inequalities for the Hilbert–Schmidt norm. First, the refined Young and its reverse inequalities for scalars are given. Then, based on these inequalities, operator inequalities and matrices inequalities on the Hilbert–Schmidt norm are established.

In this note, we use the convexity of the function phi(nu) to sharpen the matrix version of the Heinz means, where phi(nu) is defined as phi(nu) = parallel to A(nu)XB(1-nu) + A(1-nu)XB(nu)parallel to on [0,1] for A, B, X is an element of Mn such that A and B are positive semidefinite, and also give a refinement of the inequality [Theorem 6, SIAM J. Matrix Anal. Appl. 20 (1998), 466-470] which is due to Zhan.

The purpose of this paper is to give a survey of the progress, advantages and limitations of various locating methods of complex matrices' eigenvalues. Some new methods of locating complex matrices’ eigenvalues as well as new applications of locating methods will be exhibited. And theoretical research and practical application of locating methods of eigenvalues will be promoted.