This note lies in the scope of operator inequalities. We assume that the reader is familiar with the continuous functional calculus and the Kubo-Ando theory [1].

It is to be understood throughout the paper that the capital letters present bounded linear operators acting on a Hilbert space 𝓗. *A* is positive (written *A* ≥ 0) in case 〈*Ax*,*x*〉 ≥ 0 for all *x* ∈ 𝓗 also an operator *A* is said to be strictly positive (denoted by *A* > 0) if *A* is positive and invertible. If *A* and *B* are self-adjoint, we write *B* ≥ *A* in case *B*–*A* ≥ 0. As usual, by *I* we denote the identity operator.

The weighted arithmetic mean ∇_{v}, geometric mean ♯_{v}, and harmonic mean !_{v}, for *v* ∈ [0, 1] and *a*,*b* > 0, are defined as follows:

$$\begin{array}{}{\displaystyle a{\mathrm{\nabla}}_{v}b=\left(1-v\right)a+vb,\phantom{\rule{1em}{0ex}}a{\mathrm{\u266f}}_{v}b={a}^{1-v}{b}^{v},\phantom{\rule{1em}{0ex}}a{!}_{v}b={\left\{(1-v){a}^{-1}+v{b}^{-1}\right\}}^{-1}.}\end{array}$$

If *v* = $\begin{array}{}\frac{1}{2}\end{array}$, we denote the arithmetic, geometric, and harmonic means, respectively, by ∇, ♯ and !, for the simplicity. Like the scalar cases, the operator arithmetic mean, the operator geometric mean, and the operator harmonic mean for *A*,*B* > 0 can be stated in the following form:

$$\begin{array}{}{\displaystyle A{\mathrm{\nabla}}_{v}B=\left(1-v\right)A+vB,\phantom{\rule{1em}{0ex}}A{\mathrm{\u266f}}_{v}B={A}^{\frac{1}{2}}{\left({A}^{-\frac{1}{2}}B{A}^{-\frac{1}{2}}\right)}^{v}{A}^{\frac{1}{2}},\phantom{\rule{1em}{0ex}}A{!}_{v}B={\left\{(1-v){A}^{-1}+v{B}^{-1}\right\}}^{-1}.}\end{array}$$

The celebrated arithmetic-geometric-harmonic-mean inequalities for scalars assert that if *a*,*b* > 0, then

$$\begin{array}{}{\displaystyle a{!}_{v}b\le a{\mathrm{\u266f}}_{v}b\le a{\mathrm{\nabla}}_{v}b.}\end{array}$$(1)

Generalization of the inequalities (1) to operators can be seen as follows: If *A*,*B* > 0, then

$$\begin{array}{}{\displaystyle A{!}_{v}B\le A{\mathrm{\u266f}}_{v}B\le A{\mathrm{\nabla}}_{v}B.}\end{array}$$

The last inequality above is called the operator Young inequality. During the past years, several refinements and reverses were given for Young’s inequality, see for example [2,3,4].

Zuo et al. showed in [5, Theorem 7] that the following inequality holds:

$$\begin{array}{}{\displaystyle K{\left(h,2\right)}^{r}A{\mathrm{\u266f}}_{v}B\le A{\mathrm{\nabla}}_{v}B,\phantom{\rule{1em}{0ex}}\text{\hspace{0.17em}}r=min\left\{v,1-v\right\},\text{\hspace{0.17em}}K\left(h,2\right)=\frac{{\left(h+1\right)}^{2}}{4h},\text{\hspace{0.17em}}h=\frac{M}{m}}\end{array}$$(2)

whenever 0 < *m*′*I* ≤ *B* ≤ *mI* < *MI* ≤ *A* ≤ *M*′*I* or 0 < *m*′*I* ≤ *A* ≤ *mI* < *MI* ≤ *B* ≤ *M*′*I*. As the authors mentioned in [5], the inequality (2) improves the following refinement of Young’s inequality involving Specht
$\begin{array}{}S\left(t\right)=\frac{{t}^{\frac{1}{t-1}}}{e\mathrm{log}{t}^{\frac{1}{t-1}}}\left(t>0,t\ne 1\right)\end{array}$
(see [6, Theorem 2]),

$$\begin{array}{}{\displaystyle S\left({h}^{r}\right)A{\mathrm{\u266f}}_{v}B\le A{\mathrm{\nabla}}_{v}B.}\end{array}$$

Under the above assumptions, Dragomir proved in [7, Corollary 1] that

$$\begin{array}{}{\displaystyle A{\mathrm{\nabla}}_{v}B\le \mathrm{exp}\left[\frac{v\left(1-v\right)}{2}{\left(h-1\right)}^{2}\right]A{\mathrm{\u266f}}_{v}B.}\end{array}$$

We remark that there is no relationship between two constants *K*(*h*,2)^{r} and
$\begin{array}{}\mathrm{exp}\left[\frac{v\left(1-v\right)}{2}{\left(h-1\right)}^{2}\right]\end{array}$ in general.

In [3, 8] we proved some sharp multiplicative reverses of Young’s inequality. In this brief note, as the continuation of our previous works, we establish sharp bounds for the arithmetic, geometric and harmonic mean inequalities. Moreover, we shall show some additive-type refinements and reverses of Young’s inequality. We will formulate our new results in a more general setting, namely the sandwich assumption *sA* ≤ *B* ≤ *tA* (0 < *s* ≤ *t*). Additionally, we present some Young type inequalities for the wider range of *v*; i.e., *v* ∉ [0, 1].

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.