Generalized Refinements of Reversed AM-GM Operator Inequalities for Positive Linear Maps

Abstract

We shall present some more generalized and further refinements of reversed AM-GM operator inequalities for positive linear maps due to Xue’s and Ali’s publications.

1. Introduction

Let $m,M$ be scalars and I be the identity operator. $B(\mathcal{H})$ denote all bounded linear operators acting on a Hilbert space $(\mathcal{H},⟨·,·⟩)$. In addition, $A\ge 0$ means the operator A is positive. We say $A\ge B$ if $A-B\ge 0$. A linear map $\mathsf{\Phi }:B(\mathcal{H})\to B(\mathcal{H})$ is called positive (strictly positive) if $\mathsf{\Phi }(A)\ge 0$ ($\mathsf{\Phi }(A)>0$) whenever $A\ge 0$ ($A>0$), and $\mathsf{\Phi }$ is said to be unital if $\mathsf{\Phi }(I)=I$.

If $A,B\in B(\mathcal{H})$ are two positive operators, then the operator weighted arithmetic mean and geometric mean are defined as

$$\begin{array}{c}\hfill A{\nabla }_{v}B=(1-v)A+vB\mathrm{and}A{♯}_{v}B=A^{\frac{1}{2}}{\left(A^{-\frac{1}{2}}B{A}^{-\frac{1}{2}}\right)}^v{A}^{\frac{1}{2}}\end{array}$$

for $v\in [0,1]$, respectively. Denoted $A{\nabla }_{v}B$ by $A\nabla B$ and $A{♯}_{v}B$ by $A♯B$ when $v=\frac{1}{2}$ for brevity. The Kantorovich constant and Specht’s ratio are defined by $K(h,2)=\frac{{(h+1)}^2}{4h}$ and $S(h)=\frac{h^{\frac{1}{h-1}}}{eln{h}^{\frac{1}{h-1}}}$ when $h>0$. If there is no special explanations, we always default to $a,b>0$ and $v\in [0,1]$ in this paper.

It is well known that the famous Young’s inequality reads

$$\begin{array}{c}\hfill a^{1-v}{b}^v\le (1-v)a+vb.\end{array}$$

(1)

Furuichi [1] improved (1) with Specht’s ratio

$$\begin{array}{c}\hfill S\left({\left(\frac{b}{a}\right)}^r\right)a^{1-v}{b}^v\le (1-v)a+vb,\end{array}$$

(2)

where $r=min\{v,1-v\}.$ Zuo et al. [2] further improved (2) as

$$\begin{array}{c}\hfill S\left({\left(\frac{b}{a}\right)}^r\right)a^{1-v}{b}^v\le K^r\left(\frac{b}{a}\right)a^{1-v}{b}^v\le (1-v)a+vb.\end{array}$$

(3)

Sababheh and Moslehian [3] gave a refinement of (3) in the following form

$$\begin{array}{c}\hfill K{\left(\sqrt[2^N]{h},2\right)}^{{\beta }_N(v)}{a}^{1-v}{b}^v+S_N(v;b,a)\le (1-v)a+vb,\end{array}$$

(4)

where $N\in {\mathbb{N}}^{+}$, $h=\frac{a}{b}$, ${\beta }_N(v)=min\{{\alpha }_N(v),1-{\alpha }_N(v)\}$ with ${\alpha }_N(v)=1+\left[2^{N}v\right]-2^{N}v$ and

$$\begin{array}{c}\hfill S_N(v;b,a)=\sum _{j=1}^N{s}_j(v){\left(\sqrt[2^j]{a^{2^{j-1}-k_j(v)}{b}^{k_j(v)}}-\sqrt[2^j]{b^{k_j(v)+1}{a}^{2^{j-1}-k_j(v)-1}}\right)}^2\end{array}$$

(5)

for $j=1,2,\cdots ,N$, $k_j(v)=\left[2^{j-1}v\right],r_j(v)=\left[2^{j}v\right]$ and $s_j(v)={(-1)}^{r_j(v)}{2}^{j-1}v+{(-1)}^{r_j(v)+1}\left[\frac{r_j(v)+1}{2}\right].$

Taking $v=\frac{1}{2}$ in (1), we can get the following AM-GM operator inequality for any two positive operators A and B,

$$\begin{array}{c}\hfill A♯B\le A\nabla B.\end{array}$$

(6)

Lin [4] gave a reverse of inequality (6) involving unital positive linear maps $\mathsf{\Phi }:$

$$\begin{array}{c}\hfill \mathsf{\Phi }(A\nabla B)\le K(h,2)\mathsf{\Phi }(A♯B)\end{array}$$

(7)

for $0<mI\le A,B\le MI$ and $h=\frac{M}{m}$. In generally, for any two positive operators $A,B$ and $P>1$,

$$\begin{array}{c}\hfill A\ge B\nRightarrow A^P\ge B^P.\end{array}$$

(8)

For example, putting $P=2$, $A=\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$ and $B=\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$. However, Lin [4] showed that the inequality (7) can be squared under the same conditions as in it,

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2(A\nabla B)\le K^2(h,2){\mathsf{\Phi }}^2(A♯B)\end{array}$$

(9)

and

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2(A\nabla B)\le K^2(h,2){\left(\mathsf{\Phi }(A)♯\mathsf{\Phi }(B)\right)}^2.\end{array}$$

(10)

Moreover, the author [4] found that Specht’s ratio and Kantorovich constant have the following relations

$$\begin{array}{c}\hfill S(h)\le K(h,2)\le S^2(h)\end{array}$$

(11)

for $h>1.$ Also, Lin [4] conjectured $K(h)$ can be replaced by $S(h)$ in (9) and (10). Xue [5] solved Lin’s conjecture under some conditions: $0<mI\le A,B\le MI$, $\sqrt{\frac{M}{m}}\le 2.314$ and $h=\frac{M}{m}$, she got

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2(A\nabla B)\le K(h,2){\mathsf{\Phi }}^2(A♯B)\end{array}$$

(12)

and

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2(A\nabla B)\le K(h,2){(\mathsf{\Phi }(A)♯\mathsf{\Phi }(B))}^2.\end{array}$$

(13)

Recently, Ali et al. [6] gave some refinements of inequalities (12) and (13) as follows:

Theorem 1.

Let $0<m\le M$, $\sqrt{\frac{M}{m}}\le 2.314$. For every positive unital linear map Φ,

(1) 

if $0<mI\le A,B\le \frac{M+m}{2}I$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A\nabla B+\frac{M+m}{2}m\left(\frac{A^{-1}+B^{-1}}{2}-(A^{-1}♯B^{-1})\right)\right)\le {\left(\frac{M+m}{2\sqrt{Mm}}\right)}^2{\mathsf{\Phi }}^2(A{♯}_{v}B);\end{array}$$

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A\nabla B+\frac{M+m}{2}m\left(\frac{A^{-1}+B^{-1}}{2}-(A^{-1}♯B^{-1})\right)\right)\le {\left(\frac{M+m}{2\sqrt{Mm}}\right)}^2{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^2;\end{array}$$

(2) 

if $0<\frac{M+m}{2}I\le A,B\le MI$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A\nabla B+\frac{M+m}{2}M\left(\frac{A^{-1}+B^{-1}}{2}-(A^{-1}♯B^{-1})\right)\right)\le {\left(\frac{M+m}{2\sqrt{Mm}}\right)}^2{\mathsf{\Phi }}^2(A{♯}_{v}B);\end{array}$$

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A\nabla B+\frac{M+m}{2}M\left(\frac{A^{-1}+B^{-1}}{2}-(A^{-1}♯B^{-1})\right)\right)\le {\left(\frac{M+m}{2\sqrt{Mm}}\right)}^2{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^2;\end{array}$$

(3) 

if $0<mI\le A<\frac{M+m}{2}I\le B\le MI$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A\nabla B+\frac{M+m}{2}\left(\frac{m{A}^{-1}+M{B}^{-1}}{2}-(m{A}^{-1}♯M{B}^{-1})\right)\right)\le {\left(\frac{M+m}{2\sqrt{Mm}}\right)}^2{\mathsf{\Phi }}^2(A{♯}_{v}B);\end{array}$$

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A\nabla B+\frac{M+m}{2}\left(\frac{m{A}^{-1}+M{B}^{-1}}{2}-(m{A}^{-1}♯M{B}^{-1})\right)\right)\le {\left(\frac{M+m}{2\sqrt{Mm}}\right)}^2{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^2;\end{array}$$

(4) 

if $0<mI\le B\le \frac{M+m}{2}I\le A\le MI$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A\nabla B+\frac{M+m}{2}\left(\frac{M{A}^{-1}+m{B}^{-1}}{2}-(M{A}^{-1}♯m{B}^{-1})\right)\right)\le {\left(\frac{M+m}{2\sqrt{Mm}}\right)}^2{\mathsf{\Phi }}^2(A{♯}_{v}B);\end{array}$$

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A\nabla B+\frac{M+m}{2}\left(\frac{M{A}^{-1}+m{B}^{-1}}{2}-(M{A}^{-1}♯m{B}^{-1})\right)\right)\le {\left(\frac{M+m}{2\sqrt{Mm}}\right)}^2{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^2;\end{array}$$

For more information about operator inequalities involving positive linear maps, we refer the readers to [7,8,9,10,11] and references therein.

In this paper, we shall give some more generalized and further refinements of reversed AM-GM operator inequalities for positive linear maps due to Ali’s results.

2. Main Results

We give some lemmas to prove our main results.

Lemma 1.

Let $0<mI\le A\le m^{\prime }I<M^{\prime }I\le B\le MI$, $h=\frac{M}{m}$, $h^{\prime }=\frac{M^{\prime }}{m^{\prime }}$. Then we have

$$\begin{array}{c}\hfill K{\left(\sqrt[2^N]{h^{\prime }},2\right)}^{{\beta }_N(v)}A{♯}_{v}B+\sum _{j=1}^N{s}_j(v)\left(A{♯}_{{\alpha }_j(v)}B+A{♯}_{{\alpha }_j(v)+2^{1-j}}B-2A{♯}_{{\alpha }_j(v)+2^{-j}}B\right)\le A{▿}_{v}B,\end{array}$$

(14)

where ${\alpha }_j(v)=\frac{k_j(v)}{2^{j-1}},k_j(v)=\left[2^{j-1}v\right]$ and ${\beta }_N(v)=min\left\{1+\left[2^{N}v\right]-2^{N}v,2^{N}v-\left[2^{N}v\right]\right\}$, $r_j(v)=\left[2^{j}v\right]$ and $s_j(v)={(-1)}^{r_j(v)}{2}^{j-1}v+{(-1)}^{r_j(v)+1}\left[\frac{r_j(v)+1}{2}\right].$

Proof. 

Putting $a=1$ in (4), we obtain

$$\begin{array}{c}\hfill K{\left(\sqrt[2^N]{\frac{1}{b}},2\right)}^{{\beta }_N(v)}{b}^v+\sum _{j=1}^N{s}_j(v)\left(b^{{\alpha }_j(v)}+b^{{\alpha }_j(v)+2^{1-j}}-2b^{{\alpha }_j(v)+2^{-j}}\right)\le (1-v)+vb.\end{array}$$

Taking $X=A^{-\frac{1}{2}}B{A}^{-\frac{1}{2}}$, and then $Sp(X)\subseteq [h^{\prime },h]\subseteq (1,+\infty )$. By standard functional calculus and the Kantorovich constant $K(\frac{1}{t},2)$ is a decreasing function on $\frac{1}{t}\in (0,1)$, we get

$$\begin{array}{c}\hfill K{\left(\sqrt[2^N]{\frac{1}{h^{\prime }}},2\right)}^{{\beta }_N(v)}{X}^v+\sum _{j=1}^N{s}_j(v)\left(X^{{\alpha }_j(v)}+X^{{\alpha }_j(v)+2^{1-j}}-2X^{{\alpha }_j(v)+2^{-j}}\right)\le (1-v)+vX.\end{array}$$

(15)

Multiplying $A^{\frac{1}{2}}$ on both sides of inequality (15), with the fact $K(h,2)=K\left(\frac{1}{h},2\right)$, we can get (14) directly.  □

Lemma 2.

Under the same conditions as in Lemma 1, we have

$$\begin{array}{c}\hfill A{\nabla }_{v}B+MmK{\left(\sqrt[2^N]{h^{\prime }},2\right)}^{{\beta }_N(v)}{(A{♯}_{v}B)}^{-1}+Mm{S}_N(v;B^{-1},A^{-1})\le M+m,\end{array}$$

(16)

where $S_N(v;B^{-1},A^{-1})={\displaystyle \sum _{j=1}^N}{s}_j(v)\left(A^{-1}{♯}_{{\alpha }_j(v)}{B}^{-1}+A^{-1}{♯}_{{\alpha }_j(v)+2^{1-j}}{B}^{-1}-2A^{-1}{♯}_{{\alpha }_j(v)+2^{-j}}{B}^{-1}\right).$

Proof. 

If $0<mI\le A,B\le MI$, then

$$\begin{array}{c}\hfill (M-A)(A-m)A^{-1}\ge 0\mathrm{and}(M-B)(B-m)B^{-1}\ge 0,\end{array}$$

that is

$$\begin{array}{c}\hfill A+Mm{A}^{-1}\le M+m\mathrm{and}B+Mm{B}^{-1}\le M+m.\end{array}$$

(17)

So we have

$$\begin{array}{c}\hfill A{\nabla }_{v}B+Mm(A^{-1}{\nabla }_v{B}^{-1})\le M+m.\end{array}$$

(18)

Thus, we obtain

$$\begin{array}{cc}\hfill & A{\nabla }_{v}B+Mm{S}_N(v;B^{-1},A^{-1})+MmK{\left(\sqrt[2^N]{h^{\prime }},2\right)}^{{\beta }_N(v)}{(A{♯}_{v}B)}^{-1}\hfill \\ \hfill & =A{\nabla }_{v}B+Mm\left(S_N(v;B^{-1},A^{-1})+K{\left(\sqrt[2^N]{h^{\prime }},2\right)}^{{\beta }_N(v)}\left(A^{-1}{♯}_v{B}^{-1}\right)\right)\hfill \\ \hfill & \le A{\nabla }_{v}B+Mm\left(A^{-1}{\nabla }_v{B}^{-1}\right)(\mathrm{by}(14))\hfill \\ \hfill & \le M+m.(\mathrm{by}(18))\hfill \end{array}$$

 □

Lemma 3

([12]). Let $A,B\ge 0$. Then the following norm inequality holds

$$\begin{array}{c}\hfill \parallel AB\parallel \le \frac{1}{4}\parallel A+B{\parallel }^2.\end{array}$$

(19)

Lemma 4

([13]). Let Φ be a unital positive linear map and $A>0$. Then

$$\begin{array}{c}\hfill \mathsf{\Phi }{(A)}^{-1}\le \mathsf{\Phi }(A^{-1}).\end{array}$$

(20)

Lemma 5

([14]). (i) If $0\le P\le 1$ and $A\ge B\ge 0$, then

$$\begin{array}{c}\hfill A^P\ge B^P.\end{array}$$

(21)

(ii) Let Φ be a unital positive linear map and $A,B>0$. For $v\in [0,1]$, we have

$$\begin{array}{c}\hfill \mathsf{\Phi }(A{♯}_{v}B)\le \mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B).\end{array}$$

(22)

Lemma 6

([15]). Let $A,B\ge 0$. Then for $1\le P<+\infty$,

$$\begin{array}{c}\hfill \parallel A^P+B^P\parallel \le \parallel {(A+B)}^P\parallel .\end{array}$$

(23)

Theorem 2.

Let $0<mI\le MI$, $\sqrt{\frac{M}{m}}\le 2.314$ and $S_N(v;B^{-1},A^{-1})$ defined as in Lemma 2. Then for every positive unital linear map Φ and $P\in [0,2]$,

(1) 

if $0<mI\le A\le m_1^{\prime }I<M_1^{\prime }I\le B\le \frac{M+m}{2}I$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \frac{K^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)}{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

(24)

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \frac{K^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

(25)

where $h=\frac{M}{m}$, $h_1^{\prime }=\frac{M_1^{\prime }}{m_1^{\prime }}$ and $v\in [0,1]$.

(2) 

if $0<\frac{M+m}{2}I\le A\le m_2^{\prime }I<M_2^{\prime }I\le B\le MI$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}M\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \frac{K^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_2^{\prime }},2\right)}{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

(26)

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}M\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \frac{K^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_2^{\prime }},2\right)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

(27)

where $h=\frac{M}{m}$, $h_2^{\prime }=\frac{M_2^{\prime }}{m_2^{\prime }}$ and $v\in [0,1]$.

(3) 

if $0<mI\le A\le m_3^{\prime }I<\frac{M+m}{2}I\le B\le MI$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\le \frac{{\left(\frac{M}{m}\right)}^{\frac{P}{2}(1-2v)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_3^{\prime }},2\right)}{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

(28)

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\le \frac{{\left(\frac{M}{m}\right)}^{\frac{P}{2}(1-2v)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_3^{\prime }},2\right)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

(29)

where $h=\frac{M}{m}$, $h_3^{\prime }=\frac{M+m}{2m_3^{\prime }}$ and $v\in [0,\frac{1}{2}]$.

(4) 

if $0<mI\le B\le \frac{M+m}{2}I<M_4^{\prime }I\le A\le MI$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;m{B}^{-1},M{A}^{-1})\right)\right)\le \frac{{\left(\frac{M}{m}\right)}^{\frac{P}{2}(2v-1)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_4^{\prime }},2\right)}{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

(30)

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;m{B}^{-1},M{A}^{-1})\right)\right)\le \frac{{\left(\frac{M}{m}\right)}^{\frac{P}{2}(2v-1)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_4^{\prime }},2\right)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

(31)

where $h=\frac{M}{m}$, $h_4^{\prime }=\frac{2M_4^{\prime }}{M+m}$ and $v\in [\frac{1}{2},1]$.

Proof. 

If $0<mI\le A\le m_1^{\prime }I<M_1^{\prime }I\le B\le \frac{M+m}{2}I$, we obtain

$$\begin{array}{cc}& \left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\times \frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}{\mathsf{\Phi }}^{-1}(A{♯}_{v}B)\left|\right|\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)+\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}{\mathsf{\Phi }}^{-1}(A{♯}_{v}B)|{|}^2\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)+\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}\mathsf{\Phi }\left({(A{♯}_{v}B)}^{-1}\right)|{|}^2\hfill \\ \hfill & =\frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}{(A{♯}_{v}B)}^{-1}+\frac{M+m}{2}m{S}_N(v;B^{-1},A^{-1})\right)|{|}^2\hfill \\ \hfill & \le \frac{1}{4}{\left(\frac{M+m}{2}+m\right)}^2,\hfill \end{array}$$

(32)

where the first inequality is by (19), the second one is by (20), and the last inequality comes from (16). That is

$$\begin{array}{c}\hfill \left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right){\mathsf{\Phi }}^{-1}(A{♯}_{v}B)\left|\right|\le \frac{{(\frac{M+m}{2}+m)}^2}{4\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}}.\end{array}$$

Since $1\le \sqrt{\frac{M}{m}}\le 2.314$, it follows that ${\left(\sqrt{\frac{M}{m}}-1\right)}^2\left[{\left(\sqrt{\frac{M}{m}}\right)}^3-\frac{2M}{m}+\sqrt{\frac{M}{m}}-4\right]\le 0,$ which is equivalent to

$$\begin{array}{c}\hfill \frac{{\left(\frac{M+m}{2}+m\right)}^2}{4\frac{M+m}{2}m}\le \frac{M+m}{2\sqrt{Mm}}.\end{array}$$

(33)

So we have

$$\begin{array}{c}\hfill \left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right){\mathsf{\Phi }}^{-1}(A{♯}_{v}B)\left|\right|\le \frac{M+m}{2\sqrt{Mm}K{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}}.\end{array}$$

That is

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \left(\frac{K(h,2)}{K{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{2{\beta }_N(v)}}\right){\mathsf{\Phi }}^2(A{♯}_{v}B).\end{array}$$

(34)

In addition, we can get

$$\begin{array}{cc}& \left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\times \frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-1}\left|\right|\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)+\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-1}|{|}^2\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)+\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}{\mathsf{\Phi }}^{-1}(A{♯}_{v}B)|{|}^2\hfill \\ \hfill & \le \frac{1}{4}{\left(\frac{M+m}{2}+m\right)}^2,\hfill \end{array}$$

where the first inequality is by (19), the second is by (22), and the third is by (32). That is

$$\begin{array}{cc}\hfill & \left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right){\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-1}\left|\right|\hfill \\ & \le \frac{{\left(\frac{M+m}{2}+m\right)}^2}{4\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}}\le \frac{K^{\frac{1}{2}}(h,2)}{K{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}}.\hfill \end{array}$$

So we have

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \left(\frac{K(h,2)}{K{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{2{\beta }_N(v)}}\right){\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^2.\end{array}$$

(35)

We can get (24) and (25) by (34) and (35) with Lemma 5 (i), respectively.

Since $\frac{{\left(\frac{M+m}{2}+M\right)}^2}{4\frac{M+m}{2}M}\le \frac{{\left(\frac{M+m}{2}+m\right)}^2}{4\frac{M+m}{2}m}$, by 2nd case $0<\frac{M+m}{2}I\le A\le m_2^{\prime }I<M_2^{\prime }I\le B\le MI$, we can similarly obtain the inequalities (26) and (27) by (16), (17), (19) and (20). So we omit the details.

If $0<mI\le A\le m_3^{\prime }I<\frac{M+m}{2}I\le B\le MI$ and $v\in [0,\frac{1}{2}]$, then we have

$$\begin{array}{c}\hfill A+\frac{M+m}{2}m{A}^{-1}\le \frac{M+m}{2}+m\mathrm{and}B+\frac{M+m}{2}M{B}^{-1}\le \frac{M+m}{2}+M.\end{array}$$

So

$$\begin{array}{c}\hfill A{\nabla }_{v}B+\frac{M+m}{2}\left((m{A}^{-1}){\nabla }_v(M{B}^{-1})\right)\le (v+\frac{1}{2})M+(\frac{3}{2}-v)m\le M+m.\end{array}$$

(36)

Compute

$$\begin{array}{cc}& \left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\times \frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v{\mathsf{\Phi }}^{-1}(A{♯}_{v}B)\left|\right|\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)+\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v{\mathsf{\Phi }}^{-1}(A{♯}_{v}B)|{|}^2\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)+\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v\mathsf{\Phi }\left({(A{♯}_{v}B)}^{-1}\right)|{|}^2\hfill \\ \hfill & =\frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})+K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}\left((m{A}^{-1}){♯}_v(M{B}^{-1})\right)\right)\right)|{|}^2\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left((m{A}^{-1}){\nabla }_v(M{B}^{-1})\right)\right)|{|}^2\hfill \\ \hfill & \le \frac{{(M+m)}^2}{4}.\hfill \end{array}$$

(37)

where the first inequality is by (19), the second is by (20), the third is by (14), and the last one is by (36). That is

$$\begin{array}{cc}\hfill & \left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right){\mathsf{\Phi }}^{-1}(A{♯}_{v}B)\left|\right|\hfill \\ \hfill & \le \frac{{(M+m)}^2}{4\frac{M+m}{2}{m}^{1-v}{M}^{v}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}}=\frac{{\left(\frac{M}{m}\right)}^{\frac{1}{2}-v}{K}^{\frac{1}{2}}(h,2)}{K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}}.\hfill \end{array}$$

That is

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\le \frac{{\left(\frac{M}{m}\right)}^{1-2v}K(h,2)}{K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{2{\beta }_N(v)}}{\mathsf{\Phi }}^2(A{♯}_{v}B).\end{array}$$

(38)

Moreover,

$$\begin{array}{cc}\hfill & \left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\times \frac{M+m}{2}{m}^{1-v}{M}^{v}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-1}\left|\right|\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)+\frac{M+m}{2}{m}^{1-v}{M}^{v}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-1}|{|}^2\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)+\frac{M+m}{2}{m}^{1-v}{M}^{v}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{\mathsf{\Phi }}^{-1}(A{♯}_{v}B)|{|}^2\hfill \\ \hfill & \le \frac{{(M+m)}^2}{4},\hfill \end{array}$$

where the first inequality is by (19), the second is by (22), and the third is by (37). That is

$$\begin{array}{c}\hfill \left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right){\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-1}\left|\right|\le \frac{{\left(\frac{M}{m}\right)}^{\frac{1}{2}-v}{K}^{\frac{1}{2}}(h,2)}{K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}},\end{array}$$

so we have

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^2\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\le \frac{{\left(\frac{M}{m}\right)}^{1-2v}K(h,2)}{K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{2{\beta }_N(v)}}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^2.\end{array}$$

(39)

We can get (28) and (29) by (38) and (39) with Lemma 5 (i), respectively.

We can similarly obtain the inequalities (30) and (31) under the conditions $0<mI\le B\le \frac{M+m}{2}I<M_3^{\prime }I\le A\le MI$ and $v\in [\frac{1}{2},1]$. So we omit the details.

Here we complete the proof.  □

Remark 1.

Putting $v=\frac{1}{2}$, $N=1$ and $P=2$ in Theorem 2, we can get Theorem 1.

Next, we present the generalizations of Theorem 2 for $P\ge 2.$

Theorem 3.

Let $0<mI\le MI$, $\sqrt{\frac{M}{m}}\le 2.314$ and $S_N(v;B^{-1},A^{-1})$ defined as in Lemma 2. Then for every positive unital linear map Φ and $P\ge 2$,

(i) 

if $0<mI\le A\le m_1^{\prime }I<M_1^{\prime }I\le B\le \frac{M+m}{2}I$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \frac{4^{P-2}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)}{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

(40)

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \frac{4^{P-2}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

(41)

where $h=\frac{M}{m}$, $h_1^{\prime }=\frac{M_1^{\prime }}{m_1^{\prime }}$ and $v\in [0,1]$.

(ii) 

if $0<\frac{M+m}{2}I\le A\le m_2^{\prime }I<M_2^{\prime }I\le B\le MI$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}M\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \frac{4^{P-2}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_2^{\prime }},2\right)}{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

(42)

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}M\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \frac{4^{P-2}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_2^{\prime }},2\right)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

(43)

where $h=\frac{M}{m}$, $h_2^{\prime }=\frac{M_2^{\prime }}{m_2^{\prime }}$ and $v\in [0,1]$.

(iii) 

if $0<mI\le A\le m_3^{\prime }I<\frac{M+m}{2}I\le B\le MI$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\le \frac{4^{P-2}{\left(\frac{M}{m}\right)}^{\frac{P}{2}(1-2v)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}(\sqrt[2^N]{h_3^{\prime }},2)}{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

(44)

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\le \frac{4^{P-2}{\left(\frac{M}{m}\right)}^{\frac{P}{2}(1-2v)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_3^{\prime }},2\right)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

(45)

where $h=\frac{M}{m}$, $h_3^{\prime }=\frac{M+m}{2m_3^{\prime }}$ and $v\in [0,\frac{1}{2}]$.

(iv) 

if $0<mI\le B\le \frac{M+m}{2}I<M_4^{\prime }I\le A\le MI$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;m{B}^{-1},M{A}^{-1})\right)\right)\le \frac{4^{P-2}{\left(\frac{M}{m}\right)}^{\frac{P}{2}(2v-1)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}(\sqrt[2^N]{h_4^{\prime }},2)}{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

(46)

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;m{B}^{-1},M{A}^{-1})\right)\right)\le \frac{4^{P-2}{\left(\frac{M}{m}\right)}^{\frac{P}{2}(2v-1)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_4^{\prime }},2\right)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

(47)

where $h=\frac{M}{m}$, $h_4^{\prime }=\frac{2M_4^{\prime }}{M+m}$ and $v\in [\frac{1}{2},1]$.

Proof. 

The proof of the line (ii) and (iv) are similar to the one presented in (i) and (iii), respectively, thus we omit them. Under the conditions i) $0<mI\le A\le m_1^{\prime }I<M_1^{\prime }I\le B\le \frac{M+m}{2}I$, we have

$$\begin{array}{cc}\hfill & \left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\times {\left(\frac{M+m}{2}\right)}^{\frac{P}{2}}{m}^{\frac{P}{2}}{K}^{\frac{P}{2}{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right){\mathsf{\Phi }}^{-\frac{P}{2}}(A{♯}_{v}B)\left|\right|\hfill \\ \hfill & \le \frac{1}{4}\left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)+{\left(\frac{M+m}{2}\right)}^{\frac{P}{2}}{m}^{\frac{P}{2}}{K}^{\frac{P}{2}{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right){\mathsf{\Phi }}^{-\frac{P}{2}}(A{♯}_{v}B)|{|}^2\hfill \\ \hfill & \le \frac{1}{4}\left|\right|{\left(\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)+\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}{\mathsf{\Phi }}^{-1}(A{♯}_{v}B)\right)}^{\frac{P}{2}}|{|}^2\hfill \\ \hfill & =\frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)+\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}{\mathsf{\Phi }}^{-1}(A{♯}_{v}B)|{|}^P\hfill \\ \hfill & \le \frac{1}{4}{\left(\frac{M+m}{2}+m\right)}^P.\hfill \end{array}$$

where the first inequality is by (19), the second is by (23) and the third is by (32). That is

$$\begin{array}{cc}\hfill & \left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right){\mathsf{\Phi }}^{-\frac{P}{2}}(A{♯}_{v}B)\left|\right|\hfill \\ \hfill & \le \frac{{\left(\frac{M+m}{2}+m\right)}^P}{4{\left(\frac{M+m}{2}\right)}^{\frac{P}{2}}{m}^{\frac{P}{2}}{K}^{\frac{P}{2}{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)}\le \frac{4^{\frac{P}{2}-1}{\left(\frac{M+m}{2\sqrt{Mm}}\right)}^{\frac{P}{2}}}{K^{\frac{P}{2}{\beta }_N(v)}(\sqrt[2^N]{h_1^{\prime }},2)}=\frac{4^{\frac{P}{2}-1}{K}^{\frac{P}{4}}(h,2)}{K^{\frac{P}{2}{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)},\hfill \end{array}$$

where the second inequality is by (33). So we have

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \frac{4^{P-2}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)}{\mathsf{\Phi }}^P(A{♯}_{v}B).\end{array}$$

In addition, we can get

$$\begin{array}{cc}\hfill & \left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\times {\left(\frac{M+m}{2}\right)}^{\frac{P}{2}}{m}^{\frac{P}{2}}{K}^{\frac{P}{2}{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right){\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-\frac{P}{2}}\left|\right|\hfill \\ \hfill & \le \frac{1}{4}\left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)+{\left(\frac{M+m}{2}\right)}^{\frac{P}{2}}{m}^{\frac{P}{2}}{K}^{\frac{P}{2}{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right){\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-\frac{P}{2}}|{|}^2\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\left(\mathsf{\Phi }(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)+\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}{\left({(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B))}^{-1}\right)}^{\frac{P}{2}}|{|}^2\hfill \\ \hfill & =\frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)+\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-1}|{|}^P\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\mathsf{\Phi }(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right))+\frac{M+m}{2}mK{\left(\sqrt[2^N]{h_1^{\prime }},2\right)}^{{\beta }_N(v)}{\mathsf{\Phi }}^{-1}(A{♯}_{v}B)|{|}^P\hfill \\ \hfill & \le \frac{1}{4}{\left(\frac{M+m}{2}+m\right)}^P.\hfill \end{array}$$

where the first inequality is by (19), the second is by (23), the third is by (22) and the last is by (32). That is

$$\begin{array}{c}\hfill \left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right){\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-\frac{P}{2}}\left|\right|\le \frac{4^{\frac{P}{2}-1}{K}^{\frac{P}{4}}(h,2)}{K^{\frac{P}{2}{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)},\end{array}$$

so we have

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\le \frac{4^{P-2}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P,\end{array}$$

as desired.

If $0<mI\le A\le m_3^{\prime }I<\frac{M+m}{2}I\le B\le MI$ and $v\in [0,\frac{1}{2}]$, then

$$\begin{array}{cc}\hfill & \left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\times {\left[\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v\right]}^{\frac{P}{2}}{\mathsf{\Phi }}^{-\frac{P}{2}}(A{♯}_{v}B)\left|\right|\hfill \\ \hfill & \le \frac{1}{4}\left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)+{\left[\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v\right]}^{\frac{P}{2}}{\mathsf{\Phi }}^{-\frac{P}{2}}(A{♯}_{v}B)|{|}^2\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\left(\mathsf{\Phi }(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)+\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v{\mathsf{\Phi }}^{-1}(A{♯}_{v}B){)}^{\frac{P}{2}}|{|}^2\hfill \\ \hfill & =\frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)+\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v{\mathsf{\Phi }}^{-1}(A{♯}_{v}B)|{|}^P\hfill \\ \hfill & \le \frac{1}{4}{(M+m)}^P.\hfill \end{array}$$

where the first inequality is by (19), the second is by (23) and the third is by (37). That is

$$\begin{array}{cc}\hfill & \left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right){\mathsf{\Phi }}^{-\frac{P}{2}}(A{♯}_{v}B)\left|\right|\hfill \\ \hfill & \le \frac{{(M+m)}^P}{4{\left[\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v\right]}^{\frac{P}{2}}}=\frac{2^{P-2}{\left(\frac{M}{m}\right)}^{\frac{P}{4}(1-2v)}{K}^{\frac{P}{4}}(h,2)}{K^{\frac{P}{2}{\beta }_N(v)}\left(\sqrt[2^N]{h_3^{\prime }},2\right)}.\hfill \end{array}$$

So we have

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\le \frac{4^{P-2}{\left(\frac{M}{m}\right)}^{\frac{P}{2}(1-2v)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_3^{\prime }},2\right)}{\mathsf{\Phi }}^P(A{♯}_{v}B).\end{array}$$

At the same time, we can get

$$\begin{array}{cc}\hfill & \left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\times {\left[\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v\right]}^{\frac{P}{2}}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-\frac{P}{2}}\left|\right|\hfill \\ \hfill & \le \frac{1}{4}\left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)+{\left[\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v\right]}^{\frac{P}{2}}{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-\frac{P}{2}}|{|}^2\hfill \\ \hfill & \le \frac{1}{4}\left|\right|{\left(\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)+\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-1}\right)}^{\frac{P}{2}}|{|}^2\hfill \\ \hfill & =\frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)+\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-1}|{|}^P\hfill \\ \hfill & \le \frac{1}{4}\left|\right|\mathsf{\Phi }\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)+\frac{M+m}{2}K{\left(\sqrt[2^N]{h_3^{\prime }},2\right)}^{{\beta }_N(v)}{m}^{1-v}{M}^v{\mathsf{\Phi }}^{-1}(A{♯}_{v}B)|{|}^P\hfill \\ \hfill & \le \frac{1}{4}{(M+m)}^P.\hfill \end{array}$$

where the first inequality is by (19), the second is by (23), the third is by (22) and the last is by (37). That is

$$\begin{array}{cc}\hfill & \left|\right|{\mathsf{\Phi }}^{\frac{P}{2}}\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right){\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^{-\frac{P}{2}}\left|\right|\hfill \\ \hfill & \le \frac{{(M+m)}^P}{4{\left(\frac{M+m}{2}{m}^{1-v}{M}^v\right)}^{\frac{P}{2}}{K}^{\frac{P}{2}{\beta }_N(v)}\left(\sqrt[2^N]{h_3^{\prime }},2\right)}=\frac{2^{P-2}{\left(\frac{M}{m}\right)}^{\frac{P}{4}(1-2v)}{K}^{\frac{P}{4}}(h,2)}{K^{\frac{P}{2}{\beta }_N(v)}\left(\sqrt[2^N]{h_3^{\prime }},2\right)}.\hfill \end{array}$$

So we have

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P(A{\nabla }_{v}B+\frac{M+m}{2}(S_N(v;M{B}^{-1},m{A}^{-1})))\le \frac{4^{P-2}{(\frac{M}{m})}^{\frac{P}{2}(1-2v)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}(\sqrt[2^N]{h_3^{\prime }},2)}{(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B))}^P.\end{array}$$

Here we complete the proof.  □

Theorems 2 and 3 implies the following results.

Corollary 1.

Let $0<m\le M$, $\sqrt{\frac{M}{m}}\le 2.314$ and $S_N(v;B^{-1},A^{-1})$ defined as in Lemma 2. Then for every positive unital linear map Φ and $P\ge 0$,

(i) 

if $0<mI\le A\le m_1^{\prime }I<M_1^{\prime }I\le B\le \frac{M+m}{2}I$, $v\in [0,1]$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\le W_1{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}m\left(S_N(v;B^{-1},A^{-1})\right)\right)\le W_1{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

where $h=\frac{M}{m}$, $h_1^{\prime }=\frac{M_1^{\prime }}{m_1^{\prime }}$, $W_1=max\left\{\frac{K^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)},\frac{4^{P-2}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)}\right\}$.

(ii) 

if $0<\frac{M+m}{2}I\le A\le m_2^{\prime }I<M_2^{\prime }I\le B\le MI$, $v\in [0,1]$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}M\left(S_N(v;B^{-1},A^{-1})\right)\right)\le W_2{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}M\left(S_N(v;B^{-1},A^{-1})\right)\right)\le W_2{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

where $h=\frac{M}{m}$, $h_2^{\prime }=\frac{M_2^{\prime }}{m_2^{\prime }}$, $W_2=max\left\{\frac{K^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_2^{\prime }},2\right)},\frac{4^{P-2}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_1^{\prime }},2\right)}\right\}$.

(iii) 

if $0<mI\le A\le m_3^{\prime }I<\frac{M+m}{2}I\le B\le MI$, $v\in [0,\frac{1}{2}]$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\le W_3{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}\left(S_N(v;M{B}^{-1},m{A}^{-1})\right)\right)\le W_3{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

where $h=\frac{M}{m}$, $h_3^{\prime }=\frac{M+m}{2m_3^{\prime }}$, $W_3=max\left\{\frac{{\left(\frac{M}{m}\right)}^{\frac{P}{2}(1-2v)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_3^{\prime }},2\right)},\frac{4^{P-2}{\left(\frac{M}{m}\right)}^{\frac{P}{2}(1-2v)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_3^{\prime }},2\right)}\right\}$.

(iv) 

if $0<mI\le B\le \frac{M+m}{2}I<M_4^{\prime }I\le A\le MI$, $v\in [\frac{1}{2},1]$, then

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}(S_N(v;m{B}^{-1},M{A}^{-1})\right))\le W_4{\mathsf{\Phi }}^P(A{♯}_{v}B);\end{array}$$

$$\begin{array}{c}\hfill {\mathsf{\Phi }}^P\left(A{\nabla }_{v}B+\frac{M+m}{2}(S_N(v;m{B}^{-1},M{A}^{-1})\right))\le W_4{\left(\mathsf{\Phi }(A){♯}_v\mathsf{\Phi }(B)\right)}^P;\end{array}$$

where $h=\frac{M}{m}$, $h_4^{\prime }=\frac{2M_4^{\prime }}{M+m}$, $W_4=max\left\{\frac{{\left(\frac{M}{m}\right)}^{\frac{P}{2}(2v-1)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_4^{\prime }},2\right)},\frac{4^{P-2}{\left(\frac{M}{m}\right)}^{\frac{P}{2}(2v-1)}{K}^{\frac{P}{2}}(h,2)}{K^{P{\beta }_N(v)}\left(\sqrt[2^N]{h_4^{\prime }},2\right)}\right\}$.