数理解析研究所講究録
第 1996 巻 2016 年 34-44

34Some examples of operator monotone functions
Masaru Nagisa
Graduate School of Science,
Chiba University

Introduction and preliminaries

1

Let be a real valued continuous function on
if it holds $f(A)\leq f(B)$ , formonotone on$A,$with $0\leq A\leq B$ , where $A\leq B$ means$(0, \infty)$$f$$(0, \infty)$. We call$n\cross n$-matrix
self-adjoint matrices
$f$

$n$

$B$

$(A\xi, \xi)\leq(B\xi, \xi) \forall\xi\in \mathbb{C}^{n}$isfor any positive integer
is -matrix monotone on. By’s theorem, it is known thatcalled operator monotone on,
if and only if is a Pick function on
is operator monotone on
has the analytic continuation $f(z)$which means the function :on the upper half plane $\mathbb{H}+=\{z\in \mathbb{C} : \Im z>0\}$ and satisfies the condition
is contained in the closure of $\mathbb{H}+([1], [3], [7])$ . For any positive integer$\beta_{i}(1\leq i\leq n)$ with, and positive numbers, a real number
as follows:, we define the function $f(t)$ on

When

$f$

$(0, \infty)$$n$$(0, \infty)$$n\in \mathbb{N},$$L\ddot{\circ}wner$$(0, \infty)$

$f$

$f$

$(0, \infty)$$f$$(0, \infty)arrow \mathbb{R}$$f$$f(\mathbb{H}_{+})$$n\in \mathbb{N}$$\gamma\in \mathbb{R}$$\alpha_{i},$$(0, \infty)$$\alpha_{i}\neq\beta_{j}(1\leq i,j\leq n)$$f(t)=t^{\gamma} \prod_{i=1}^{n}\frac{\beta_{i}}{\alpha_{i}}\frac{t^{\alpha_{i}}-1}{t^{\beta_{i}}-1} (t\neq 1)$

and $f(1)=1$ . In [9], the author gave the method to investigate the operator
monotonicity of functions $f(t)$ . Using this result, we consider the operator
monotonicity of the function $f(t)$ with some special form.
In section 2, we treat the following functions related to the power diﬀerence mean:$h(t)= \frac{b}{a}\frac{t^{a}-1}{t^{b}-1}, t\in(0, \infty)$,

for any real number and . In section 3, we treat the following functions(extended Petz-Hasegawa’s functions) :
$a$

$b$

$h(t)= \frac{ab(t-1)^{2}}{(t^{a}-1)(t^{b}-1)}, t\in(0, \infty)$,

35
for any real number$a$and , where we use the notation$b$$\frac{t^{0}-1}{0}=\log t(=\lim_{aarrow 0}\frac{t^{a}-1}{a})$.

We remark that the point-wise limit function $f(t)$ of
matrix monotone if
is -matriX monotone for all $m.$
with
Let, for,and$\{f_{m}(t)\}_{m=1}^{\infty}$$f_{m}(t)$$n$$\alpha=(\alpha_{1}, \ldots, \alpha_{n})$$\gamma\in \mathbb{R}$is n-$\beta=(\beta_{1}, \ldots, \beta_{n})$$\alpha_{i},$$\beta_{i}>0,$$f(t)=t^{\gamma} \prod_{i=1}^{n}\frac{\beta_{i}}{\alpha_{i}}\frac{t^{\alpha_{i}}-1}{t^{\beta_{i}}-1} (t\neq 1)$

for $f(t)$ .
andand $f(1)=1$ . We introduce two quantities
The following two lemmas related to these quantities are used to determine
the operator monotonicity of functions in section 3.
, we defineWhen$F(\alpha, \beta)$$F_{0}(\alpha, \beta)$$0<\alpha_{i},$ $\beta_{i}\leq 2$

for $z\in(O, \infty)$$\arg\frac{z^{\alpha_{i}}-1}{z^{\beta_{i}}-1}=0$

and continuously define the argument of$\beta_{i}\leq 2(i=1, \ldots, n)$ ,for$\frac{z^{\alpha_{\grave{l}}}-1}{z^{\beta_{i}}-1}$on$z\in \mathbb{H}+\cdot$

So we can define,$\alpha_{i},$$\arg f(z)=\gamma\arg z+\sum_{i=1}^{n}\arg\frac{z^{\alpha_{i}}-1}{z^{\beta_{i}}-1}$,

for$z\in \mathbb{H}_{+},$

and $\arg f(t)=0$ for $t\in(O, \infty)$ .
If $f(t)$ is non-constant operator $monotone_{\}}$ then its analytic continuation$f(z)$ has no zeros and no singular points on
since f is Pick function. It
is known (see, [9]:Proposition 3.1) that $f(z)$ has no zeros and no singular$(1\leq i\leq n)$ . Whenpoints on
if and only if
and$|\gamma|>2$ or $\max\{\alpha_{i}, \beta_{i}:1\leq i\leq n\}>2,$ $f(t)$ is not operator monotone.
$\mathbb{H}+$$0<\alpha_{i},$ $\beta_{i}\leq 2.$$|\gamma|\leq 2$$\mathbb{H}_{+}$

Lemma X.1 ([9]:Theorem 1.1, Lemma 2.3 and Proposition 3.2). Let$\beta_{i}\leq 2(1\leq i\leq n)$ .

$|\gamma|\leq 2,$$0<\alpha_{i},$(1) $f(t)$ is operator monotone on$(O, \infty)$$\gamma+G_{0}(\alpha, \beta)\geq 0$

where weand

if and only if$\gamma+F_{0}(\alpha, \beta)\leq 1,$$\mathcal{S}et$$g(t)= \prod_{i=1}^{n}\frac{\beta_{i}}{\alpha_{i}}\frac{t^{\alpha_{i}}-1}{t^{\beta_{i}}-1}$

and define $G_{0}( \alpha, \beta)=\inf\{\arg g(re^{\pi i}) : r\in(0, \infty)\}/7r$ and$\sup\{\arg g(re^{\pi i}) :r\in(O, \infty)\}/\pi.$$F_{0}(\alpha, \beta)=$

36
(2)

$G_{0}(\alpha, \beta)=-F_{0}(\beta, \alpha)$(3)$F_{0}( \alpha, \beta)+G_{0}(\alpha, \beta)=\sum_{i=1}^{n}(\alpha_{i}-\beta_{i})$and$F_{0}(\alpha, \beta)\geq 0.$.

(4) When $0<b<a\leq 2_{J}$$0<b\leq 1\Leftrightarrow G_{0}(a, b)\geq 0\Leftrightarrow F_{0}(a, b)\leq a-b,$

where we use the notation$a$(resp. b) instead $of\alpha=(a)$ (resp. $\beta=(b)$ ).

For $0\leq a,$ $b\leq 2$ , we define$F(a, b)=\{\begin{array}{ll}a-b if a\geq b, 0\leq b\leq 1a-1 if 1<a, b\leq 20 if a<b, 0\leq a\leq 1\end{array}$Let$\alpha=(\alpha_{1}, \ldots, \alpha_{n})$andpermutations on 1, . . . ,$\{$$\beta=(\beta_{1}, \ldots, \beta_{n})(0<\alpha_{i}, \beta_{i}\leq 2)$

satisfying with. Then we define$n\}$$\beta_{\tau(1)}\leq\beta_{\tau(2)}\leq\cdots\leq\beta_{\tau(n)}$$\sigma$andand$\tau$$\alpha_{\sigma(1)}\leq\alpha_{\sigma(2)}\leq\cdots\leq\alpha_{\sigma(n)}$$F( \alpha, \beta)=\sum_{i=1}^{n}F(\alpha_{\sigma(i)}, \beta_{\tau(i)})$

Lemma 1.2 ([9]:Theorem 1.2). For

and$|\gamma|\leq 2,$.

$0<\alpha_{i},$ $\beta_{i}\leq 2$, the function$f(t)=t^{\gamma} \prod_{i=1}^{n}\frac{\beta_{i}}{\alpha_{i}}\frac{t^{\alpha_{i}}-1}{t^{\beta_{i}}-1}$$become\mathcal{S}$operator monotone on$(0, \infty)$$\gamma-F(\beta, \alpha)\geq 0$

2

if

and

$\gamma+F(\alpha, \beta)\leq 1.$

Functions related the power diﬀerence mean

The following characterization is well-known:

Lemma 2.1 ([6]:Theorem 2.4.3). The function
monotone if and only if is inand$h$$h:(0, \infty)arrow \mathbb{R}i_{\mathcal{S}}2$$C^{1}(0, \infty)$$(_{h^{[1]}(\lambda_{2},\lambda_{1})}^{h^{[1]}(\lambda_{1},\lambda_{1})} h^{[1]}(\lambda_{2},\lambda_{2})h^{[1]}(\lambda_{1},\lambda_{2}))\geq 0$

for any$\lambda_{1},$$\lambda_{2}\in(0, \infty)$, where$h^{[1]}(\lambda, \mu)=\{\begin{array}{ll}\frac{h(\lambda)-h(\mu)}{\lambda-\mu} \lambda\neq\mu h’(\lambda) \lambda=\mu\end{array}$-matrix

37
Theorem 2.2. Let$a_{\rangle}b$be real numbers and$h(t)= \frac{b}{a}\frac{t^{a}-1}{t^{b}-1}, t\in(O, oo)$.

Then we have
(1)

$h$

(2)

$h$

is increasing on$(0, \infty)$

if $a>b$ and decreasing Of $a<b.$becomes operator monotone on
belongs to the set$(0, \infty)$

if and only if the point $(a, b)$$\Omega=\{a=b\}\cup\{(a, b):0\leq a-b\leq 1, a\geq-1, b\leq 1\}\cup([0,1]\cross[-1,0])$in the $(a, b)$ -plane:(3)

is 2-matrix monotone on
.
on

$h$

$(0, \infty)$

if and only if$h$is operator monotone$(0, \infty)$

Proof (1) We set$\frac{dh(t)}{dt}=\frac{t^{b-1}}{(t^{b}-1)^{2}}k(t)$,where $k(t)= \frac{b}{a}((a-b)t^{a}-at^{(l-b}+b)$ . Since$k(1)=0, \frac{dk(t)}{dt}=b(a-b)t^{a-1}(1-t^{-b})$ ,we have $k(t)\geq 0$ for $t\in(0, \infty)$ if $a>b$ . This means $h(t)$ is positive andif $a>b.$increasing on$(0, \infty)$

Remarking the fact$h(t)= \frac{b}{a}\frac{t^{a}-1}{t^{b}-1}=\frac{1}{\frac{a}{b}\frac{t^{b}-1}{t^{a}-1}},$38
is decreasing on
if $a<b.$(2) This has been proved in [9]:Example 3.4(1).
(3) It suﬃces to show that $(a, b)\in\Omega$ if $h(t)$ is 2-matrix monotone on$h(t)$$(0, \infty)$$(0, \infty)$.

We assume thathave $a>b$ and$h$is 2-matrix monotone and not constant. By (1) we$h’(t)= \frac{b}{a}\frac{k_{1}(t)}{(t^{b}-1)^{2}}\geq 0$

for all $t\in(O, \infty)$ ,where $k_{1}(t)=(a-b)t^{a+b-1}-at^{a-1}+bt^{b-1}$ . So we have, for any$s,$$t>0,$$(_{h^{[1]}(s,t)}h’(s) h^{[1]}(s,t)h’(t))\geq 0,$

equivalently, $h’(s)h’(t)-(h^{[1]}(s, t))^{2}\geq 0$ . Then we set$D(s, t)=h’(s)h’(t)-(h^{[1]}(s, t))^{2}$$= \frac{b^{2}}{a^{2}}\frac{1}{(s^{b}-1)^{2}(t^{b}-1)^{2}(s-t)^{2}}\cross(k_{1}(\mathcal{S})k_{1}(t)(s-t)^{2}-k_{2}\{s, t$

where$k_{2}(s, t)=((s^{a}-1)(t^{b}-1)-(s^{b}-1)(t^{a}-1))^{2}$$=((s^{a}-1)t^{b}-(\mathcal{S}^{b_{-1)t^{a}-s^{a}+S^{b})^{2}}},$

and we remark $k_{1}(s)k_{1}(t)(s-t)^{2},$ $k_{2}(s, t)\geq 0$ for all
When $b>0$ and $a+b+1<2a$ , we have$\lim_{tarrow\infty}D(s, t)<0$

for some$s,$ $t\in(O, \infty)$.

$s$

because$\lim_{tarrow\infty}t^{-2a}(k_{1}(s)k_{1}(t)(s-t)^{2}-k_{2}(s, t))$$= \lim_{tarrow\infty}t^{-2a+(a+b+1)}(((a-b)-at^{-b}+bt^{-a})k_{1}(s)(st^{-1}-1)^{2})$$-((s^{a}-1)t^{b-a}-(s^{b}-1)-(s^{a}-s^{b})t^{-a})^{2}$$=-(s^{b}-1)^{2}<$ O.

This contradicts to the assumption of . The highest degreein(resp.
) is
$h$

$t$

$k_{1}(s)k_{1}(t)(\mathcal{S}-t)^{2}$$k_{2}(t)$$(d_{1}, d_{2})=\{\begin{array}{ll}(a+b+1,2a) (0<b<a)(a+1,2a) (b<0<a) ,(a+1,0) (b<a<0)\end{array}$$d_{1}$(resp.

$d_{2}$) of39
and the lowest degree$d_{1}’$(resp.

$d_{2}^{1}$) of in $k_{1}(s)k_{1}(t)(s-t)^{2}$ (resp.
$t$$k_{2}(t)$) is$(d_{1}’, d_{2}’)=\{\begin{array}{ll}(b-1, O) (0<b<a)(b-1,2b) (b<0<a)(a+b-1,2b) (b<a<0)\end{array}$By using the similar argument as above, when $d_{1}<d_{2}$ or$d_{1}^{l}>d_{2}’$, we have$D(s,t)<0$ for a suﬃciently large $t(>0)$ and some fixed$s,$$D(s, t)<0$ for a suﬃciently small $t(>0)$ and some fixed

$s.$

or
So 2-matrix monotonicity of$h$implies$d_{1}\geq d_{2}$This means that $(a, b)\in\Omega$ if

3

$h$

and

$d_{1}’\leq d_{2}’.$is 2-matrix monotone.

$\square$

Extended Petz-Hasegawa’s functions

We consider the operator monotonicity of the function$h(t)= \frac{ab(t-1)^{2}}{(t^{a}-1)(t^{b}-1)}, \#\in(0, \infty)$,

for any real number and . When $b=1-a$ and $-1\leq a\leq 2$ , this function iscalled Petz-Hasegawa’s function and becomes operator monotone on(see [4], [5], [7]).
$b$

$a$

$(0, \infty)$

Theorem 3.1. Let

$a,$

$b$

be real numbers and$h(t)= \frac{ab(t-1)^{2}}{(t^{a}-1)(t^{b}-1)}.$

Then becomes operator monotone on
belongs to the following set:$h$$(0, \infty)$

if and only if the point $(a, b)$$\Omega=\{(a, b):a\in[-1, 2], g_{1}(a)\leq b\leq g_{2}(a)\},$

where$g_{1}(a)=\{\begin{array}{ll}1, a\in[-1, 0]1-a, a\in[0, 1 )-1, a\in[1_{\}}2]\end{array}$40
and $92(a)$ is satisfying $1-a\leq g_{2}(a)\leq 2-a$ and the following equation:$( \frac{a-92(a)}{a}\frac{\sin a\pi}{\sin(a+92(a))\pi})^{a}=(\frac{g_{2}(a)-a}{g_{2}(a)}\frac{\sin g_{2}(a)\pi}{\sin(92(a)+a)\pi})^{g_{2}(a)}.$This set$\Omega$in the $(a, b)$ -plane is as follows:

where the boundary curve$g_{2}(a)$is given by computations of approximate val-$ues.$

Proof. The function

is symmetric for$a\geq b$ . We can rewrite $h(t)$ as follows:
$h$

$a$

and . So we may assume that$b$$h(t)=ab \cdot\frac{(t-1)^{2}}{(t^{a}-1)(t^{b}-1)} a\geq b\geq 0$$=a(-b)t^{-b} \cdot\frac{(t-1)^{2}}{(t^{a}-1)(t^{-b}-1)} b<0\leq a$$=(-a)(-b)t^{-a-b} \cdot\frac{(t-1)^{2}}{(t^{-a}-1)(t^{-b}-1)} b\leq a<0$$|b|\leq 2$ if
By the remark before Lemma 1.1, we haveis operator monotone.
We can consider $\gamma=0,$ $\alpha=(1,1)$ , andCase (1) $0<b\leq a\leq 2$ :$\beta=(a, b)$ . Since$|a|,$$h$$\lim_{rarrow\infty}\arg h(re^{\pi i})=\lim_{rarrow\infty}\arg abr^{2-a-b}\cdot\frac{(e^{\pi i}-1/r)^{2}}{(e^{a\pi i}-1/r^{a})(e^{b\pi i}-1/r^{b})}=(2-a-b)\pi$

and $G_{0}(\alpha, \beta)\leq 2-a-b\leq F_{0}(\alpha, \beta)$ , it follows that, by Lemma 1.1,
$1\leq a+b\leq 2$

if

$h$

is operator monotone.

41
When $0<b\leq a\leq 1$ , we have$F(\alpha_{\}}\beta)=F(1, a)+F(1, b)=(1-a)+(1-b)=2-a-b,$$-F(\beta, \alpha)=-(F(a, 1)+F(b, 1))=-(0+0)=0.$

By Lemma 1.2, $\alpha+b\geq 1$ and $0<b\leq a\leq 1$ implies that is operator
monotone.
Case (2) $-2\leq b\leq 0<a\leq 2$ : We can consider $\gamma=-b,$ $\alpha=(1,1)$ , and$\beta=(a, -b)$ . Since$h$$\lim_{rarrow\infty}\arg h(re^{\pi i})=\lim_{rarrow\infty}\arg a(-b)r^{2-a}e^{-b\pi i}\cdot\frac{(e^{\pi i}-1/r)^{2}}{(e^{a\pi i}-1/r^{a})(e^{-b\pi i}-1/r^{-b})}$$=(-b+2-a-(-b))\pi=(2-a)\pi$
and$\lim_{rarrow 0+}\arg h(re^{7fi})=\lim_{rarrow 0+}\arg a(-b)r^{-b}e^{-b\prime\kappa i}\cdot\frac{(re^{\pi i}-1)^{2}}{(r^{a}e^{a\pi i}-1)(r^{-b}e^{-b\prime\kappa i}-1)}$$=-b7r,$we have $1\leq a\leq 2$ and $-1\leq b\leq 0$ if is operator monotone.
When $1\leq a\leq 2$ and $-1\leq b\leq 0$ , we have$h$$-b+F(\alpha, \beta)=-b+F(1, a)+F(1, -b)=-b+0+(1-(-b))=1,$$-b+G(\alpha, \beta)=-b-F(a, 1)-F(-b, 1)=-b-(a-1)-0=1-(a+b)$ .

By Lemma 1.2, $a+b\geq 1,$ $1\leq a\leq 2$ , and $-1\leq b\leq 0$ implies that is
operator monotone.
Case (3) $-2\leq a,$ $b<0$ : We can consider $\gamma=-a-b,$ $\alpha=(1,1)$ , and$\beta=(-a, -b)$ . Since$h$$\lim_{rarrow\infty}\arg h(re^{\pi i})=\lim_{rarrow\infty}\arg abr^{2}e^{(-a-b)\pi i}\cdot\frac{(e^{\pi i}-1/r)^{2}}{(e^{-a\pi i}-1/r^{-(l})(e^{-b\pi i}-1/r^{-b})}=2\pi,$$h$is not operator monotone.
So we have that is contained in$\Omega$$\{(a, b):-1\leq a\leq 0, g_{1}(a)\leq b\leq 2\}\cup\{(a, b):0\leq a\leq 2, g_{1}(a)\leq b\leq 2-a\}$is operator monotone if the point $(a, b)$ is contained in the following
three triangles:

and

$h$

42

By the numerical computation of, we can replace theandabove figure to the figure in the statement of Theorem 3.1.
We consider the function. By the symmetry of and , we only
$1\leqa\leq
2$
consider the caseand $1-a\leq b\leq 2-a$ . We remark that $f(t)$is operator monotone on
if and only if $\Im f(re^{\pi i})\geq 0$ for all $r\geq 0$ by
Lemma 1.1 (1). Since$F_{0}(\alpha, \beta)$$G_{0}(\alpha, \beta)$$g_{2}(a)$

$b$

$a$

$(0, \infty)$$\Im f(re^{\pi i})$$= \frac{(r+1)^{2}}{|(r^{a}e^{a\pi i}-1)(r^{b}r^{b\pi i}-1)|^{2}}\Im ab(r^{a}e^{-a\pi i}-1)(r^{-b\pi i}-1)$$= \frac{(r+1)^{2}r^{b}}{|(r^{a}e^{a\pi i}-1)(r^{b}r^{b\pi i}-1)|^{2}}ab(-r^{a}\sin(a+b)\pi+\sin b\pi+r^{a-b}\sin a\pi)$the signature of$\Im f(re^{\pi i})$,is equal to that of$k(r)=ab(-r^{a}\sin(a+b)\pi+\sin b\pi+r^{a-b}\sin a\pi)$for all $r>0$ . We can see that the solution$r_{0}$

of $k’(r)=0$ is$( \frac{(a-b)\sin a\pi}{a\sin(a+b)\pi})^{1/b},$

and $k(r)$ is decreasing on $(0, r_{0})$ and increasing on. So we have $f(t)$is operator monotone on
if and only if $k(r_{0})\geq 0$ . As a relation ofand satisfying with $k(r_{0})=0$ , we can get the following:$(r_{0}, \infty)$$(0, \infty)$

$a$

$b$

$( \frac{a-b}{a}\frac{\sin a\pi}{\sin(a+b)\pi})^{a}=(\frac{b-}{b}a\frac{\sin b\pi}{\sin(b+a)\pi})^{b}.$So we can get the desired relation.

$\square$

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by Mathematica.
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43
;

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References[1] R. Bhatia, Matrix Analysis, Springer, New York, 1997.

[2] R. Bhatia, Positive Definite Matrices, Princeton University Press, 2007.

44[3] W. F. Donoghue, Jr Monotone matrix functions and analytic continuation, Springer-Verlag, 1974.
[4] H. Hasegawa and D. Petz, On the Riemannian metric of a-entropies ofdensity matrices, Lett. Math. Phys., 38(1996), 221-225.

[5] H. Hasegawa and D. Petz, Non-commutative extension of the information geometw II, In O. Hirota, editor, Quantum Communication andMeasurement, pages 109-118. Plenum, New York, 1997.
[6] F. Hiai, Matrix Analysis: Matrix monotone functions, matrix means,
and majorization, Interdecip. Inform. Sci. 16 (2010), 139-248.
[7] F. Hiai and D. Petz, introduction to matrix analysis and applications,
Springer, 2014.

[8] A. N. Imam, M. Nagisa and H. Watanabe, Some classes of operator
monotone functions, in preparation.
[9] M. Nagisa and S. Wada, 0perator monotonicity of some functions,Linear Algebra Appl. 486(2015), 389-408.
Department of Mathematics and Informatics
Graduate School of Science
Chiba University
Chiba 263-8522

JAPAN
$E$

-mail address: nagisa@math.s.chiba-u.ac.jp
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