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## Linear Algebra and its Applications

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## SOC-monotone and SOC-convex functions vs.

## matrix-monotone and matrix-convex functions<

### Shaohua Pan

^{a}

### , Yungyen Chiang

^{b}

### , Jein-Shan Chen

^{c,}

^{∗}

^{,1,2}

a*Department of Mathematics, South China University of Technology, Wushan Road 381, Tianhe District, Guangzhou 510641, China*
b*Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan*

c*Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan*

A R T I C L E I N F O A B S T R A C T

*Article history:*

Received 30 September 2010
Accepted 9 April 2012
Available online 12 May 2012
Submitted by R.A. Brualdi
*AMS classification:*

26B05 26B35 90C33 65K05

*Keywords:*

Hilbert space Second-order cone SOC-monotonicity SOC-convexity

The SOC-monotone function (respectively, SOC-convex function) is a scalar valued function that induces a map to preserve the monotone order (respectively, the convex order), when imposed on the spectral factorization of vectors associated with second-order cones (SOCs) in general Hilbert spaces. In this paper, we provide the sufficient and necessary characterizations for the two classes of functions, and particularly establish that the set of continuous SOC-monotone (re- spectively, SOC-convex) functions coincides with that of continuous matrix monotone (respectively, matrix convex) functions of order 2.

© 2012 Elsevier Inc. All rights reserved.

**1. Introduction**

Let

### H

be a real Hilbert space of dimension dim*(H) ≥*

3 endowed with an inner product*·, ·*

and its induced norm

### ·

*. Fix a unit vector e*

### ∈ H

and denote by

^{e}^{⊥}the orthogonal complementary

< This work was supported by National Young Natural Science Foundation (No. 10901058) and the Fundamental Research Funds for the Central Universities.

∗ Corresponding author.

*E-mail addresses:*shhpan@scut.edu.cn(S. Pan),chiangyy@math.nsysu.edu.tw(Y. Chiang),jschen@math.ntnu.edu.tw(J.-S. Chen).

1 The author’s work is supported by National Science Council of Taiwan, Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan.

2 Also a Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office.

0024-3795/$ - see front matter © 2012 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.laa.2012.04.030

*space of e, i.e.,*

^{e}^{⊥}

### = {

^{x}### ∈ H |

^{x}*,*

^{e}### =

^{0}

*} .*

*Then each x can be written as*

*x*

### =

*x*

*e*

### +

*x*0

*e for some x*

*e*

### ∈

*e*

^{⊥}

*and x*0

*∈ R.*

The second-order cone (SOC) in

### H

, also called the Lorentz cone, is a set defined by*K*

### :=

*x*

### ∈ H |

^{x}*,*

^{e}### ≥ √

^{1}2

^{x}

### =

^{}

^{x}*e*

### +

*0*

^{x}*e*

### ∈ H |

*0*

^{x}### ≥

^{x}*e*

^{}

*.*

From [7, Section 2], we know that K is a pointed closed convex self-dual cone. Hence,

### H

^{becomes a}partially ordered space via the relation

*K*

*. In the sequel, for any x*

*,*

*y*

### ∈ H

*, we always write x*

*K*

*y*

*(respectively, x*

*K*

*y) when x*

### −

^{y}### ∈

*K (respectively, x*

### −

^{y}### ∈

**intK); and denote x***by the vector*

_{e}_{}

^{x}

_{x}

^{e}*e*
*if x*_{e}

### =

0, and otherwise by any unit vector from

^{e}^{⊥}

^{.}

*Associated with the second-order cone K, each x*

### =

^{x}*e*

### +

*0*

^{x}*e*

### ∈H

can be decomposed as*x*

*= λ*

1*(*

^{x}*)*

*1*

^{u}*(*

^{x}*) + λ*

2*(*

^{x}*)*

*2*

^{u}*(*

^{x}*),*

^{(1)}

where

*λ*

*i*

*(*

*x*

*) ∈ R*

*and u*

*i*

*(*

*x*

*) ∈ H*

*for i*

### =

1*,*

2 are the spectral values and the associated spectral vectors
*of x, defined by*

*λ*

*i*

*(*

^{x}*) =*

*0*

^{x}*+ (−*

^{1}

*)*

^{i}

^{x}*e*

*,*

^{u}*i*

*(*

^{x}*) =*

^{1}2

*e*

*+ (−*

^{1}

*)*

^{i}

^{x}*e*

*.*

^{(2)}

*Clearly, when x**e*

### =

*0, the spectral factorization of x is unique by definition.*

*Let f*

### :

^{J}### ⊆ R → R

*be a scalar valued function, where J is an interval (finite or infinite, closed or*open) in

### R

*. Let S be the set of all x*

### ∈ H

whose spectral values*λ*

1*(*

^{x}*)*

^{and}

*λ*

2*(*

^{x}*)*

*belong to J. Unless*

*otherwise stated, in this paper S is always taken in this way. By the spectral factorization of x in (1)*and (2), it is natural to define f

^{soc}

### :

^{S}### ⊆ H → H

^{by}

*f*^{soc}

*(*

^{x}*) :=*

^{f}*(λ*

1*(*

^{x}*))*

*1*

^{u}*(*

^{x}*) +*

^{f}*(λ*

2*(*

^{x}*))*

*2*

^{u}*(*

^{x}*), ∀*

^{x}### ∈

^{S}*.*

^{(3)}

*It is easy to see that the function f*

^{soc}

*is well defined whether x*

_{e}### =

0 or not. For example, by taking*f*

*(*

*t*

*) =*

*t*

^{2}

*, we have that f*

^{soc}

*(*

*x*

*) =*

*x*

^{2}

### =

*x*

### ◦

*x, where “*

### ◦

” means the Jordan product and the detailed definition is see in the next section. Note that*(λ*

1*(*

^{x}*) − λ*

1*(*

^{y}*))*

^{2}

*+ (λ*

2*(*

^{x}*) − λ*

2*(*

^{y}*))*

^{2}

### =

^{2}

*(*

^{x}^{2}

### +

^{y}^{2}

### −

*0*

^{2x}*y*0

### −

^{2}

^{x}*e*

^{y}*e*

*)*

### ≤

^{2}

^{}

^{x}^{2}

### +

^{y}^{2}

### −

^{2}

^{x}*,*

^{y}^{ }

### =

^{2}

^{x}### −

^{y}^{2}

*.*

*We may verify that the domain S of f*^{soc}is open in

### H

*if and only if J is open in*

### R

*. Also, S is always*

*convex since, for any x*

### =

^{x}*e*

### +

*0*

^{x}*e*

*,*

^{y}### =

^{y}*e*

### +

*0*

^{y}*e*

### ∈

^{S and}*β ∈ [*

^{0}

*,*

^{1}

### ]

^{,}

*λ*

1[*β*

^{x}*+ (*

^{1}

*− β)*

^{y]}### =

^{}

*β*

*0*

^{x}*+ (*

^{1}

*− β)*

*0*

^{y}*− β*

^{x}*e*

*+ (*

^{1}

*− β)*

^{y}*e*

### ≥

^{min}

*{λ*

1*(*

^{x}*), λ*

1*(*

^{y}*)},* *λ*

2[*β*

^{x}*+ (*

^{1}

*− β)*

^{y]}### =

^{}

*β*

*0*

^{x}*+ (*

^{1}

*− β)*

*0*

^{y}*+ β*

^{x}*e*

*+ (*

^{1}

*− β)*

^{y}*e*

### ≤

^{max}

*{λ*

2*(*

^{x}*), λ*

2*(*

^{y}*)},*

which implies that*β*

^{x}*+ (*

^{1}

*− β)*

^{y}### ∈

^{S. Thus, f}^{soc}

*(β*

^{x}*+ (*

^{1}

*− β)*

^{y}*)*

is well defined.
In this paper we are interested in two classes of special scalar valued functions that induce the maps via (3) to preserve the monotone order and the convex order, respectively.

**Definition 1.1. A function f**

### :

^{J}### → R

*is said to be SOC-monotone if for any x*

*,*

^{y}### ∈

^{S,}*x*

*K*

*y*

### ⇒

^{f}^{soc}

*(*

^{x}*) *

*K*

*f*

^{soc}

*(*

^{y}*);*

^{(4)}

*and f is said to be SOC-convex if, for any x*

*,*

^{y}### ∈

^{S and any}*β ∈ [*

^{0}

*,*

^{1}

### ]

^{,}

*f*^{soc}

*(β*

^{x}*+ (*

^{1}

*− β)*

^{y}*) *

*K*

*β*

^{f}^{soc}

*(*

^{x}*) + (*

^{1}

*− β)*

^{f}^{soc}

*(*

^{y}*).*

^{(5)}From Definition1.1and Eq. (3), it is easy to see that the set of SOC-monotone and SOC-convex functions are closed under positive linear combinations and pointwise limits.

The concept of SOC-monotone (respectively, SOC-convex) functions above is a direct extension of those given by [5,6] to general Hilbert spaces, and is analogous to that of matrix monotone (respec- tively, matrix convex) functions and more general operator monotone (respectively, operator convex) functions; see, e.g., [17,15,14,2,11,23]. Just as the importance of matrix monotone (respectively, matrix convex) functions to the solution of convex semidefinite programming [19,4], SOC-monotone (respec- tively, SOC-convex) functions also play a crucial role in the design and analysis of algorithms for convex second-order cone programming [3,22]. For matrix monotone and matrix convex functions, after the seminal work of Löwner [17] and Kraus [15], there have been systematic studies and perfect char- acterizations for them; see [8,16,4,13,12,21,20] and the references therein. However, the study on SOC-monotone and SOC-convex functions just begins with [5], and the characterizations for them are still imperfect. Particularly, it is not clear what is the relation between the SOC-monotone (respectively, SOC-convex) functions and the matrix monotone (respectively, matrix convex) functions.

In this work, we provide the sufficient and necessary characterizations for SOC-monotone and SOC- convex functions in the setting of Hilbert spaces, and show that the set of continuous SOC-monotone (SOC-convex) functions coincides with that of continuous matrix monotone (matrix convex) functions of order 2. Some of these results generalize those of [5,6] (see Propositions3.2and4.2), and some are new, which are difficult to achieve by using the techniques of [5,6] (see, for example, Proposition 4.4). In addition, we also discuss the relations between SOC-monotone functions and SOC-convex functions, verify Conjecture 4.2 in [5] under a little stronger condition (see Proposition6.2), and present a counterexample to show that Conjecture 4.1 in [5] generally does not hold. It is worthwhile to point out that the analysis in this paper depends only on the inner product of Hilbert spaces, whereas most of the results in [5,6] are obtained with the help of matrix operations.

*Throughout this paper, all differentiability means Fréchet differentiability. If F*

### : H → H

^{is (twice)}

*differentiable at x*

### ∈ H

*, we denote by F*

^{}

*(*

^{x}*)*

^{(F}^{}

*(*

^{x}*)*

) the first-order F-derivative (the second-order
*F-derivative) of F at x. In addition, we use C*

^{n}*(*

*J*

*)*

*and C*

^{∞}

*(*

*J*

*)*

*to denote the set of n times and infinite*

*times continuously differentiable real functions on J, respectively. When f*

### ∈

^{C}^{1}

*(*

^{J}*)*

*, we denote by f*

^{[}

^{1}

^{]}

*the function on J*

### ×

*J defined by*

*f*^{[}^{1}^{]}

*(λ, μ) :=*

⎧⎨

⎩

*f**(λ)−*^{f}*(μ)*

*λ−μ* if

*λ = μ,*

*f*

^{}

*(λ)*

^{if}

*λ = μ;*

*and when f*

### ∈

^{C}^{2}

*(*

^{J}*)*

*, denote by f*

^{[}

^{2}

^{]}

*the function on J*

### ×

^{J}### ×

*J defined by*

*f*

^{[}

^{2}

^{]}

*(τ*

1*, τ*

2*, τ*

3*) :=*

^{f}^{[}

^{1}

^{]}

*(τ*

1*, τ*

2*) −*

^{f}^{[}

^{1}

^{]}

*(τ*

1*, τ*

3*)*

*τ*

2*− τ*

3
if

*τ*

1*, τ*

2*, τ*

3are distinct, and for other values of*τ*

1*, τ*

2*, τ*

3*, f*

^{[}

^{2}

^{]}is defined by continuity; e.g.,

*f*

^{[}

^{2}

^{]}

*(τ*

1*, τ*

1*, τ*

3*) =*

^{f}*(τ*

3*) −*

*f*

*(τ*

1*) −*

*f*

^{}

*(τ*

1*)(τ*

3*− τ*

1*)*

*(τ*

3*− τ*

1*)*

^{2}

*,*

^{f}^{[}

^{2}

^{]}

*(τ*

1*, τ*

1*, τ*

1*) =*

^{1}2

*f*

^{}

*(τ*

1*).*

For a linear operator*L from*

### H

^{into}

### H

^{, we write}

*L*

### ≥

0 (respectively,*L*

*>*

0) to mean that*L is positive*semidefinite (respectively, positive definite), i.e.,

*h*

*,*

*Lh*

### ≥

*0 for any h*

### ∈ H

(respectively,*h*

*,*

*Lh*

* >*

0
for any 0### =

^{h}### ∈ H

^{).}

**2. Preliminaries**

This section recalls some background material and gives several lemmas that will be used in the subsequent sections. We start with the definition of Jordan product [9]. For any x

### =

*x*

*e*

### +

*x*0

*e*

*,*

*y*

### =

*y*

_{e}### +

*0*

^{y}*e*

### ∈ H

*, the Jordan product of x and y is defined as*

*x*

### ◦

*y*

*:= (*

*x*0

*y*

*e*

### +

*y*0

*x*

*e*

*) + *

*x*

*,*

*y*

*e*

*.*

*A simple computation can verify that for any x*

*,*

^{y}*,*

^{z}### ∈ H

*and the unit vector e, (i) e*

### ◦

^{e}### =

^{e and e}### ◦

^{x}### =

^{x;}*(ii) x*

### ◦

*y*

### =

*y*

### ◦

*x; (iii) x*

*◦ (*

*x*

^{2}

### ◦

*y*

*) =*

*x*

^{2}

*◦ (*

*x*

### ◦

*y*

*)*

*, where x*

^{2}

### =

*x*

### ◦

*x; (iv)*

*(*

*x*

### +

*y*

*) ◦*

*z*

### =

*x*

### ◦

*z*

### +

*y*

### ◦

*z. For*

*any x*

### ∈ H

, define its determinant bydet

*(*

^{x}*) := λ*

1*(*

^{x}*)λ*

2*(*

^{x}*) =*

^{x}_{0}

^{2}

### −

^{x}*e*

^{2}

*.*

*Then each x*

### =

*x*

*e*

### +

*x*0

*e with det*

*(*

*x*

*) =*

0 is invertible with respect to the Jordan product, i.e., there is
*a unique x*

^{−}

^{1}

*= (−*

^{x}*e*

### +

*0*

^{x}*e*

*)/*

^{det}

*(*

^{x}*)*

*such that x*

### ◦

^{x}^{−}

^{1}

### =

^{e.}We next give several lemmas where Lemma 2.1 is used in Section 3 to characterize SOC- monotonicity, and Lemmas2.2and2.3are used in Section4to characterize SOC-convexity.

**Lemma 2.1. Let**

### B := {

^{z}### ∈

^{e}^{⊥}

### |

^{z}### ≤

^{1}

### }

*. Then, for any given u*

### ∈

^{e}^{⊥}

^{with}

^{u}### =

^{1 and}*θ, λ ∈ R*

^{,}*the following results hold.*

*(a)*

*θ + λ*

^{u}*,*

^{z}### ≥

*0 for any z*

### ∈ B

*if and only if*

*θ ≥ |λ|*

^{.}*(b)*

*θ − λ*

^{z}^{2}

*≥ (θ − λ*

^{2}

*)*

^{u}*,*

^{z}^{2}

^{for any z}### ∈ B

*if and only if*

*θ − λ*

^{2}

### ≥

^{0.}**Proof. (a) Suppose that**

*θ + λ*

^{u}*,*

^{z}### ≥

*0 for any z*

### ∈ B

^{. If}

*λ =*

^{0, then}

*θ ≥ |λ|*

clearly holds. If*λ =*

^{0,}

*take z*

### = −

^{sign}

*(λ)*

^{u. Since}

^{u}### =

*1, we have z*

### ∈ B

, and consequently,*θ + λ*

^{u}*,*

^{z}### ≥

0 reduces to*θ − |λ| ≥*

0. Conversely, if*θ ≥ |λ|*

, then using the Cauchy–Schwartz inequality yields*θ + λ*

*u*

*,*

*z*

### ≥

0*for any z*

### ∈ B

^{.}

(b) Suppose that

*θ − λ*

^{z}^{2}

*≥ (θ − λ*

^{2}

*)*

^{u}*,*

^{z}^{2}

^{for any z}### ∈ B

. Then we must have*θ − λ*

^{2}

### ≥

^{0. If not,}

*for those z*

### ∈ B

with*z*

### =

1 but*u*

*,*

*z*

### =

*u*

*z*, it holds that

*(θ − λ*

^{2}

*)*

^{u}*,*

^{z}^{2}

*> (θ − λ*

^{2}

*)*

^{u}^{2}

^{z}^{2}

*= θ − λ*

^{z}^{2}

*,*

which contradicts the given assumption. Conversely, if

*θ − λ*

^{2}

### ≥

0, the Cauchy–Schwartz inequality implies that*(θ − λ*

^{2}

*)*

^{u}*,*

^{z}^{2}

*≤ θ − λ*

^{z}^{2}

^{for any z}### ∈ B

^{.}

**Lemma 2.2. For any given a**

*,*

^{b}*,*

^{c}### ∈ R

^{and x}### =

^{x}*e*

### +

*0*

^{x}*e with x*

_{e}### =

*0, the inequality*

*a*

^{h}*e*

^{2}

### −

^{h}*e*

*,*

^{x}*e*

^{2}

^{}

### +

^{b}^{}

*0*

^{h}### +

^{x}*e*

*,*

^{h}*e*

^{}

^{2}

### +

^{c}^{}

*0*

^{h}### −

^{x}*e*

*,*

^{h}*e*

^{}

^{2}

### ≥

^{0}

^{(6)}

*holds for all h*

### =

*h*

*e*

### +

*h*0

*e*

### ∈ H

*if and only if a*

### ≥

0*,*

*b*

### ≥

*0 and c*

### ≥

*0.*

**Proof. Suppose that (6) holds for all h**

### =

^{h}*e*

### +

*0*

^{h}*e*

### ∈ H

*. By letting h*

_{e}### =

^{x}*e*

*,*

*0*

^{h}### =

^{1 and h}*e*

### =

### −

*x*

*e*

*,*

*h*0

### =

*1, respectively, we get b*

### ≥

*0 and c*

### ≥

0 from (6). If a### ≥

0 does not hold, then by taking*h*

_{e}### =

^{}

^{b}^{+}

_{|}

^{c}

_{a}^{+}

_{|}

^{1}

_{}

^{z}

_{z}

^{e}

_{e}_{}

^{with}

^{z}*e*

*,*

^{x}*e*

### =

*0*

^{0 and h}### =

1, (6) gives a contradiction### −

^{1}

### ≥

0. Conversely, if*a*

### ≥

^{0}

*,*

^{b}### ≥

^{0 and c}### ≥

0, then (6) clearly holds for all h### ∈ H

^{.}

**Lemma 2.3. Let f**

### ∈

^{C}^{2}

*(*

^{J}*)*

^{and u}*e*

### ∈

^{e}^{⊥}

^{with}

^{u}*e*

### =

*1. For any h*

### =

^{h}*e*

### +

*0*

^{h}*e*

### ∈ H

^{, define}*μ*

1*(*

^{h}*) :=*

^{h}^{0}

### −

*u*

*e*

*,*

*h*

*e*

### √

2*, μ*

2*(*

^{h}*) :=*

^{h}^{0}

### +

*u*

*e*

*,*

*h*

*e*

### √

2*, μ(*

^{h}*) :=*

^{}

^{h}*e*

^{2}

### −

^{u}*e*

*,*

^{h}*e*

^{2}

*.*

*Then, for any given a*

*,*

^{d}### ∈ R

^{and}*λ*

1*, λ*

2### ∈

*J, the following inequality*

*4f*

^{}

*(λ*

1*)*

^{f}^{}

*(λ*

2*)μ*

1*(*

^{h}*)*

^{2}

*μ*

2*(*

^{h}*)*

^{2}

### +

^{2}

*(*

^{a}### −

^{d}*)*

^{f}^{}

*(λ*

2*)μ*

2*(*

^{h}*)*

^{2}

*μ(*

^{h}*)*

^{2}

### +

^{2}

*(*

^{a}### +

^{d}*)*

^{f}^{}

*(λ*

1*)μ*

1*(*

^{h}*)*

^{2}

*μ(*

^{h}*)*

^{2}

### +

^{}

^{a}^{2}

### −

^{d}^{2}

^{ }

*μ(*

^{h}*)*

^{4}

### −

^{2 [}

*(*

^{a}### −

^{d}*) μ*

1*(*

^{h}*) + (*

^{a}### +

^{d}*) μ*

2*(*

^{h}*)*

^{]}

^{2}

*μ(*

^{h}*)*

^{2}

### ≥

^{0}

^{(7)}

*holds for all h*

### =

*h*

*e*

### +

*h*0

*e*

### ∈ H

*if and only if*

*a*^{2}

### −

^{d}^{2}

### ≥

^{0}

*,*

^{f}^{}

*(λ*

2*)(*

^{a}### −

^{d}*) ≥ (*

^{a}### +

^{d}*)*

^{2}

^{and f}^{}

*(λ*

1*)(*

^{a}### +

^{d}*) ≥ (*

^{a}### −

^{d}*)*

^{2}

*.*

^{(8)}

**Proof. Suppose that (7) holds for all h**

### =

^{h}*e*

### +

*0*

^{h}*e*

### ∈ H

*0*

^{. Taking h}### =

^{0 and h}*e*

### =

^{0 with}

^{h}*e*

*,*

^{u}*e*

### =

^{0,}we have

*μ*

1*(*

*h*

*) =*

0*, μ*

2*(*

*h*

*) =*

0 and*μ(*

*h*

*) = *

*h*

*e*

* >*

0, and then (7) gives a^{2}

### −

*d*

^{2}

### ≥

0. Taking*h*

_{e}### =

0 such that### |

^{u}*e*

*,*

^{h}*e*

*| < *

^{h}*e*

*0*

^{and h}### =

^{u}*e*

*,*

^{h}*e*

### =

^{0, we have}

*μ*

1*(*

^{h}*) =*

^{0}

*, μ*

2*(*

^{h}*) =* √

*2h*_{0}and

*μ(*

^{h}*) >*

0, and then (7) reduces to the following inequality
4

*(*

^{a}### −

^{d}*)*

^{f}^{}

*(λ*

2*) − (*

^{a}### +

^{d}*)*

^{2}

^{}

^{h}^{2}0

*+ (*

^{a}^{2}

### −

^{d}^{2}

*)(*

^{h}*e*

^{2}

### −

^{h}^{2}0

*) ≥*

^{0}

*.*

This implies that

*(*

^{a}### −

^{d}*)*

^{f}^{}

*(λ*

2*) − (*

^{a}### +

^{d}*)*

^{2}

### ≥

*0. If not, by letting h*

_{0}be sufficiently close to

^{h}*e*

^{, the}

*last inequality yields a contradiction. Similarly, taking h with h*

*e*

### =

0 satisfying### |

^{u}*e*

*,*

^{h}*e*

*| < *

^{h}*e*

^{and}

*h*

_{0}

### = −

^{u}*e*

*,*

^{h}*e*

^{, we get f}^{}

*(λ*

1*)(*

^{a}### +

^{d}*) ≥ (*

^{a}### −

^{d}*)*

^{2}

^{from (7).}

Next, suppose that (8) holds. Then, the inequalities f^{}

*(λ*

2*)(*

^{a}### −

^{d}*) ≥ (*

^{a}### +

^{d}*)*

^{2}

^{and f}^{}

*(λ*

1*)(*

^{a}### +

^{d}*) ≥* *(*

^{a}### −

^{d}*)*

^{2}imply that the left-hand side of (7) is greater than

*4f*^{}

*(λ*

1*)*

^{f}^{}

*(λ*

2*)μ*

1*(*

^{h}*)*

^{2}

*μ*

2*(*

^{h}*)*

^{2}

### −

^{4}

*(*

^{a}^{2}

### −

^{d}^{2}

*)μ*

1*(*

^{h}*)μ*

2*(*

^{h}*)μ(*

^{h}*)*

^{2}

### +

^{}

^{a}^{2}

### −

^{d}^{2}

^{ }

*μ(*

^{h}*)*

^{4}

*,*

which is obviously nonnegative if

*μ*

1*(*

^{h}*)μ*

2*(*

^{h}*) ≤*

0. Now assume that*μ*

1*(*

^{h}*)μ*

2*(*

^{h}*) >*

^{0. If a}^{2}

### −

^{d}^{2}

### =

^{0,}

*then the last expression is clearly nonnegative, and if a*

^{2}

### −

^{d}^{2}

*>*

0, then the last two inequalities in
(8) imply that f^{}

*(λ*

1*)*

*f*

^{}

*(λ*

2*) ≥ (*

*a*

^{2}

### −

*d*

^{2}

*) >*

0*,*

and therefore,
*4f*^{}

*(λ*

1*)*

^{f}^{}

*(λ*

2*)μ*

1*(*

^{h}*)*

^{2}

*μ*

2*(*

^{h}*)*

^{2}

### −

^{4}

*(*

^{a}^{2}

### −

^{d}^{2}

*)μ*

1*(*

^{h}*)μ*

2*(*

^{h}*)μ(*

^{h}*)*

^{2}

### +

^{}

^{a}^{2}

### −

^{d}^{2}

^{ }

*μ(*

^{h}*)*

^{4}

### ≥

^{4}

*(*

^{a}^{2}

### −

^{d}^{2}

*)μ*

1*(*

^{h}*)*

^{2}

*μ*

2*(*

^{h}*)*

^{2}

### −

^{4}

*(*

^{a}^{2}

### −

^{d}^{2}

*)μ*

1*(*

^{h}*)μ*

2*(*

^{h}*)μ(*

^{h}*)*

^{2}

### +

^{}

^{a}^{2}

### −

^{d}^{2}

^{ }

*μ(*

^{h}*)*

^{4}

*= (*

^{a}^{2}

### −

^{d}^{2}

*)*

^{ }

^{2}

*μ*

1*(*

^{h}*)μ*

2*(*

^{h}*) − μ(*

^{h}*)*

^{2}

^{}

^{2}

### ≥

^{0}

*.*

Thus, we prove that inequality (7) holds. The proof is complete.

To close this section, we introduce the regularization of a locally integrable real function. Let

*ϕ*

^{be a}

*real function of class C*

^{∞}with the following properties:

*ϕ ≥*

^{0,}

*ϕ*

is even, the support supp*ϕ = [−*

^{1}

*,*

^{1}

### ]

^{,}and

R

*ϕ =*

1. For each*ε >*

^{0, let}

*ϕ*

_{ε}*(*

^{t}*) =*

^{1}

_{ε}*ϕ(*

_{ε}

^{t}*)*

. Then supp*ϕ*

_{ε}*= [−ε, ε]*

^{and}

*ϕ*

*has all the properties of*

_{ε}*ϕ*

*listed above. If f is a locally integrable real function, we define its regularization of order*

*ε*

^{as the}function

*f _{ε}*

*(*

^{s}*) :=*

^{}

^{f}*(*

^{s}### −

^{t}*)ϕ*

_{ε}*(*

^{t}*)*

^{dt}### =

^{}

^{f}*(*

^{s}*− ε*

^{t}*)ϕ(*

^{t}*)*

^{dt}*.*

^{(9)}

*Note that f*

_{ε}is a C^{∞}function for each

*ε >*

^{0, and lim}

*ε→*0

*f*

_{ε}*(*

^{x}*) =*

^{f}*(*

^{x}*)*

*if f is continuous.*

**3. Characterizations of SOC-monotone functions**

In this section we present some characterizations for SOC-monotone functions, by which the set of continuous SOC-monotone functions is shown to coincide with that of continuous matrix monotone functions of order 2. To this end, we need the following technical lemma.

**Lemma 3.1. For any given f**

### :

*J*

### → R

*with J open, let f*

^{soc}

### :

*S*

### → H

*be defined by (3).*

*(a) f*^{soc}*is continuous on S if and only if f is continuous on J.*

*(b) f*^{soc}*is (continuously) differentiable on S iff f is (continuously) differentiable on J. Also, when f is*
*differentiable on J, for any x*

### =

^{x}*e*

### +

*0*

^{x}*e*

### ∈

^{S and v}### =

^{v}*e*

### +

*0*

^{v}*e*

### ∈ H

^{,}*(*

^{f}^{soc}

*)*

^{}

*(*

^{x}*)*

^{v}### =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*f*^{}

*(*

*x*0

*)*

*v*

*if x*

*e*

### =

0### ;

*(*

*1*

^{b}*(*

^{x}*) −*

*0*

^{a}*(*

^{x}*))*

^{x}*e*

*,*

^{v}*e*

^{x}*e*

### +

*1*

^{c}*(*

^{x}*)*

*0*

^{v}*x*

_{e}### +

*a*0

*(*

*x*

*)*

*v*

*e*

### +

*b*1

*(*

*x*

*)*

*v*0

*e*

### +

*c*1

*(*

*x*

*)*

*x*

*e*

*,*

*v*

*e*

*e if x*

*e*

### =

0*,*

(10)

*where a*_{0}

*(*

^{x}*) =*

^{f}

^{(λ}_{λ}^{2}

^{(}_{2}

^{x}

_{(}^{))−}

_{x}

_{)−λ}

^{f}

^{(λ}_{1}

_{(}^{1}

_{x}

^{(}_{)}

^{x}*1*

^{))}^{, b}*(*

^{x}*) =*

^{f}^{}

^{(λ}^{2}

^{(}

^{x}

^{))+}_{2}

^{f}^{}

^{(λ}^{1}

^{(}

^{x}*1*

^{))}^{, c}*(*

^{x}*) =*

^{f}^{}

^{(λ}^{2}

^{(}

^{x}

^{))−}_{2}

^{f}^{}

^{(λ}^{1}

^{(}

^{x}

^{))}^{.}*(c) If f is differentiable on J, then for any given x*

### ∈

*S and all v*

### ∈ H

^{,}*(*

*f*

^{soc}

*)*

^{}

*(*

*x*

*)*

*e*

*= (*

*f*

^{}

*)*

^{soc}

*(*

*x*

*)*

and *e*

*, (*

*f*

^{soc}

*)*

^{}

*(*

*x*

*)*

*v*

### =

^{}

*v*

*, (*

*f*

^{}

*)*

^{soc}

*(*

*x*

*)*

^{}

*.*

*(d) If f*

^{}

*is nonnegative (respectively, positive) on J, then for each x*

### ∈

*S,*

*(*

^{f}^{soc}

*)*

^{}

*(*

^{x}*) ≥*

^{0}

*(*

respectively*, (*

^{f}^{soc}

*)*

^{}

*(*

^{x}*) >*

^{0}

*).*

**Proof. (a) Suppose that f**^{soc}is continuous. Let*be the set composed of those x*

### =

^{te with t}### ∈

^{J.}Clearly,

### ⊆

^{S, and f}^{soc}is continuous on

*. Noting that f*

^{soc}

*(*

^{x}*) =*

^{f}*(*

^{t}*)*

*e for any x*

### ∈

, it follows that*f is continuous on J. Conversely, if f is continuous on J, then f*

^{soc}

*is continuous at any x*

### =

^{x}*e*

### +

*0*

^{x}*e*

### ∈

^{S}*with x*

_{e}### =

^{0 since}

*λ*

*i*

*(*

^{x}*)*

^{and u}*i*

*(*

^{x}*)*

^{for i}### =

^{1}

*,*

*2 are continuous at such points. Next, let x*

### =

^{x}*e*

### +

*0*

^{x}*e*

*be an arbitrary element from S with x*

_{e}### =

*0, and we prove that f*

^{soc}

*is continuous at x. Indeed, for any*

*z*

### =

^{z}*e*

### +

*0*

^{z}*e*

### ∈

*S sufficiently close to x, it is not hard to verify that*

*f*

^{soc}

*(*

*z*

*) −*

*f*

^{soc}

*(*

*x*

*) ≤* |

^{f}*(λ*

2*(*

^{z}*)) −*

^{f}*(*

*0*

^{x}*)|*

2

### + |

^{f}*(λ*

1*(*

^{z}*)) −*

^{f}*(*

*0*

^{x}*)|*

2

### + |

^{f}*(λ*

2*(*

^{z}*)) −*

^{f}*(λ*

1*(*

^{z}*))|*

2

*.*

*Since f is continuous on J, and*

*λ*

1*(*

*z*

*), λ*

2*(*

*z*

*) →*

*x*0

*as z*

### →

*x, it follows that*

*f*

*(λ*

1*(*

^{z}*)) →*

^{f}*(*

*0*

^{x}*)*

^{and f}*(λ*

2*(*

^{z}*)) →*

^{f}*(*

*0*

^{x}*)*

^{as z}### →

^{x}*.*

*The last two equations imply that f*^{soc}*is continuous at x.*

*(b) When f*^{soc}is (continuously) differentiable, using the similar arguments as in part (a) can show that
*f is (continuously) differentiable. Next assume that f is differentiable. Fix any x*

### =

*x*

*e*

### +

*x*0

*e*

### ∈

*S. We first*

*consider the case where x*

_{e}### =

^{0. Since}

*λ*

*i*

*(*

^{x}*)*

^{for i}### =

^{1}

*,*

^{2 and}

_{}

^{x}

_{x}

^{e}

_{e}_{}are continuously differentiable at such

*x, it follows that f*

*(λ*

*i*

*(*

^{x}*))*

^{and u}*i*

*(*

^{x}*)*

are differentiable and continuously differentiable, respectively, at
*x. Then f*

^{soc}

*is differentiable at such x by the definition of f*

^{soc}. Also, an elementary computation shows that

*[λ*

*i*

*(*

^{x}*)]*

^{}

^{v}### =

^{v}*,*

^{e}* + (−*

^{1}

*)*

^{i}

^{x}*e*

*,*

^{v}### −

^{v}*,*

^{e}

^{e}

^{x}*e*

### =

*0*

^{v}*+ (−*

^{1}

*)*

^{i}

^{x}*e*

*,*

^{v}*e*

^{x}*e*

### *,*

^{(11)}

*x**e*

^{x}*e*

_{}

*v*

### =

^{v}### −

^{v}*,*

^{e}

^{e}

^{x}*e*

### −

^{x}*e*

*,*

^{v}### −

^{v}*,*

^{e}

^{e}

^{x}*e*

^{x}*e*

^{3}

### =

^{v}

^{e}

^{x}*e*

### −

^{x}*e*

*,*

^{v}*e*

^{x}*e*

^{x}*e*

^{3}

^{(12)}

*for any v*

### =

^{v}*e*

### +

*0*

^{v}*e*

### ∈ H

, and consequently,*[f*

*(λ*

*i*

*(*

^{x}*))*

^{]}

^{}

^{v}### =

^{f}^{}

*(λ*

*i*

*(*

^{x}*))*

*v*_{0}

*+ (−*

^{1}

*)*

^{i}

^{x}*e*

*,*

^{v}*e*

^{x}*e*

*,*

*[u*

_{i}*(*

^{x}*)*

^{]}

^{}

^{v}### =

^{1}

2

*(−*

^{1}

*)*

^{i} *v**e*

^{x}*e*

### −

*x*

*e*

*,*

*v*

*e*

*x*

*e*

^{x}*e*

^{3}

*.*

*Together with the definition of f*^{soc}, we calculate that

*(*

*f*

^{soc}

*)*

^{}

*(*

*x*

*)*

*v is equal to*

*f*

^{}

*(λ*

1*(*

*x*

*))*

2

*v*_{0}

### −

*x*

*e*

*,*

*v*

*e*

*x*

*e*

*e*

### −

^{x}x

^{e}*e*

### −

^{f}*(λ*

1*(*

*x*

*))*

2
*v**e*

*x*

*e*

### −

*x*

*e*

*,*

*v*

*e*

*x*

*e*

*x*

*e*

^{3}

### +

^{f}^{}

*(λ*

2*(*

^{x}*))*

2

*v*0

### +

^{x}*e*

*,*

^{v}*e*

^{x}*e*

*e*

### +

^{x}^{x}

^{e}*e*

### +

^{f}*(λ*

2*(*

^{x}*))*

2
*v*_{e}

^{x}*e*

### −

^{x}*e*

*,*

^{v}*e*

^{x}*e*

^{x}*e*

^{3}

### =

*1*

^{b}*(*

^{x}*)*

*0*

^{v}*e*

### +

*1*

^{c}*(*

^{x}*) *

^{x}*e*

*,*

^{v}*e*

^{e}### +

*1*

^{c}*(*

^{x}*)*

*0*

^{v}*x*

_{e}### +

*1*

^{b}*(*

^{x}*)*

^{x}*e*

*,*

^{v}*e*

^{x}*e*

### +

*0*

^{a}*(*

^{x}*)*

^{v}*e*

### −

*0*

^{a}*(*

^{x}*)*

^{x}*e*

*,*

^{v}*e*

^{x}*e*

*,*

where

*λ*

2*(*

^{x}*) − λ*

1*(*

^{x}*) =*

^{2}

^{x}*e*is used for the last equality. Thus, we get (10) for x

_{e}### =

^{0. We next}

*consider the case where x*

*e*

### =

*0. Under this case, for any v*

### =

^{v}*e*

### +

*0*

^{v}*e*

### ∈ H

^{,}

*f*^{soc}

*(*

^{x}### +

^{v}*) −*

^{f}^{soc}

*(*

^{x}*) =*

^{f}*(*

*0*

^{x}### +

*0*

^{v}### −

^{v}*e*

*)*

2

*(*

^{e}### −

^{v}*e*

*) +*

^{f}*(*

*0*

^{x}### +

*0*

^{v}### +

^{v}*e*

*)*

2

*(*

^{e}### +

^{v}*e*

*) −*

^{f}*(*

*0*

^{x}*)*

^{e}### =

^{f}^{}

*(*

*0*

^{x}*)(*

*0*

^{v}### −

^{v}*e*

*)*

2 *e*

### +

^{f}^{}

*(*

*0*

^{x}*)(*

*0*

^{v}### +

^{v}*e*

*)*

2 *e*

### +

^{f}^{}

*(*

*0*

^{x}*)(*

*0*

^{v}### +

^{v}*e*

*)*

2 *v**e*

### −

^{f}^{}

*(*

*0*

^{x}*)(*

*0*

^{v}### −

^{v}*e*

*)*

2 *v**e*

### +

^{o}*(*

^{v}*)*

### =

*f*

^{}

*(*

*x*0

*)(*

*v*0

*e*

### +

*v*

*e*

*v*

*e*

*) +*

*o*

*(*

*v*

*),*

*where v*_{e}

### =

_{}

^{v}

_{v}

^{e}

_{e}_{}

^{if v}*e*

### =

*0, and otherwise v*

*is an arbitrary unit vector from*

_{e}

^{e}^{⊥}

^{. Hence,}

^{f}^{soc}

*(*

^{x}### +

^{v}*) −*

^{f}^{soc}

*(*

^{x}*) −*

^{f}^{}

*(*

*0*

^{x}*)*

^{v}### =

^{o}*(*

^{v}*).*

*This shows that f*^{soc}*is differentiable at such x with*

*(*

^{f}^{soc}

*)*

^{}

*(*

^{x}*)*

^{v}### =

^{f}^{}

*(*

*0*

^{x}*)*

^{v.}*Assume that f is continuously differentiable. From (10), it is easy to see that*

*(*

^{f}^{soc}

*)*

^{}

*(*

^{x}*)*

is continuous
*at every x with x*

*e*

### =

0. We next argue that*(*

^{f}^{soc}

*)*

^{}

*(*

^{x}*)*

*is continuous at every x with x*

*e*

### =

^{0. Fix any}

*x*

### =

*x*0

*e with x*0

### ∈

*J. For any z*

### =

*z*

*e*

### +

*z*0

*e with z*

*e*

### =

0, we have*(*

^{f}^{soc}

*)*

^{}

*(*

^{z}*)*

^{v}*− (*

^{f}^{soc}

*)*

^{}

*(*

^{x}*)*

^{v}### ≤ |

*1*

^{b}*(*

^{z}*) −*

*0*

^{a}*(*

^{z}*)|*

^{v}*e*

### + |

*1*

^{b}*(*

^{z}*) −*

^{f}^{}

*(*

*0*

^{x}*)||*

*0*

^{v}### |

### +|

*0*

^{a}*(*

^{z}*) −*

^{f}^{}

*(*

*0*

^{x}*)|*

^{v}*e*

### + |

*1*

^{c}*(*

^{z}*)|(|*

*0*

^{v}### | +

^{v}*e*

*).*

^{(13)}

*Since f is continuously differentiable on J and*

*λ*

2*(*

^{z}*) →*

*0*

^{x}*, λ*

1*(*

^{z}*) →*

*0*

^{x}*as z*

### →

^{x, we have}*a*_{0}

*(*

^{z}*) →*

^{f}^{}

*(*

*0*

^{x}*),*

*1*

^{b}*(*

^{z}*) →*

^{f}^{}

*(*

*0*

^{x}*)*

*1*

^{and c}*(*

^{z}*) →*

^{0}

*.*

Together with Eq. (13), we obtain that

*(*

*f*

^{soc}

*)*

^{}

*(*

*z*

*) → (*

*f*

^{soc}

*)*

^{}

*(*

*x*

*)*

*as z*

### →

*x.*

(c) The result is direct by the definition of

*(*

^{f}^{}

*)*

^{soc}and a simple computation from (10).

*(d) Suppose that f*^{}

*(*

^{t}*) ≥*

*0 for all t*

### ∈

*J. Fix any x*

### =

^{x}*e*

### +

*0*

^{x}*e*

### ∈

^{S. If x}*e*

### =

0, the result is direct. It*remains to consider the case x*

*e*

### =

^{0. Since f}^{}

*(*

^{t}*) ≥*

*0 for all t*

### ∈

*J, we have b*1

*(*

^{x}*) ≥*

*1*

^{0, b}*(*

^{x}*) −*

*1*

^{c}*(*

^{x}*) =*

*f*

^{}

*(λ*

1*(*

^{x}*)) ≥*

*1*

^{0, b}*(*

^{x}*) +*

*1*

^{c}*(*

^{x}*) =*

^{f}^{}

*(λ*

2*(*

^{x}*)) ≥*

*0*

^{0 and a}*(*

^{x}*) ≥*

0. From part (b) and the definitions of
*b*

_{1}

*(*

^{x}*)*

*1*

^{and c}*(*

^{x}*)*

*, it follows that for any h*

### =

^{h}*e*

### +

*0*

^{h}*e*

### ∈ H

^{,}

*h*

*, (*

*f*

^{soc}

*)*

^{}

*(*

*x*

*)*

*h*

* = (*

*b*1

*(*

*x*

*) −*

*a*0

*(*

*x*

*))*

*x*

*e*

*,*

*h*

*e*

^{2}

### +

*2c*1

*(*

*x*

*)*

*h*0

*x*

*e*

*,*

*h*

*e*

### +

*b*1

*(*

*x*

*)*

*h*

^{2}

_{0}

### +

*a*0

*(*

*x*

*)*

*h*

*e*

^{2}

### =

*0*

^{a}*(*

^{x}*)*

^{ }

^{h}*e*

^{2}

### −

^{x}*e*

*,*

^{h}*e*

^{2}

^{}

### +

^{1}

2

*(*

*1*

^{b}*(*

^{x}*) −*

*1*

^{c}*(*

^{x}*))*

*0*

^{[h}### −

^{x}*e*

*,*

^{h}*e*

^{]}

^{2}

### +

^{1}

2

*(*

*b*1

*(*

*x*

*) +*

*c*1

*(*

*x*

*))*

*0*

^{[h}### +

*x*

*e*

*,*

*h*

*e*

^{]}

^{2}

### ≥

0*.*

This implies that the operator

*(*

*f*

^{soc}

*)*

^{}

*(*

*x*

*)*

*is positive semidefinite. Particularly, if f*

^{}

*(*

*t*

*) >*

*0 for all t*

### ∈

*J,*we have that

^{h}*, (*

^{f}^{soc}

*)*

^{}

*(*

^{x}*)*

^{h}* >*

*0 for all h*

### =

0. The proof is complete.Lemma3.1(d) shows that the differential operator

*(*

^{f}^{soc}

*)*

^{}

*(*

^{x}*)*

corresponding to a differentiable non-
*decreasing f is positive semidefinite. So, the differential operator*

*(*

*f*

^{soc}

*)*

^{}

*(*

*x*

*)*

associated with a differen-
tiable SOC-monotone function is also positive semidefinite.
**Proposition 3.1. Assume that f**

### ∈

*C*

^{1}

*(*

*J*

*)*

*with J open. Then f is SOC-monotone if and only if*

*(*

*f*

^{soc}

*)*

^{}

*(*

*x*

*)*

*h*

### ∈

*K*

*for any x*

### ∈

^{S and h}### ∈

^{K.}**Proof. If f is SOC-monotone, then for any x**

### ∈

*S, h*

### ∈

*K and t*

*>*

0, we have
*f*

^{soc}

*(*

^{x}### +

^{th}*) −*

^{f}^{soc}

*(*

^{x}*) *

*K*0

*,*

*which, by the continuous differentiability of f*^{soc}*and the closedness of K, implies that*

*(*

*f*

^{soc}

*)*

^{}

*(*

*x*

*)*

*h*

*K*0