AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 836804 10.1155/2012/836804 836804 Research Article A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions Kum Sangho 1 Lim Yongdo 2 Lee Gue 1 Department of Mathematics Education Chungbuk National University Cheongju 361-763 Republic of Korea cnu.ac.kr 2 Department of Mathematics Sungkyunkwan University Suwon 440-746 Republic of Korea skku.edu 2012 30 12 2012 2012 06 11 2012 14 12 2012 2012 Copyright © 2012 Sangho Kum and Yongdo Lim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matrices A and B. Moreover, an interesting generalization of the geometric mean A#B of A and B to convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.

1. Introduction

The notion of geometric means is extended by Ando  to the case of positive semidefinite matrices A and B as the maximum A#B of all X0 for which (AXXB) is positive semidefinite. If A is invertible, then A#B=A1/2(A-1/2BA-1/2)1/2A1/2. The geometric mean A#B appears in the literature with many applications in matrix inequalities, semidefinite programming (scaling point [2, 3]), geometry (geodesic middle [4, 5]), statistical shape analysis (intrinsic mean [6, 7]), and symmetric matrix word equations . The most important property of the geometric mean is that it has a Riccati matrix equation as the defining equation. The geometric mean is the unique positive definite solution of the Riccati matrix equation XA-1X=B.

An interesting generalization of the geometric mean A#B to convex functions was introduced by Atteia and Raïssouli  with a different viewpoint of the convex analysis. The natural idea to make an extension from positive semidefinite matrices to convex functions is nothing but the association of a positive semidefinite matrix A with the quadratic convex function qA(x)=(1/2)Ax,x. Atteia and Raïssouli  provided a general algorithm to construct the (self-dual) geometric mean and the square root of convex functions. As pointed out in , self-dual operators are important in convex analysis and also arise in PDE.

The present work aims at providing a further development of the geometric mean of the convex functions mentioned above. We develop a new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” by exploiting the proximal average of convex functions by Bauschke et al.  and investigate its essential properties such as limiting behaviors, self-duality, and monotonicity with respect to parameters. While doing so, we will see that the geometric mean due to Atteia and Raïssouli  can be interpreted as an element of “the geometric mean of parameterized arithmetic and harmonic means of convex functions” with the particular parameter μ=0.

In fact, this work is motivated by a recent result due to Kim et al.  concerned with a new matrix mean. Actually, the geometric mean of parameterized arithmetic and harmonic means of convex functions is an extension of the new matrix mean to a convex function mean under a standard setting with two convex functions.

2. Geometric Mean and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M19"><mml:mi>𝒜</mml:mi><mml:mo> </mml:mo><mml:mo mathvariant="bold">#</mml:mo><mml:mo> </mml:mo><mml:mi>ℋ</mml:mi></mml:math></inline-formula>-Mean of Parameter <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M20"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:math></inline-formula>

We begin with the algorithm of finding the geometric mean of two proper convex lower semicontinuous functions f and g introduced by Atteia and Raïssouli [11, Proposition 4.4] and some comments on the procedure. Let f,gΓ with domfdomg where Γ denotes the class of proper convex lower semicontinuous functions from the Euclidean space n to (-,+]. Set two sequences of convex functions βn(f,g) and βn*(f,g) recursively: (2.1)β0(f,g)=12(f+g),β0*(f,g)=(12(f*+g*))*,βn+1(f,g)=12(βn(f,g)+βn*(f,g))whereβn*(f,g)=(βn(f*,g*))*, where f* stands for the Fenchel conjugate of f.

It is claimed that all the βn(f,g) and βn*(f,g) do belong to Γ [11, Proposition 4.4]. However, to ensure this property, we need more. Indeed, we see (2.2)β0*(f,g)=(12(f*+g*))*=(12(fg)*)*, where stands for the infimal convolution. As is well known, fg can take - as a value so it may not be proper. This happens for two simple linear functionals f(x)=x and g(x)=-x in the one-dimensional case. So the properness of β0*(f,g) equivalent to that of fg is not safe. Exactly the same problem may occur whenever βn*(f,g) is defined. Moreover, it is not sure that βn+1(f,g) is proper because domβn(f,g)domβn*(f,g) can be empty. Thus the basic necessity that βn(f,g) and βn*(f,g) belong to Γ is not guaranteed under the general assumption only that f,gΓ with domfdomg in . Hence it is necessary to impose a suitable condition to meet this demand. For that purpose, recall that a function fΓ is called cofinite if the recession function f0+ of f satisfies (f0+)(y)=+, for all y0 (see [15, page 116]). Then f is cofinite if and only if domf*=n by means of [15, Corollary 13.3.1]. The terminology “cofinite” is renewed as “coercive” in [16, 3.26 Theorem].

Now we take a look at Atteia and Raïssouli [11, Proposition 4.4] with a refined proof.

Proposition 2.1 (See Atteia and Raïssouli [<xref ref-type="bibr" rid="B11">11</xref>, Proposition 4.4]).

Let domfdomg. If either f or g is cofinite, then all βn(f,g) and βn*(f,g) belong to Γ and βn(f,g) is cofinite for all n0. Hence the geometric mean f#g due to Atteia and Raïssouli , that is, the limit (2.3)f#g=limnβn(f,g), is well defined and proper convex on domβ0(f,g). In particular, it belongs to Γ under the assumption that either domβ0(f,g)=domβ0*(f,g) or domβ0(f,g) is closed. Moreover, f#g=(f*#g*)* under the condition domβ0(f,g)=domβ0*(f,g).

Proof.

Without loss of generality, we may assume that g is cofinite. Clearly, β0(f,g)=(1/2)(f+g)Γ since domβ0(f,g)=domfdomg. In addition, β0(f,g) is still cofinite by [15, Theorem 9.3]. Then β0*(f,g)=((1/2)(f*+g*))*=(1/2)(fg)Γ by virtue of [15, Corollary 9.2.2]. Thus domβ0*(f,g)=(1/2)(domf+domg)domβ0(f,g). By induction, assume that (2.4)βn(f,g),βn*(f,g)Γ,βn(f,g)  iscofinite,domβn(f,g)domβn*(f,g). Then domβn+1(f,g)=domβn(f,g)domβn*(f,g)=domβn(f,g), so βn+1(f,g)Γ. Moreover, βn+1(f,g) is cofinite because βn(f,g) is cofinite. It is readily checked that (2.5)βn+1*(f,g)=(βn+1(f*,g*))*=(12(βn(f,g))*+12(βn*(f,g))*)*. Hence βn+1*(f,g)=(1/2)(βn(f,g)βn*(f,g))Γ. In this case, domβn+1*(f,g)=(1/2)(domβn(f,g)+domβn*(f,g))domβn(f,g)=domβn+1(f,g). Thus we obtain that (2.6)n,domβn(f,g)=domfdomg=domβ0(f,g),n,domβn*(f,g)domβn(f,g)=domβ0(f,g). According to Atteia and Raïssouli [11, Proposition 4.4], we have (2.7)βn+1(f,g)-βn+1*(f,g)12(βn(f,g)-βn*(f,g)),n0;β0*(f,g)·βn*(f,g)βn+1*(f,g)·βn+1(f,g)βn(f,g)·β0(f,g). Hence the geometric mean f#g is well defined and belongs to Γ under the given hypothesis. (If domβ0(f,g) is closed, we define an increasing sequence γn(f,g)Γ by (2.8)γn(f,g)=βn*(f,g)+δC, where δC denotes the indicator function of the closed convex set C=domβ0(f,g). Obviously, f#g is the common limit of βn(f,g) and γn(f,g), hence, belongs to Γ.)

For the equality f#g=(f*#g*)*, we have (2.9)(f*#g*)*(x)=supyn[y,x-(f*#g*)(y)]=supyn[y,x-limnβn(f*,g*)(y)]=supyn[y,x-limnβn*(f*,g*)(y)]=supyn[y,x-limn(βn(f,g))*(y)]  supyn[y,x-(βn(f,g))*(y)],n=(βn(f,g))**(x)=βn(f,g)(x),n. Hence (2.10)(f*#g*)*(x)limnβn(f,g)(x)=(f#g)(x). On the other hand, (2.11)(f*#g*)*(x)=supyn[y,x-(f*#g*)(y)]=supyn[y,x-limnβn(f*,g*)(y)]supyn[y,x-βn(f*,g*)(y)],n=(βn(f*,g*))*(x)=βn*(f,g)(x),  n. Thus (2.12)(f*#g*)*(x)limnβn*(f,g)(x)=(f#g)(x). Therefore we get (2.13)f#g=(f*#g*)*.

Remark 2.2.

( 1 ) The well definedness of f*#g* is readily checked by the assumption g is cofinite. (Without this condition, f*#g* may not be well defined so that the identity f#g=(f*#g*)* breaks down.) With the additional property that domf* is closed, we have f*#g*Γ. Hence (2.14)(f#g)*=f*#g*.(2) Proposition 2.1 provides a sufficient condition to entail the validity of [11, Proposition 4.4]. It is also mentioned in [11, Remark 4.5] that if f and g are finite-valued, domβ0(f,g)=domβ0*(f,g) is satisfied. But even though it is true, β0*(f,g) can be identically - as shown in the case of f(x)=x and g(x)=-x in so that the limiting process using (2.7) may not be available any more. So some restrictions should be imposed to properly define the geometric mean of two convex functions f and gΓ. Of course, for an fΓ, the geometric mean f#f and the convex square root f1/2 of f (see [11, Definition 4.7]) are always well defined because q is cofinite. What is a minimal assumption? That is a question to be answered.

Throughout this paper, we adopt the following modified definition of proximal average for the convenience of presentation. For μ0, with q=(1/2)·2, (2.15)pμ(f,λ)=(λ1(f1+μq)*++λm(fm+μq)*)*-μq, where f=(f1,,fm), g=(g1,,gm), each fi:n(-,+] belongs to Γ, and λi’s are positive real numbers with λ1++λm=1.

From now on, we consider the simple case where m=2, λ1=λ2=1/2, and f,gΓ with domfdomg. Define two sequences of convex functions αn(f,g) and αn(f,g) recursively as follows: (2.16)α0(f,g)=12(f+g),α0(f,g)=pμ(f,g;12,12),αn+1(f,g)=12(αn(f,g)+αn(f,g)),αn+1(f,g)=pμ(αn(f,g),αn(f,g);12,12).

Theorem 2.3.

For μ>0, one has

αn(f,g)Γ and αn(f,g)Γ,  for all n0;

αn(f,g)αn(f,g),αn+1(f,g)αn(f,g) and αn(f,g)αn+1(f,g),  for all n0;

αn+1(f,g)-αn+1(f,g)(1/2)(αn(f,g)-αn(f,g)),  for all n0;

there exists a limit τμ(f,g)=limnαn(f,g) which is a proper convex function with domτμ(f,g)=domfdomg=domα0(f,g). Furthermore, if either domα0(f,g)=domα0(f,g) or domα0(f,g) is closed, τμ(f,g) is the common limit of αn(f,g) and γn(f,g) for some increasing sequence γn(f,g)Γ. In this case, τμ(f,g)Γ.

Proof.

(i) Since α0(f,g)=pμ(f,g;  1/2,1/2), by Bauschke et al. [13, Theorem 4.6], (2.17)domα0(f,g)=12  domf+12  domg12(domfdomg)+12(domfdomg)=domfdomg=domα0(f,g) because domfdomg is a convex set. By induction, assume that domαn(f,g)domαn(f,g). Then (2.18)domαn+1(f,g)=domαn(f,g)domαn(f,g)=domαn(f,g),domαn+1(f,g)=12domαn(f,g)+12  domαn(f,g)domαn(f,g)=domαn+1(f,g). Thus we obtain that (2.19)n,domαn(f,g)=domfdomg=domα0(f,g),n,domαn(f,g)domαn(f,g)=domα0(f,g). This implies that, for all n0, αn(f,g)Γ and αn(f,g)Γ with the help of [13, Corollary 5.2].

(ii) The first assertion αn(f,g)αn(f,g) is a direct consequence of [13, Theorem 5.4]. For the second, by definition and the first assertion, we see (2.20)αn+1(f,g)=12(αn(f,g)+αn(f,g))12(αn(f,g)+αn(f,g))=αn(f,g). For the last, observe that (2.21)αn(f,g)αn+1(f,g)αn(f,g)+μqαn+1(f,g)+μq(αn+1(f,g)+μq)*(αn(f,g)+μq)*12(αn(f,g)+μq)*+12(αn(f,g)+μq)*(αn(f,g)+μq)*(αn(f,g)+μq)*(αn(f,g)+μq)*αn(f,g)+μqαn(f,g)+μqαn(f,g)αn(f,g), which is nothing but the first assertion. Note that all the arithmetics are safe because both (αn(f,g)+μq)* and (αn(f,g)+μq)* are finite-valued.

(iii) By (ii) and the extended arithmetic +(-)=(-)+= (see ), we get (2.22)αn+1(f,g)-αn+1(f,g)12(αn(f,g)+αn(f,g))-αn(f,g)=12(αn(f,g)-αn(f,g)).

(iv) From (ii), we have (2.23)α0(f,g)·αn(f,g)αn+1(f,g)·αn+1(f,g)αn(f,g)·α0(f,g). Hence if xdomα0(f,g)=domfdomg=domαn(f,g) by (2.19), αn(f,g)(x) converges to a real number r. If xdomα0(f,g), αn(f,g)(x)=. Let the limit function be τμ(f,g). Clearly, τμ(f,g) is proper convex because αn(f,g) is convex. Moreover, if domα0(f,g)=domα0(f,g), by (iii) and (2.23), it is the common limit of αn(f,g) and αn(f,g), so τμ(f,g)Γ since it is a supremum of αn(f,g)Γ. If domα0(f,g) is closed, we define an increasing sequence γn(f,g)Γ by (2.24)γn(f,g)=αn(f,g)+δC, where δC denotes the indicator function of the closed convex set C=domα0(f,g). Obviously, τμ(f,g) is the common limit of αn(f,g) and γn(f,g), hence belongs to Γ.

Remark 2.4.

If both f and g are finite-valued, the condition domα0(f,g)=domα0(f,g) is automatically satisfied.

Corollary 2.5.

For μ>0 and f,gΓ with domfdomg,

τμ(f,g)=τμ(g,f),

((1/2)(f*+g*))*α0(f,g)τμ(f,g)α0(f,g)=(1/2)(f+g).

Proof.

(i) Trivially, α0(f,g)=α0(g,f) and α0(f,g)=α0(g,f). Again using the induction argument yields that (2.25)αn(f,g)=αn(g,f),αn(f,g)=αn(g,f),n0. Hence τμ(f,g)=τμ(g,f).

(ii) This is immediate from (2.23) and [13, Theorem 5.4].

Now we express τμ(f,g) in terms of a geometric mean.

Theorem 2.6.

Let μ>0. For f,gΓ with domfdomg, one has (2.26)τμ(f,g)=(12(f+μq)+12(g+μq))#(12(f+μq)*+12(g+μq)*)*-μq=(f+μq)#(g+μq)-μq.

Proof.

Claim 1. We have (2.27)τμ(f,g)=(12(f+μq)+12(g+μq))#(12(f+μq)*+12(g+μq)*)*-μq. Indeed, put f0=(1/2)(f+μq)+(1/2)(g+μq) and g0=((1/2)(f+μq)*+(1/2)(g+μq)*)*. Then f0,g0Γ because (f+μq)* and (g+μq)* are finite-valued, and f0 is cofinite by [15, Theorem 9.3]. By Proposition 2.1, we obtain (2.28)limnβn(f0,g0)=f0#g0, where βn(f0,g0) and βn*(f0,g0) are defined as in (2.1). Set, for each n0, (2.29)βn(f0,g0)=βn(f0,g0)-μq,(βn*)(f0,g0)=βn*(f0,g0)-μq. Then by (2.5) (2.30)βn+1(f0,g0)=βn+1(f0,g0)-μq=βn(f0,g0)+βn*(f0,g0)2-μq=βn(f0,g0)-μq+βn*(f0,g0)-μq2=βn(f0,g0)+(βn*)(f0,g0)2(βn+1*)(f0,g0)=βn+1*(f0,g0)  -μq=(12(βn(f0,g0))*+12(βn*(f0,g0))*)*-μq=(12(βn(f0,g0)+μq)*+12((βn*)(f0,g0)+μq)*)*-μq=pμ(βn(f0,g0),(βn*)(f0,g0);12,12). Put α0(f,g)=(1/2)(f+g) and α0(f,g)=pμ(f,g,;  1/2,1/2). Also define (2.31)αn+1(f,g)=βn(f0,g0),αn+1(f,g)=(βn*)(f0,g0),n0. Then we have (2.32)α1(f,g)=β0(f0,g0)=β0(f0,g0)-μq=12(f0-μq+g0-μq)=12(12(f+g)+pμ(f,g,;12,12))=12(α0(f,g)+α0(f,g)),α1(f,g)=(β0*)(f0,g0)=β0*(f0,g0)-μq=(12(f0*+g0*))*-μq=(12(12(f+g)+μq)*+12(12(f+μq)*+12(g+μq)*))*-μq=(12(α0(f,g)+μq)*+12(α0(f,g)+μq)*)*-μq=pμ(α0(f,g),α0(f,g);12,12). Moreover, it follows from (2.30) that αn(f,g) and αn(f,g) satisfy the recursion formula in (2.1). From Theorem 2.3 and (2.28), we get (2.33)τμ(f,g)=limnαn(f,g)=limnβn(f0,g0)=limnβn(f0,g0)-μq=f0#g0-μq.

Claim 2. τμ(f,g)=(f+μq)#(g+μq)-μq.

Set two cofinite functions f1=f+μq and g1=g+μq. It sufficies to check that (2.34)(12(f1+g1))#(12(f1*+g1*))*=f1#g1. In fact, let F=β0(f1,g1) and G=β0*(f1,g1). Then F and G belong to Γ, and F is cofinite by Proposition 2.1. Clearly, we have (2.35)βn(F,G)=βn+1(f1,g1),βn*(F,G)=βn+1*(f1,g1),n0. Again appealing to (2.6) yields that (2.36)f1#g1=limnβn(f1,g1)=limnβn(F,G)=F#G=(12(f1+g1))#(12(f1*+g1*))*. This completes the proof.

Now we give the following name to τμ(f,g) by Theorem 2.6 above.

Definition 2.7.

For f,gΓ, one defines (2.37)τμ(f,g)=(τ-μ(f*,g*))*,forμ<0,τ0(f,g)=f#g,forμ=0. This τμ(f,g) is called the geometric mean of parameterized arithmetic and harmonic means of f and g and abbreviated by “𝒜#-mean of parameter μ”.

3. Properties of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M265"><mml:mi>𝒜</mml:mi><mml:mo> </mml:mo><mml:mo mathvariant="bold">#</mml:mo><mml:mo> </mml:mo><mml:mi>ℋ</mml:mi></mml:math></inline-formula>-Mean of Parameter <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M266"><mml:mrow><mml:mi>μ</mml:mi></mml:mrow></mml:math></inline-formula>

To deal with τμ(f,g) (for all μ), in what follows, we assume the following for the simplicity of arguments.

3.1. Constraint Qualifications

Consider

f,gΓ with domfdomg,

domα0(f,g)=domα0(f,g),

either f is cofinite and domg* is closed or g is cofinite and domf* is closed.

With these hypotheses, for all μ, τμ(f,g) is well-defined and is in Γ.

Theorem 3.1.

One has the limiting property: (3.1)limμτμ(f,g)=12(f+g),limμ-τμ(f,g)=(12(f*+g*))*.

Proof.

For μ>0, by Corollary 2.5, we get (3.2)limμα0(f,g)limμτμ(f,g)limμα0(f,g)=12(f+g). By Bauschke et al. [13, Theorem 8.5], (3.3)limμα0(f,g)=limμpμ(f,g;12,12)=12(f+g). Thus (3.4)limμτμ(f,g)=12(f+g). Again appealing to Corollary 2.5 yields that (3.5)α0(f*,g*)τμ(f*,g*)α0(f*,g*)=12(f*+g*);thatis,(12(f*+g*))*(τμ(f*,g*))*(α0(f*,g*))*. By the self-duality of the proximal average [13, Theorem 5.1], we have (3.6)(α0(f*,g*))*=(pμ(f*,g*;12,12))*=pμ-1(f,g;12,12). Taking the limit in (3.5), we see from (3.6) that (3.7)(12(f*+g*))*limμ(τμ(f*,g*))*limμpμ-1(f,g;12,12)=12(fg), where the equality comes from [13, Theorem 8.5]. By (CQ3), fgΓ; hence we have (3.8)12(fg)=(12(f*+g*))*. Therefore it follows from (3.7) and (3.8) that (3.9)limμ-τμ(f,g)=limμ(τμ(f*,g*))*=(12(f*+g*))*. This completes the proof.

Theorem 3.2.

One has

(i)  pμ(f,g;  1/2,1/2)τμ(f,g), for μ0,

(ii) (self-duality) (τμ(f,g))*=τ-μ(f,g), for all μ.

Proof.

(i) According to Corollary 2.5 (ii), pμ(f,g;  1/2,1/2)=α0(f,g)τμ(f,g) for μ>0. For μ=0, pμ(f,g;  1/2,1/2)=((1/2)(f*+g*))*=β0*(f,g)f#g=τ0(f,g) by Definition 2.7.

(ii) If -<μ<0, by definition, τμ(f,g)=(τ-μ(f*,g*))*, so (τμ(f,g))*=τ-μ(f*,g*) because τ-μ(f*,g*)Γ. If μ=0, then (τ0(f,g))*=(f#g)*=f*#g*=τ0(f*,g*) by virtue of Proposition 2.1 and Remark 2.2. Let μ>0. Then by definition, (τμ(f,g))*=τ-μ(f*,g*), as desired.

Proposition 3.3.

Let fi,giΓ and figi for each i=1,,m. Then, for μ0, (3.10)pμ(f,λ)pμ(g,λ), where f=(f1,,fm), g=(g1,,gm) and λi’s are positive real numbers with λ1++λm=1.

Proof.

For each i, clearly (3.11)fi+μqgi+μqλi(fi+μq)*λi(gi+μq)*i=1mλi(fi+μq)*i=1mλi(gi+μq)*(i=1mλi(fi+μq)*)*(i=1mλi(gi+μq)*)*pμ(f,λ)pμ(g,λ).

Theorem 3.4 (monotonicity).

One has, for -μν, (3.12)(12(f*+g*))*=τ-(f,g)τμ(f,g)τν(f,g)τ(f,g)=12(f+g).

Proof.

Let 0<μν<. Clearly (3.13)12(f+g)=(α0μ)(f,g)α0ν(f,g)=12(f+g),pμ(f,g;12,12)  =(α0)μ(f,g)(α0)ν(f,g)=pν(f,g;12,12) by [13, Theorem 8.5]. To use induction, assume that (3.14)αnμ(f,g)αnν(f,g),(αn)μ(f,g)(αn)ν(f,g). Then (3.15)αn+1μ(f,g)=12(αnμ(f,g)+(αn)μ(f,g))12(αnν(f,g)+(αn)ν(f,g))=αn+1ν(f,g),(αn+1)μ(f,g)=pμ(αnμ(f,g),(αn)μ(f,g);12,12)pμ(αnν(f,g),(αn)ν(f,g);12,12)pν(αnν(f,g),(αn)ν(f,g);12,12)=(αn+1)ν(f,g) by (3.14), Proposition 3.3, and [13, Theorem 8.5]. Thus (3.14) holds for all n. Hence, we get (3.16)τμ(f,g)=limnαnμ(f,g)limnαnν(f,g)=τν(f,g). On the other hand, for -<-μ-ν<0, (3.17)τ-μ(f,g)=(τμ(f*,g*))*(τν(f*,g*))*=τ-ν(f,g) by means of (3.16). Now let μ>0. Recall that α0(f,g)=β0(f,g) and α0(f,g)β0*(f,g) (see (2.16), (2.1), and Corollary 2.5 (ii)). Assume that (3.18)αn(f,g)βn(f,g),αn(f,g)βn*(f,g). Then (3.19)αn+1(f,g)=12(αn(f,g)+αn(f,g))12(βn(f,g)+βn*(f,g))  =βn+1(f,g),αn+1(f,g)=pμ(αn(f,g),αn(f,g);12,12)pμ(βn(f,g),βn*(f,g);12,12)(12(βn(f,g))*+12(βn*(f,g))*)*=βn+1*(f,g) by virtue of (3.18), Proposition 3.3, [13, Theorem 5.4], and (2.5). Hence (3.18) holds for all n. This implies that (3.20)f#g=τ0(f,g)=limnβn(f,g)limnαn(f,g)=τμ(f,g). So, we get (3.21)τ-μ(f,g)=(τμ(f*,g*))*(τ0(f*,g*))*=τ0(f,g) by (3.20) and Proposition 2.1. Therefore, the result follows from (3.16), (3.17), (3.20), (3.21), and Theorem 3.1.

Corollary 3.5.

Let A and B be two (symmetric) positive definite matrices. Then, for 0μν<, one has (3.22)μ(A,B)ν(A,B), where (3.23)μ(A,B)=[12(A+μI)+12(B+μI)]#[12(A+μI)-1+12(B+μI)-1]-1-μI. Here # denotes the matrix geometric mean of two positive definite matrices.

Proof.

For a positive definite matrix A, define the convex quadratic function (3.24)qA(x)=12Ax,x. Put f(x)=qA(x) and g(x)=qB(x), then qA and qB clearly satisfy the constraint qualifications (CQ1)(CQ3). Applying Theorem 2.6 to these functions yields that (3.25)τμ(f,g)=q(1/2)(A+μI)+(1/2)(B+μI)  #  q[(1/2)(A+μI)-1+(1/2)(B+μI)-1]-1-μqI=q[(1/2)(A+μI)+(1/2)(B+μI)]#[(1/2)(A+μI)-1+(1/2)(B+μI)-1]-1-μqI=q[(1/2)(A+μI)+(1/2)(B+μI)]#[(1/2)(A+μI)-1+(1/2)(B+μI)-1]-1-μI=qμ(A,B), where the second equality comes from Atteia and Raïssouli [11, Proposition 3.5 (v) and (vii)]. Since τμ(f,g)τν(f,g) by Theorem 3.4, we have (3.26)qμ(A,B)qν(A,B),whichisequivalenttoμ(A,B)ν(A,B).

Remark 3.6.

Corollary 3.5 is a particular case of Kim et al. [14, Theorem 3.6] and is based on a different proof using a convex analytic technique in the case of two variables with no weights. To prove the monotonicity of μ w.r.t. the parameter μ, Kim et al.  exploited a well-known variational characterization of the geometric mean of two positive definite matrices.

We close this section with one more observation.

Definition 3.7 (See Bauschke et al. [<xref ref-type="bibr" rid="B13">13</xref>, Definition 9.1]).

Let g and (gk)k be functions from n to (,+]. Then (gk)kepiconverges to g, in symbols, gkeg, if the following hold for every xX:

(i) (for  all  (xk)k)  xkx    g(x)  liminfgk(xk),

(ii) ((yk)k)  ykx and limsupgk(yk)g(x),

The epitopology is the topology induced by epiconvergence.

Proposition 3.8.

One has (3.27)τμ(f,g)e12(f+g) as μ+,τμ(f,g)e(12(f*+g*))* as μ-.

Proof.

By Theorems 3.1 and 3.4 with [16, 7.4 Proposition] or the proof of [13, Corollary 9.6], we can easily get the result.

Acknowledgment

The first author was supported by the Basic Science Research Program through the NRF Grant no. 2012-0001740.

Ando T. Topics on Operator Inequalities 1978 Sapporo, Japan Hokkaido University Lecture Notes MR0482378 ZBL0423.73082 Hauser R. A. Lim Y. Self-scaled barriers for irreducible symmetric cones SIAM Journal on Optimization 2002 12 3 715 723 10.1137/S1052623400370953 MR1884913 ZBL1008.90046 Nesterov Yu. E. Todd M. J. Self-scaled barriers and interior-point methods for convex programming Mathematics of Operations Research 1997 22 1 1 42 10.1287/moor.22.1.1 MR1436572 ZBL0871.90064 Bhatia R. On the exponential metric increasing property Linear Algebra and its Applications 2003 375 211 220 10.1016/S0024-3795(03)00647-5 MR2013466 ZBL1052.15013 Bhatia R. Holbrook J. Riemannian geometry and matrix geometric means Linear Algebra and its Applications 2006 413 2-3 594 618 10.1016/j.laa.2005.08.025 MR2198952 ZBL1088.15022 Moakher M. A differential geometric approach to the geometric mean of symmetric positive-definite matrices SIAM Journal on Matrix Analysis and Applications 2005 26 3 735 747 10.1137/S0895479803436937 MR2137480 ZBL1079.47021 Moakher M. On the averaging of symmetric positive-definite tensors Journal of Elasticity 2006 82 3 273 296 10.1007/s10659-005-9035-z MR2231065 ZBL1094.74010 Hillar C. J. Johnson C. R. Symmetric word equations in two positive definite letters Proceedings of the American Mathematical Society 2004 132 4 945 953 10.1090/S0002-9939-03-07163-6 MR2045408 ZBL1038.15005 Johnson C. R. Hillar C. J. Eigenvalues of words in two positive definite letters SIAM Journal on Matrix Analysis and Applications 2002 23 4 916 928 10.1137/S0895479801387073 MR1920925 ZBL1007.68139 Lawson J. Lim Y. Solving symmetric matrix word equations via symmetric space machinery Linear Algebra and its Applications 2006 414 2-3 560 569 10.1016/j.laa.2005.10.035 MR2214408 ZBL1105.15013 Atteia M. Raïssouli M. Self dual operators on convex functionals; geometric mean and square root of convex functionals Journal of Convex Analysis 2001 8 1 223 240 MR1829063 ZBL1003.90030 Johnstone J. A. Koch V. R. Lucet Y. Convexity of the proximal average Journal of Optimization Theory and Applications 2011 148 1 107 124 10.1007/s10957-010-9747-5 MR2747748 ZBL1215.26010 Bauschke H. H. Goebel R. Lucet Y. Wang X. The proximal average: basic theory SIAM Journal on Optimization 2008 19 2 766 785 10.1137/070687542 MR2425040 ZBL1172.26003 Kim S. Lawson J. Lim Y. The matrix geometric mean of parameterized, weighted arithmetic and harmonic means Linear Algebra and its Applications 2011 435 9 2114 2131 10.1016/j.laa.2011.04.010 MR2810556 ZBL1222.15036 Rockafellar R. T. Convex Analysis 1970 Princeton, NJ, USA Princeton University Press MR0274683 Rockafellar R. T. Wets R. J.-B. Variational Analysis 1998 317 Berlin, Germany Springer 10.1007/978-3-642-02431-3 MR1491362