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 [1] to the case of positive semidefinite matrices A and B as the maximum A#B of all X≥0 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 [8–10]. 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 [11] 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 [11] provided a general algorithm to construct the (self-dual) geometric mean and the square root of convex functions. As pointed out in [12], 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. [13] 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 [11] 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. [14] 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,g∈Γ with domf∩domg≠∅ where Γ denotes the class of proper convex lower semicontinuous functions from the Euclidean space ℝn 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(f□g)*)*,
where □ stands for the infimal convolution. As is well known, f□g 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 f□g 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,g∈Γ with domf∩domg≠∅ in [11]. Hence it is necessary to impose a suitable condition to meet this demand. For that purpose, recall that a function f∈Γ is called cofinite if the recession function f0+ of f satisfies (f0+)(y)=+∞, for all y≠0 (see [15, page 116]). Then f is cofinite if and only if domf*=ℝn 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 domf∩domg≠∅. 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 n≥0. Hence the geometric mean f#g due to Atteia and Raïssouli [11], that is, the limit
(2.3)f#g=limn→∞βn(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)=domf∩domg≠∅. In addition, β0(f,g) is still cofinite by [15, Theorem 9.3]. Then β0*(f,g)=((1/2)(f*+g*))*=(1/2)⋆(f□g)∈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)=domf∩domg=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)),∀n≥0;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)=supy∈ℝn[〈y,x〉-(f*#g*)(y)]=supy∈ℝn[〈y,x〉-limn→∞βn(f*,g*)(y)]=supy∈ℝn[〈y,x〉-limn→∞βn*(f*,g*)(y)]=supy∈ℝn[〈y,x〉-limn→∞(βn(f,g))*(y)]≤supy∈ℝn[〈y,x〉-(βn(f,g))*(y)],∀n=(βn(f,g))**(x)=βn(f,g)(x),∀n.
Hence
(2.10)(f*#g*)*(x)≤limn→∞βn(f,g)(x)=(f#g)(x).
On the other hand,
(2.11)(f*#g*)*(x)=supy∈ℝn[〈y,x〉-(f*#g*)(y)]=supy∈ℝn[〈y,x〉-limn→∞βn(f*,g*)(y)]≥supy∈ℝn[〈y,x〉-βn(f*,g*)(y)],∀n=(βn(f*,g*))*(x)=βn*(f,g)(x),∀n.
Thus
(2.12)(f*#g*)*(x)≥limn→∞βn*(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 ℝ 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 g∈Γ. Of course, for an f∈Γ, 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 μ≥0, 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:ℝn→(-∞,+∞] 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,g∈Γ with domf∩domg≠∅. 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 n≥0;

αn•(f,g)≤αn(f,g),αn+1(f,g)≤αn(f,g) and αn•(f,g)≤αn+1•(f,g), for all n≥0;

αn+1(f,g)-αn+1•(f,g)≤(1/2)(αn(f,g)-αn•(f,g)), for all n≥0;

there exists a limit τμ(f,g)=limn→∞αn(f,g) which is a proper convex function with domτμ(f,g)=domf∩domg=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)=12domf+12domg⊇12(domf∩domg)+12(domf∩domg)=domf∩domg=domα0(f,g)
because domf∩domg 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)+12domαn•(f,g)⊇domαn(f,g)=domαn+1(f,g).
Thus we obtain that
(2.19)∀n,domαn(f,g)=domf∩domg=domα0(f,g),∀n,domαn•(f,g)⊇domαn(f,g)=domα0(f,g).
This implies that, for all n≥0, 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)+μq≤αn+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)+μq≤αn(f,g)+μq⟺αn•(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 [16]), 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 x∈domα0(f,g)=domf∩domg=domαn(f,g) by (2.19), αn(f,g)(x) converges to a real number r. If x∉domα0(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.

(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),∀n≥0.
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,g∈Γ with domf∩domg≠∅, 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,g0∈Γ 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)limn→∞βn(f0,g0)=f0#g0,
where βn(f0,g0) and βn*(f0,g0) are defined as in (2.1). Set, for each n≥0,
(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),∀n≥0.
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)=limn→∞αn(f,g)=limn→∞βn′(f0,g0)=limn→∞βn(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),∀n≥0.
Again appealing to (2.6) yields that
(2.36)f1#g1=limn→∞βn(f1,g1)=limn→∞βn(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,g∈Γ, 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#ℋ-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∈ℝ), in what follows, we assume the following for the simplicity of arguments.

3.1. Constraint Qualifications

Consider

f,g∈Γ with domf∩domg≠∅,

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∈ℝ, τμ(f,g) is well-defined and is in Γ.

Theorem 3.1.

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

Proof.

For μ>0, by Corollary 2.5, we get
(3.2)limμ→∞α0•(f,g)≤limμ→∞τμ(f,g)≤limμ→∞α0(f,g)=12(f+g).
By Bauschke et al. [13, Theorem 8.5],
(3.3)limμ→∞α0•(f,g)=limμ→∞pμ(f,g;12,12)=12(f+g).
Thus
(3.4)limμ→∞τμ(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*))*≤limμ→∞pμ-1(f,g;12,12)=12⋆(f□g),
where the equality comes from [13, Theorem 8.5]. By (CQ3), f□g∈Γ; hence we have
(3.8)12⋆(f□g)=(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 μ≥0,

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

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,gi∈Γ and fi≤gi for each i=1,…,m. Then, for μ≥0,
(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+μq≤gi+μq⇒λi(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 -∞≤μ≤ν≤∞,
(3.12)(12(f*+g*))*=τ-∞(f,g)≤τμ(f,g)≤τν(f,g)≤τ∞(f,g)=12(f+g).

Proof.

Let 0<μ≤ν<∞. 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)=limn→∞αnμ(f,g)≤limn→∞αnν(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)=limn→∞βn(f,g)≤limn→∞αn(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 0≤μ≤ν<∞, one has
(3.22)ℒμ(A,B)≤ℒν(A,B),
where
(3.23)ℒμ(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)=12〈Ax,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=qℒμ(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)qℒμ(A,B)≤qℒν(A,B),whichisequivalenttoℒμ(A,B)≤ℒν(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 ℒμ w.r.t. the parameter μ, Kim et al. [14] 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∈ℕ be functions from ℝn to (∞,+∞]. Then (gk)k∈ℕepiconverges to g, in symbols, gk→eg, if the following hold for every x∈X:

(i) (forall(xk)k∈ℕ)xk→x⇒g(x)≤liminfgk(xk),

(ii) (∃(yk)k∈ℕ)yk→x 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.

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