Majorization and refined Jensen–Mercer type inequalities for self-adjoint operators

Linear Algebra and its Applications 467 (2015) 1–14

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Majorization and refined Jensen–Mercer type inequalities for self-adjoint operators Marek Niezgoda Department of Applied Mathematics and Computer Science, University of Life Sciences in Lublin, Akademicka 13, 20-950 Lublin, Poland

a r t i c l e

i n f o

Article history: Received 4 September 2014 Accepted 28 October 2014 Available online xxxx Submitted by R. Brualdi MSC: 47A63 15B48 26D15

a b s t r a c t In this paper, Jensen–Mercer’s inequality is generalized by applying the method of pre-majorization used for comparing two tuples of self-adjoint operators. A general result in a matrix setting is established. Special cases of the main theorem are studied to recover other inequalities of Mercer type. © 2014 Elsevier Inc. All rights reserved.

Keywords: Convex function Jensen’s inequality Jensen–Mercer’s inequality Majorization Pre-majorization Doubly stochastic matrix Column stochastic matrix Self-adjoint operator

E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.laa.2014.10.040 0024-3795/© 2014 Elsevier Inc. All rights reserved.

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

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1. Introduction and summary Mercer [11] established a variant of Jensen’s inequality as follows. Theorem A. (See [11, Theorem 1.2].) Let f be a real convex function on an interval [a1 , a2 ], a1 < a2 , and a1 ≤ bi ≤ a2 for i = 1, . . . , n. Then  f a1 + a2 −

n 

 wi bi

≤ f (a1 ) + f (a2 ) −

i=1

where

n i=1

n 

wi f (bi ),

(1)

i=1

wi = 1 with wi > 0.

Relation (1) is referred as Jensen–Mercer’s inequality. An m-tuple b = (b1 , . . . , bm ) ∈ Rm is said to be majorized by an m-tuple a = (a1 , . . . , am ) ∈ Rm , written as b ≺ a if k 

b[l] ≤

l=1

k 

a[l]

for k = 1, . . . , m,

and

l=1

m 

bj =

l=1

m 

aj ,

l=1

where a[1] ≥ · · · ≥ a[m] and b[1] ≥ · · · ≥ b[m] are the entries of a and b, respectively, in nonincreasing order [9, p. 8]. A majorization approach to Jensen–Mercer’s inequality was shown in [14]. Theorem B. (See [14, Theorem 2.1].) Let f : J → R be a continuous convex function on interval J ⊆ R. Suppose a = (a1 , . . . , am ) with al ∈ J, l = 1, . . . , m, and B = (bij ) is a real n × m matrix such that bij ∈ J for i = 1, . . . , n, j = 1, . . . , m. If a majorizes each row of B, that is bi· = (bi1 , . . . , bim ) ≺ (a1 , . . . , am ) = a

for each i = 1, . . . , n,

then  f

m  l=1

where

n i=1

al −

m−1 n  j=1 i=1

 wi bij

≤

m  l=1

f (al ) −

m−1 n 

wi f (bij ),

(2)

j=1 i=1

wi = 1 with wi ≥ 0.

The symbol B(H) stands for the linear space of all bounded linear operators on a Hilbert space H. For selfadjoint operators A, B ∈ B(H), we write A ≤ B if B − A is positive, i.e., Ah, h ≤ Bh, h for all h ∈ H. A linear mapping Φ : B(H) → B(K) is said to be a positive linear map if Φ(A) ≤ Φ(B) for all A, B ∈ B(H) such that A ≤ B. A continuous function f : J → R defined on an interval J ⊆ R is said to be operator convex if f (λA + (1 − λ)B) ≤ λf (A) + (1 − λ)f (B) for any λ ∈ [0, 1] and all self-adjoint operators A, B with spectra in J.

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

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Theorem C. (See [5, Theorem 2.1, pp. 63–64].) Let f : J → R be an operator convex function on interval J ⊆ R. Then the inequality  f

n 

 Φi (Ai )

i=1

≤

n 

Φi f (Ai )

(3)

i=1

holds for self-adjoint operators Ai ∈ B(H) with spectra in J, i = 1, . . . , n, and positive n linear mappings Φi : B(H) → B(K), i = 1, . . . , n, such that i=1 Φi (IH ) = IK , where IH and IK are the identity mappings on Hilbert spaces H and K, respectively. Operator version of (1) has been done in [8] (cf. Remark 3.9(b)). See also [4,7,10,12] for extensions and generalizations of Jensen–Mercer’s inequality. In the present paper our goal is to demonstrate further results related to Theorems A, B and C. The paper is organized as follows. Section 2 is expository. Here we collect some needed definitions concerning the notions of vector pre-majorization and majorization aimed for comparing two tuples of vectors (in place of tuples of scalars) (see [16]). These relations are induced by column stochastic matrices. In Section 3 we prove operator inequalities similar to (1) and (2) by using the method of pre-majorization. In Theorem 3.1 we show a general result of this type in a matrix setting. We focus on the pre-majorization relations for corresponding rows (or columns) of the involved matrices. We indicate that the Jensen–Mercer type inequalities are consequences of the pre-majorization and the Sherman type inequalities (cf. [15]). Next, we study some special cases of Theorem 3.1. In particular, we obtain some Jensen type inequalities (see Theorem 3.6 and Corollary 3.7). Some further results are also provided (see Theorems 3.8 and 3.13 and Corollaries 3.10, 3.12, 3.13 and 3.14). 2. Pre-majorization for tuples of vectors An m × m real matrix S = (slj ) is said to be doubly stochastic if slj ≥ 0 for l, j = m 1, 2, . . . , m, and all row and column sums of S are equal to 1, i.e., j=1 slj = 1 for m l = 1, 2, . . . , m, and l=1 slj = 1 for j = 1, 2, . . . , m. Clearly, an m × m matrix S is doubly stochastic iff slj ≥ 0 for l, j = 1, 2, . . . , m, and eS = e = eST , where e = (1, . . . , 1) is the vector of ones of dimension m. It is well-known for given two real row m-tuples a = (a1 , . . . , am ) and b = (b1 , . . . , bm ) that b≺a

iff

b = aS

(4)

for some doubly stochastic m × m matrix S = (slj ) (see [9, p. 33]). Moreover, if f : J → R is a continuous convex function on interval J ⊆ R then for a = (a1 , . . . , am ) and b = (b1 , . . . , bm ) with al , bl ∈ J for l = 1, . . . , m,

4

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

b≺a

implies

m 

f (bl ) ≤

l=1

m 

f (al )

(5)

l=1

(see [6, p. 75], [9, p. 156]). An m × k real matrix S = (slj ) is said to be column stochastic if slj ≥ 0 for l = m 1, 2, . . . , m, j = 1, 2, . . . , k, and all column sums of S are equal to 1, i.e., l=1 slj = 1 for j = 1, 2, . . . , k. It is evident that an m × k matrix S is column stochastic iff slj ≥ 0 for l = 1, 2, . . . , m, j = 1, 2, . . . , k, and eS = e, where e = (1, . . . , 1) is the vector of ones of an appropriate dimension. Given a vector space V, we say that a vector k-tuple B = (b1 , b2 , . . . , bk ) ∈ Vk is pre-majorized (resp. majorized) by a vector m-tuple A = (a1 , a2 , . . . , am ) ∈ Vm , written as B ≺p A (resp. B ≺ A), if there exists an m × k column stochastic matrix (resp. an m × m doubly stochastic matrix) S = (slj ) such that (b1 , b2 , . . . , bk ) = (a1 , a2 , . . . , am )S

(6)

(see [16], cf. [3,17]). Here notation (6) means bj = s1j a1 + s2j a2 + . . . + smj am

for j = 1, 2, . . . , k

(see [16]). More precisely, in order to emphase the role of S, in place of (6) we also write (b1 , b2 , . . . , bk ) ≺p (a1 , a2 , . . . , am )

by S.

It is not hard to check that B ≺p A is equivalent to bj ∈ conv{a1 , a2 , . . . , am } for j = 1, 2, . . . , k (cf. [17, Proposition 3.3]). In particular, if V = Rn (n-column space) and (a1 , a2 , . . . , am ) is identified with n × m matrix A, and (b1 , b2 , . . . , bk ) is identified with n × k matrix B, then (6) can be interpreted as B = AS, which is equivalent to BT = ST AT with row stochastic k × m matrix ST . In other words, the pre-majorization (6) means the weak matrix majorization for matrices AT and BT in terminology of [17]. 3. Jensen–Mercer type inequalities Unless stated otherwise, throughout the paper we assume that H and K are Hilbert spaces and f : J → R is an operator convex function. The symbol J = Bsa (H, J) stands for the set of all self-adjoint operators in B(H) with spectra in a given interval J ⊆ R. In addition, all Ail for i = 1, . . . , n, l = 1, . . . , m and Bij for i = 1, . . . , n, j = 1, . . . , k, are self-adjoint operators in B(H) with spectra in J, that is Ail ∈ J and Bij ∈ J.

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

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For given matrices A = (Ail ), B = (Bij ), C = (cil ) and D = (dij ), the symbols Ai· , Bi· , ci· and di· stand for the ith rows of A, B, C and D, respectively (i = 1, . . . , n). On the other hand, A·l and c·l denote the lth columns of the matrices A and C, respectively, (l = 1, . . . , m), and B·j and d·j denote the jth columns of the matrices B and D, respectively, (j = 1, . . . , k). We begin our discussion of Jensen–Mercer type inequalities with the following result in a matrix setting. Theorem 3.1. Let f : J → R be an operator convex function on interval J ⊆ R and let J = Bsa (H, J). Suppose that (i) A = (Ail ) and C = (cil ) are n × m matrices such that Ail ∈ J and cil ∈ R for i = 1, . . . , n, l = 1, . . . , m, (ii) B = (Bij ) and D = (dij ) are n × k matrices such that Bij ∈ J and dij ∈ R for i = 1, . . . , n, j = 1, . . . , k − 1, k (iii) j=1 dij ≥ 0 with dik = 1 and there exist α, β ∈ R such that Bik ∈ [αI, βI] ⊆ J for i = 1, . . . , n. Then the following implications hold: (a) ⇒ (b) ⇒ (c) ⇒ (d), where the statements (a)–(d) are defined as follows: (a) There exists an m × k column stochastic matrix S = (slj ) such that B = (B·1 , . . . , B·k ) ≺p (A·1 , . . . , A·m ) = A C = (c·1 , . . . , c·m ) ≺p (d·1 , . . . , d·k ) = D

by S,

(7)

T

by S .

(8) (i)

(b) For each i = 1, . . . , n there exists an m × k column stochastic matrix Si = (slj ) such that Bi· = (Bi1 , . . . , Bik ) ≺p (Ai1 , . . . , Aim ) = Ai·

by Si ,

(9)

ci· = (ci1 , . . . , cim ) ≺p (di1 , . . . , dik ) = di·

STi .

(10)

by

(c) For each i = 1, . . . , n,

Bik =

m 

cil Ail −

l=1

k−1 

dij Bij ,

(11)

j=1

and the Sherman type inequality holds f (Bik ) ≤

m  l=1

cil f (Ail ) −

k−1  j=1

dij f (Bij ).

(12)

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

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(d) The refined Jensen–Mercer type inequality holds  f

n m  

cil Φi (Ail ) −

l=1 i=1

≤

n 

k−1 n 

 dij Φi (Bij )

j=1 i=1

m  n n       k−1   Φi f (Bik ) ≤ cil Φi f (Ail ) − dij Φi f (Bij ) ,

i=1

l=1 i=1

(13)

j=1 i=1

where Φi : B(H) → B(K) are positive linear maps satisfying

n i=1

Φi (I) = I.

If function f is operator concave, then inequalities (12)–(13) are reversed. Proof. (a) ⇒ (b). Utilizing (7)–(8) we can write B·j =

m 

A·l slj

for j = 1, . . . , k,

d·j slj

for l = 1, . . . , m.

l=1 k 

c·l =

j=1

In other words, for each i = 1, . . . , n, Bij =

m 

Ail slj

for j = 1, . . . , k,

dij slj

for l = 1, . . . , m.

l=1

cil =

k  j=1

Thus for each i = 1, . . . , n, (Bi1 , . . . , Bik ) ≺p (Ai1 , . . . , Aim ) by S, (ci1 , . . . , cim ) ≺p (di1 , . . . , dik ) by ST . Now, it is observed that conditions (9)–(10) are fulfilled for S1 = . . . = Sn = S. (b) ⇒ (c). It follows from (9) and (10) that for each i = 1, . . . , n, Bij =

m 

(i)

Ail slj

for j = 1, . . . , k,

(14)

(i)

for l = 1, . . . , m.

(15)

l=1

cil =

k  j=1

dij slj

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

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From (14)–(15) we derive k 

dij Bij =

j=1

k 

 dij

m 

j=1

=

 k m  

 (i) Ail slj

l=1 (i)

dij slj

 (i)  dij Ail slj

j=1 l=1

 Ail =

j=1

l=1

=

k  m 

m 

cil Ail

l=1

for i = 1, . . . , n. In this way, we have k 

dij Bij =

m 

j=1

cil Ail

for i = 1, . . . , n.

(16)

l=1

Since dik = 1, (16) ensures Bik =

m 

cil Ail −

k−1 

dij Bij

for i = 1, . . . , n,

(17)

j=1

l=1

as claimed in (11). Next, from (14)–(15) and (iii) and by the Jensen operator inequality (see Theorem C), we deduce that   m  m k k k      (i) (i) dij f (Bij ) = dij f Ail slj ≤ dij slj f (Ail ) j=1

j=1

=

 k m   l=1

l=1

j=1



(i)

dij slj

f (Ail ) =

j=1

m 

l=1

cil f (Ail )

l=1

for i = 1, . . . , n. Hence, by dik = 1, f (Bik ) ≤

m 

cil f (Ail ) −

l=1

k−1 

dij f (Bij ) for i = 1, . . . , n,

(18)

j=1

which ends the proof of (12). (c) ⇒ (d). We start with the proof of the observation that inequality (13) is welldefined. To do so, we denote Y =

m  n  l=1 i=1

cil Φi (Ail ) −

k−1 n  j=1 i=1

It is enough to show that Y lies in the domain J.

dij Φi (Bij ).

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

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It is clear that Y =

n 



m 

Φi

i=1

cil Ail −

k−1 

 dij Bij .

(19)

j=1

l=1

Thanks to (11) we can see that n 

 Φi

i=1

m 

cil Ail −

k−1 

 dij Bij

=

j=1

l=1

n 

Φi (Bik ) for i = 1, . . . , n.

(20)

i=1

Since Bik ∈ [αI, βI] ⊆ J for i = 1, . . . , n, we find that αI ≤ Bik ≤ βI, whence αΦi (I) ≤ Φi (Bik ) ≤ βΦi (I) for i = 1, . . . , n, and further αI =

n 

αΦi (I) ≤

i=1

n 

Φi (Bik ) ≤

i=1

n 

βΦi (I) = βI.

i=1

n Thus i=1 Φi (Bik ) ∈ J. This and (19)–(20) imply Y ∈ J, as desired. By the Jensen operator inequality (see Theorem C), we can write  f

n m  

cil Φi (Ail ) −

l=1 i=1



=f

n 

Φi

≤

n 

 

Φi f

i=1

 dij Φi (Bij )

j=1 i=1



i=1

k−1 n 

m 

k−1 

cil Ail −

l=1

j=1

m 

k−1 

cil Ail −

 dij Bij  dij Bij

.

(21)

j=1

l=1

On account of (21) and (11), we have  f

n m  

cil Φi (Ail ) −

k−1 n 

l=1 i=1

 dij Φi (Bij )

j=1 i=1

≤

n 

  Φi f (Bik ) ,

i=1

which proves the left-hand side inequality in (13). Since Φi are positive, it follows from (12) that n  i=1

n    Φi f (Bik ) ≤ Φi i=1



m  l=1

cil f (Ail ) −

k−1  j=1

 dij f (Bij ) ,

(22)

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

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which yields the right-hand side inequality in (13), as follows n  i=1

n m  n     k−1     Φi f (Bik ) ≤ cil Φi f (Ail ) − dij Φi f (Bij ) . l=1 i=1

(23)

j=1 i=1

Finally, by combining (22) and (23), we establish (13). This completes the proof. 2 Remark 3.2. In Theorem 3.1, since S is column stochastic, condition (7) implies that each column of the matrix B is a convex combination of all columns of the matrix A, i.e., B·j ∈ conv{A·1 , . . . , A·m } for j = 1, . . . , k.

(24)

Likewise, whenever ST is column stochastic, condition (8) implies that each column of the matrix C is a convex combination of all columns of the matrix D, i.e., c·l ∈ conv{d·1 , . . . , d·k } for l = 1, . . . , m.

(25)

However, it should be emphased that conditions (7) and (8) are connected each other by the matrices S and ST . This is not visible in (24)–(25). Thus (7)–(8) are stronger than (24)–(25). Remark 3.3. In the case when the matrix Si is doubly stochastic, conditions (9)–(10) define the so-called weighted majorization of vectors (rows) Ai· , Bi· , ci· and di· (see [1,2,18]). The role of (9)–(10) is to select quadruples of vectors satisfying the refined Jensen–Mercer type inequality (13). Remark 3.4. In the literature of Jensen–Mercer type inequalities, usually the authors k use sums of type j=1 (·) for “minus parts” of their inequalities. In our majorization k−1 k approach, we prefer to employ j=1 (·) instead of j=1 (·) (see (13)). The reason is that the “missing” index j = k appears in the middle inequality of (13), which gives a refinement of the Jensen–Mercer type inequality under consideration. This convention will be in force throughout the present paper. Remark 3.5. As can be seen in the above proof, Theorem 3.1 remains valid for any convex function f on J (without operator convexity) with the additional assumptions: [mAil , MAil ] ∩ [mBij , MBij ] = ∅ for i = 1, . . . , n, j = 1, . . . , k and l = 1, . . . , m, [mA , MA ] ∩ [mBik , MBik ] = ∅ for i = 1, . . . , n, n where A = i=1 Φi (Bik ), and mB and MB denote the lower and upper bounds, respectively, of an operator B (see [12, p. 68]).

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

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We continue to study special cases of Theorem 3.1. m A real m-tuple (c1 , . . . , cm ) is called probabilistic if l=1 cl = 1 with cl ≥ 0, l = 1, . . . , m. It easy to see that for the k-tuples di· = (0, . . . , 0, 1) ∈ Rk

for i = 1, . . . , n,

the “minus parts” of all sides of inequalities (13) disappear. Therefore we obtain a Jensen type inequality, as follows. Theorem 3.6. Let f : J → R be an operator convex function on interval J ⊆ R. Suppose that Ail are self-adjoint operators in B(H) with spectra in J for i = 1, . . . , n, l = 1, . . . , m, that is Ail ∈ J = Bsa (H, J). If for i = 1, . . . , n, ci· = (ci1 , . . . , cim ) is a given probabilistic m-tuple, and Bi = m l=1 cil Ail ∈ [αI, βI] ⊆ J for some α, β ∈ R, then we have the Jensen type inequality  f

n m  

 cil Φi (Ail )

≤

l=1 i=1

n 

n m       Φi f (Bi ) ≤ cil Φi f (Ail ) ,

i=1

l=1 i=1

where Φi : B(H) → B(K) are positive linear maps satisfying

n i=1

(26)

Φi (I) = I.

Proof. In order to use Theorem 3.1, for the probability m-tuples ci· , i = 1, . . . , n, and an arbitrary positive integer k (e.g., k = 1), we define Si to be the m × k column stochastic matrix with all columns equal to (ci1 , . . . , cim )T . Thus the last row of STi is (ci1 , . . . , cim ). Then for di· = (di1 , . . . , dik ) = (0, . . . , 0, 1), the condition ci· = (ci1 , . . . , cim ) ≺p (0, . . . , 0, 1) = di· is fulfilled. Furthermore, as Bi = Bik =

m

l=1 cil Ail

by STi

(27)

for i = 1, . . . , n, we get

Bi· = (Bi1 , . . . , Bik ) ≺p (Ai1 , . . . , Aim ) = Ai·

by Si ,

(28)

where Bi1 = . . . = Bik . We now appeal to Theorem 3.1 with (27)–(28) for the n × m matrices A = (Ail ) and C = (cil ) and for the n × k matrices B = (Bij ) and D = (dij ), to obtain the required result (26). 2 Remark 3.7. To justify that inequality (26) is of Jensen type, observe that n m   l=1 i=1

cil Φi (I) =

n  i=1

 Φi

m 

 cil I

l=1

with cil Φi ≥ 0 for l = 1, . . . , m and i = 1, . . . , n.

=

n  i=1

Φi (I) = I

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

11

We return to discussion of Jensen–Mercer type inequalities. By taking n = 1 and abbreviating notation by A1l = Al ,

B1j = Bj ,

c1l = cl ,

d1j = dj

and Φ1 = Φ,

for l = 1, . . . , m, j = 1, . . . , k, we get Theorem 3.8. Let f : J → R be an operator convex function on interval J ⊆ R and let J = Bsa (H, J). Suppose that (i) A = (Al ) and C = (cl ) are 1 × m matrices such that Al ∈ J and cl ∈ R for l = 1, . . . , m, (ii) B = (Bj ) and D = (dj ) are 1 × k matrices such that Bj ∈ J and dj ∈ R for j = 1, . . . , k − 1, k (iii) j=1 dj ≥ 0 with dk = 1 and there exist α, β ∈ R such that Bk ∈ [αI, βI] ⊆ J. If there exists an m × k column stochastic matrix S = (slj ) such that (B1 , . . . , Bk ) ≺p (A1 , . . . , Am ) (c1 , . . . , cm ) ≺p (d1 , . . . , dk ) then Bk =  f

m 

m

l=1 cl Al

cl Φ(Al ) −

l=1

−

k−1 

k−1 j=1

by S,

(29)

T

by S ,

(30)

dj Bj and 

dj Φ(Bj )

m       k−1   ≤ Φ f (Bk ) ≤ cl Φ f (Al ) − dj Φ f (Bj ) ,

j=1

(31)

j=1

l=1

where Φ : B(H) → B(K) is a unital positive linear map. Proof. Apply Theorem 3.1.

2

Remark 3.9. (a) Since the matrix S in Theorem 3.8 is column stochastic, condition (29) reads as Bj ∈ conv{A1 , . . . , Am },

j = 1, . . . , k.

(32)

(b) In the case when m = 2 and Φ is the identity map, inequality (31) corresponds to that in [8, Theorem 3.1]. By letting (c1 , . . . , cm ) = (d1 , . . . , dm ) = (1, . . . , 1), Theorem 3.8 implies the following result.

(33)

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

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Corollary 3.10. Let f : J → R be an operator convex function on J ⊆ R. Suppose that (i) A = (Al ) is a 1 × m matrix such that Al ∈ J for l = 1, . . . , m, (ii) B = (Bj ) is a 1 × k matrix such that Bj ∈ J for j = 1, . . . , m − 1, (iii) there exist α, β ∈ R such that Bm ∈ [αI, βI] ⊆ J. If (B1 , . . . , Bm ) ≺ (A1 , . . . , Am ), then Bm =  f

m  l=1

Φ(Al ) −

m−1 

 Φ(Bj )

m l=1

Al −

m−1 j=1

Bj and

m        m−1  ≤ Φ f (Bm ) ≤ Φ f (Al ) − Φ f (Bj ) ,

j=1

l=1

(34)

j=1

where Φ : B(H) → B(K) is a unital positive linear map. Proof. Use Theorem 3.8. 2 Remark 3.11. The right-hand side inequality in (34) remains valid for any convex function f on J with the additional assumption: [mAl , MAl ] ∩ [mBj , MBj ] = ∅ for l, j = 1, . . . , m, where mA and MA denote the lower and upper bounds, respectively, of an operator A (see [12, p. 68], cf. Remark 3.5). A special case of Corollary 3.10 for m = 2 corresponds to [13, Theorem 1]. Corollary 3.12. (Cf. [13, Theorem 1].) Let f : J → R be an operator convex function. Suppose that (i) A1 , A2 ∈ J, (ii) B1 ∈ J, (iii) B2 ∈ [αI, βI] ⊆ J for some α, β ∈ R. If (B1 , B2 ) ≺ (A1 , A2 ) then B2 = A1 + A2 − B1 and           f Φ(A1 ) + Φ(A2 ) − Φ(B1 ) ≤ Φ f (B2 ) ≤ Φ f (A1 ) + Φ f (A2 ) − Φ f (B1 ) , where Φ : B(H) → B(K) is a unital positive linear map. Proof. Apply Corollary 3.10 for m = 2. 2 In the forthcoming theorem we explicitly show the role of the involved matrices Si in Jensen–Mercer type inequalities. Theorem 3.13. Let f : J → R be an operator convex function and let J = Bsa (H, J). Suppose that (i) A = (Ail ) is n × m matrix such that Ail ∈ J for i = 1, . . . , n, l = 1, . . . , m, (ii) B = (Bij ) is n × k matrix such that Bij ∈ J for i = 1, . . . , n, j = 1, . . . , k − 1, (iii) there exist α, β ∈ R such that Bik ∈ [αI, βI] ⊆ J for i = 1, . . . , n.

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

13

(i)

If for i = 1, . . . , n, there exists an m × k column stochastic matrix Si = (slj ) such that (Bi1 , . . . , Bik ) ≺p (Ai1 , . . . , Aim ) then Bik =

m l=1

 f

(i)

Sl Ail −

n m  

k−1 j=1

(i) Sl Φi (Ail )

l=1 i=1

≤

n 

by Si ,

(35)

Bij for i = 1, . . . , n, and

−

k−1 n 

 Φi (Bij )

j=1 i=1

m  n n     k−1    (i)  Φi f (Bik ) ≤ Sl Φi f (Ail ) − Φi f (Bij ) ,

i=1

l=1 i=1

(36)

j=1 i=1

where Φi : B(H) → B(K) are positive linear maps satisfying k (i) (i) Sl = j=1 slj is the lth row sum of the matrix Si .

n i=1

Φi (I) = I, and

(i)

Proof. We introduce the n × m real matrix C = (cil ) with cil = Sl for i = 1, . . . , n, (i) (i) l = 1, . . . , m. That is, ci· = (ci1 , . . . , cim ) = (S1 , . . . , Sm ). Likewise, we define the n × k matrix D = (dij ) such that dij = 1 for i = 1, . . . , n, j = 1, . . . , k. That is, di· = (1, . . . , 1). It is not hard to verify that (10) is met for each i = 1, . . . , n. In fact, we have (i)

cil = Sl

=

k 

(i)

(i)

(i)

slj = sl1 di1 + . . . + slk dik

for l = 1, . . . , m.

j=1

In other words, ci· = (ci1 , . . . , cim ) ≺p (di1 , . . . , dik ) = di·

by STi ,

as wanted. So, in light of Theorem 3.1, the required result (36) is due to (13). 2 Corollary 3.14. Under the assumptions of Theorem 3.13, with condition (35) replaced by (Bi1 , . . . , Bim ) ≺ (Ai1 , . . . , Aim ) it holds that Bim =  f

n m   l=1 i=1

≤

n  i=1

m l=1

Ail −

Φi (Ail ) −

m−1 j=1

m−1 n 

for i = 1, . . . , n,

(37)

Bij for i = 1, . . . , n, and  Φi (Bij )

j=1 i=1

m  n n       m−1   Φi f (Bik ) ≤ Φi f (Ail ) − Φi f (Bij ) , l=1 i=1

j=1 i=1

where Φi : B(H) → B(K) are positive linear maps satisfying

n i=1

Φi (I) = I.

(38)

14

M. Niezgoda / Linear Algebra and its Applications 467 (2015) 1–14

Proof. For i = 1, . . . , n, it follows from (37) that there exists an m × m doubly stochastic (i) matrix Si = (slj ) satisfying (35). (i)

Since for i = 1, . . . , n, the matrix Si is doubly stochastic we have m = k and Sl = 1 for l = 1, . . . , m. Now, the required inequality (38) follows by making use of (36) in Theorem 3.13. 2 Acknowledgement The author would like to thank anonymous referee for careful reading and giving valuable comments. References [1] J. Borcea, Equilibrium points of logarithmic potentials, Trans. Amer. Math. Soc. 359 (2007) 3209–3237. [2] A.-M. Burtea, Two examples of weighted majorization, An. Univ. Craiova Ser. Mat. Inform. 37 (2) (2000) 92–99. [3] G. Dahl, Matrix majorization, Linear Algebra Appl. 288 (1999) 53–73. [4] A. Guessab, Direct and converse results for generalized multivariate Jensen-type inequalities, J. Nonlinear Convex Anal. 13 (4) (2012) 777–797. [5] F. Hansen, J. Pečarić, I. Perić, Jensen’s operator inequality and its converses, Math. Scand. 100 (2007) 61–73. [6] G.M. Hardy, J.E. Littlewood, G. Pólya, Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952. [7] S. Ivelić, A. Matković, J.E. Pečarić, On a Jensen–Mercer operator inequality, Banach J. Math. Anal. 5 (1) (2011) 19–28. [8] M. Kian, M.S. Moslehian, Refinements of the operator Mercer–Jensen’s inequality, Electron. J. Linear Algebra 26 (2013) 742–753. [9] A.W. Marshall, I. Olkin, B.C. Arnold, Inequalities: Theory of Majorization and Its Applications, second edition, Springer, New York, 2011. [10] A. Matković, J. Pečarić, I. Perić, A variant of Jensen’s inequality of Mercer’s type for operators with applications, Linear Algebra Appl. 418 (2006) 551–564. [11] A. McD Mercer, A variant of Jensen’s inequality, JIPAM. J. Inequal. Pure Appl. Math. 4 (4) (2003), Article 73. [12] J. Mićić, J. Pečarić, J. Perić, Extension of the refined Jensen’s operator inequality with condition on spectra, Ann. Funct. Anal. 3 (1) (2012) 67–85. [13] M.S. Moslehian, J. Mićić, M. Kian, An operator inequality and its consequences, Linear Algebra Appl. 439 (2013) 584–591. [14] M. Niezgoda, A generalization of Mercer’s result on convex functions, Nonlinear Anal. 71 (2009) 2771–2779. [15] M. Niezgoda, Remarks on Sherman like inequalities for (α, β)-convex functions, Math. Inequal. Appl. 17 (4) (2014) 1579–1590. [16] M. Niezgoda, Vector majorization and Schur-concavity of some sums generated by the Jensen and Jensen–Mercer functionals, Math. Inequal. Appl. (2015), in print. [17] F.D. Peria, P.G. Massey, L.E. Silvestre, Weak matrix majorization, Linear Algebra Appl. 403 (2005) 343–368. [18] S. Sherman, On a theorem of Hardy, Littlewood, Pólya, and Blackwell, Proc. Natl. Acad. Sci. USA 37 (1) (1957) 826–831.