This is a submission to one of journals of TMRG: BJMA/AFA EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY WITH CONDITION ON SPECTRA

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1 This is a submission to one of journals of TMRG: BJMA/AFA EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY WITH CONDITION ON SPECTRA JADRANKA MIĆIĆ, JOSIP PEČARIĆ AND JURICA PERIĆ3 Abstract. We give an extension of the refined Jensen s operator inequality for n tuples of self-adjoint operators, unital n tuples of positive linear mappings and real valued continuous convex functions with conditions on the spectra of the operators. We also study the order among quasi-arithmetic means under similar conditions.. Introduction We recall some notations and definitions. Let BH be the C -algebra of all bounded linear operators on a Hilbert space H and H stands for the identity operator. We define bounds of a self-adjoint operator A BH by m A = inf Ax, x and M A = sup Ax, x x = x = for x H. If SpA denotes the spectrum of A, then SpA is real and SpA [m A, M A ]. For an operator A BH we define operators A, A +, A by A = A A /, A + = A + A/, A = A A/. Obviously, if A is self-adjoint, then A = A / and A +, A 0 called positive and negative parts of A = A + A. B. Mond and J. Pečarić in [9] proved Jensen s operator inequality f w i Φ i A i w i Φ i fa i,. for operator convex functions f defined on an interval I, where Φ i : BH BK, i =,..., n, are unital positive linear mappings, A,..., A n are self-adjoint operators with the spectra in I and w,..., w n are non-negative real numbers with n w i =. Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. Corresponding author. 00 Mathematics Subject Classification. Primary 47A63; Secondary 47B5. Key words and phrases. Self-adjoint operator, positive linear mapping, convex function, Jensen s operator inequality, quasi-arithmetic mean.

2 J. MIĆIĆ, J. PEČARIĆ, J. PERIĆ F. Hansen, J. Pečarić and I. Perić gave in [3] a generalization of. for a unital field of positive linear mappings. The following discrete version of their inequality holds f Φ i A i Φ i fa i,. for operator convex functions f defined on an interval I, where Φ i : BH BK, i =,..., n, is a unital field of positive linear mappings i.e. n Φ i H = K, A,..., A n are self-adjoint operators with the spectra in I. Recently, J. Mićić, Z. Pavić and J. Pečarić proved in [5, Theorem ] that. stands without operator convexity of f : I R if a condition on spectra m A, M A [m i, M i ] = Ø for i =,..., n holds, where m i and M i, m i M i are bounds of A i, i =,..., n; and m A and M A, m A M A, are bounds of A = n Φ ia i provided that the interval I contains all m i, M i. Next, they considered in [6, Theorem.] the case when m A, M A [m i, M i ] = Ø is valid for several i {,..., n}, but not for all i =,..., n and obtain an extension of. as follows. Theorem A. Let A,..., A n be an n tuple of self-adjoint operators A i BH with the bounds m i and M i, m i M i, i =,..., n. Let Φ,..., Φ n be an n tuple of positive linear mappings Φ i : BH BK, such that n Φ i H = K, n Φ i H = K, where n < n,, > 0 and + =. Let m = min{m,..., m n } and M = max{m,..., M n }. If m, M [m i, M i ] = Ø for i = n +,..., n, and one of two equalities n Φ i A i = is valid, then n Φ i fa i Φ i A i = Φ i fa i Φ i A i Φ i fa i,.3 holds for every continuous convex function f : I R provided that the interval I contains all m i, M i, i =,..., n,. If f : I R is concave, then the reverse inequality is valid in.3. Very recently, J. Mićić, J. Pečarić and J. Perić gave in [7, Theorem 3] the following refinement of. with condition on spectra, i.e. a refinement of [5, Theorem 3] see also [5, Corollary 5]. Theorem B. Let A,..., A n be an n tuple of self-adjoint operators A i BH with the bounds m i and M i, m i M i, i =,..., n. Let Φ,..., Φ n be an n tuple of positive linear mappings Φ i : BH BK, i =,..., n, such that n Φ i H = K. Let m A, M A [m i, M i ] = Ø for i =,..., n, and m < M,

3 EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY 3 where m A and M A, m A M A, are the bounds of the operator A = n Φ ia i and m = max {M i : M i m A, i {,..., n}}, M = min {m i : m i M A, i {,..., n}}. If f : I R is a continuous convex resp. concave function provided that the interval I contains all m i, M i, then f Φ i A i Φ i fa i δ f Ã Φ i fa i.4 resp. f Φ i A i Φ i fa i + δ f Ã Φ i fa i holds, where δ f δ f m, M = f m + f M m+ f resp. δ f δ f m, M m+ = f f m f M, à ÃA m, M = K M m+ M m A K and m [m, m A ], M [MA, M], m < M, are arbitrary numbers. There is an extensive literature devoted to Jensens inequality concerning different refinements and extensive results, see, for example [], [], [4], [0] [4]. In this paper we study an extension of Jensen s inequality given in Theorem B and a refinement of Theorem A. As an application of this result to the quasiarithmetic mean with a weight, we give an extension of results given in [7] and a refinement of ones given in [6].. Main results To obtain our main result we need a result [7, Lemma ] given in the following lemma. Lemma C. Let A be a self-adjoint operator A BH with SpA [m, M], for some scalars m < M. Then f A M H A M m fm + A m H M m fm δ fã. resp. f A M H A M m fm + A m H M m fm + δ fã holds for every continuous convex resp. concave function f : [m, M] R, where δ f = fm + fm f m+m resp. δ f = f m+m fm fm, and à = H M m A m+m H. We shall give the proof for the convenience of the reader. Proof of Lemma C. We prove only the convex case.

4 4 J. MIĆIĆ, J. PEČARIĆ, J. PERIĆ In [8, Theorem, p. 77] is prove that min{p, p } [ fx + fy f ] x+y p fx + p fy fp x + p y. holds for every convex function f on an interval I and x, y I, p, p [0, ] such that p + p =. Putting x = m, y = M in. it follows that f p m + p M p fm + p fm min{p, p } fm + fm f m+m.3 holds for every p, p [0, ] such that p + p =. For any t [m, M] we can write M t f t = f M m m + t m M m M. Then by using.3 for p = M t and p M m = t m we get M m ft M t M m fm + t m M m fm M m t m + M m + M fm + fm f,.4 since { } M t min M m, t m = M m M m t m + M. Finally we use the continuous functional calculus for a self-adjoint operator A: f, g CI, SpA I and f g implies fa ga; and ht = t implies ha = A. Then by using.4 we obtain the desired inequality.. In the following theorem we give an extension of Jensen s inequality given in Theorem B and a refinement of Theorem A. Theorem.. Let A,..., A n be an n tuple of self-adjoint operators A i BH with the bounds m i and M i, m i M i, i =,..., n. Let Φ,..., Φ n be an n tuple of positive linear mappings Φ i : BH BK, such that n Φ i H = K, n Φ i H = K, where n < n,, > 0 and + =. Let m L = min{m,..., m n }, M R = max{m,..., M n } and { ml, if {M m = i : M i m L, i {n +,..., n}} = Ø, { max {M i : M i m L, i {n +,..., n}}, otherwise, MR, if {m M = i : m i M R, i {n +,..., n}} = Ø, min {m i : m i M R, i {n +,..., n}}, otherwise. If m L, M R [m i, M i ] = Ø for i = n +,..., n, m < M, and one of two equalities n Φ i A i = Φ i A i = Φ i A i

5 is valid, then EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY 5 n Φ i fa i n Φ i fa i + δ f Ã Φ i fa i Φ i fa i δ f Ã Φ i fa i,.5 holds for every continuous convex function f : I R provided that the interval I contains all m i, M i, i =,..., n, where δ f δ f m, M = f m + f M m + M f à ÃA,Φ,n, m, M = n m + M.6 K M Φ i A i H m and m [m, m L ], M [MR, M], m < M, are arbitrary numbers. If f : I R is concave, then the reverse inequality is valid in.5. Proof. We prove only the convex case. Let us denote A = n Φ i A i, B = Φ i A i, C = Φ i A i. It is easy to verify that A = B or B = C or A = C implies A = B = C. Since f is convex on [ m, M] and SpA i [m i, M i ] [ m, M] for i =,..., n, it follows from Lemma C that f A i M H A i M m f m + A i m H holds, where δ f = f m+f M f m+ M and Ãi = H M m Applying a positive linear mapping Φ i and summing, we obtain n Φ i fa i M K n Φ ia i f m + M m δ f K M n m Φ i M m f M δ f à i, i =,..., n A i n Φ ia i m K M m Ai f M m+ M H, m+ M H. since n Φ i H = K. It follows that n Φ i fa i M K A M m f m + A m K M m f M δ f Ã,.7 Ai where à = K n M m Φ i m+ M H. In addition, since f is convex on all [m i, M i ] and m, M [m i, M i ] = Ø for i = n +,..., n, then fa i M H A i M m f m + A i m H M m f M, i = n +,..., n.

6 6 J. MIĆIĆ, J. PEČARIĆ, J. PERIĆ It follows Φ i fa i δ f à M K B M m f m + B m K M m f M δ f Ã..8 Combining.7 and.8 and taking into account that A = B, we obtain n Φ i fa i Φ i fa i δ f Ã..9 Next, we obtain = n n Φ i fa i n Φ i fa i + n Φ i fa i + = Φ i fa i δ f à + Φ i fa i by + = Φ i fa i δ f à by.9 Φ i fa i δ f à by + =, which gives the following double inequality n Φ i fa i Φ i fa i δ f à Adding δ f à in the above inequalities, we get n Φ i fa i + δ f Ã Φ i fa i Φ i fa i δ f à by.9 Φ i fa i δ f Ã. Φ i fa i δ f Ã..0 Now, we remark that δ f 0 and à 0. Indeed, since f is convex, then f m + M/ f m + f M/, which implies that δ f 0. Also, since SpA i [ m, M] A i M + m H M m H, for i =,..., n, then which gives n 0 K Φ i A i M + m H M m n M m K, Φ i A i M + m H = Ã.

7 EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY 7 Consequently, the following inequalities n Φ i fa i n Φ i fa i δ f Ã Φ i fa i + δ f Ã, Φ i fa i, hold, which with.0 proves the desired series inequalities.5. Example.. We observe the matrix case of Theorem. for ft = t 4, which is the convex function but not operator convex, n = 4, n = and the bounds of matrices as in Figure. Figure. An example a convex function and the bounds of four operators We show an example such that Φ A 4 + Φ A 4 < Φ A 4 + Φ A 4 + δ f à < Φ A 4 + Φ A 4 + Φ 3 A Φ 4 A 4 4. < Φ 3A Φ 4 A 4 4 δ f à < Φ 3A Φ 4 A 4 4 holds, where δ f = M 4 + m 4 M + m 4 /8 and à = I Φ M A M + m I h + Φ A M + m I 3. m We define mappings Φ i : M 3 C M C as follows: Φ i a jk j,k 3 = a 4 jk j,k, i =,..., 4. Then 4 Φ ii 3 = I and = =. Let A = 9/8 9/8 0, A = 3 9/8 0 9/8 0, A 3 = 3 4 / 5/3 / 0 / 4 0, A 4 = / 3/ Then m =.8607, M = , m = , M = 5.46, m 3 = , M 3 = 4.707, m 4 =.974, M 4 = 36., so m L = m, M R = M,

8 8 J. MIĆIĆ, J. PEČARIĆ, J. PERIĆ m = M 3 and M = m 4 rounded to five decimal places. Also, and Φ A + Φ A = Φ 3A 3 + Φ 4 A 4 = A f Φ A 4 + Φ A 4 = C f Φ A 4 + Φ A 4 + Φ 3 A Φ 4 A 4 4 = Then B f Φ3 A Φ 4 A 4 4 = 4 9/4 9/4 3, , , A f < C f < B f. holds which is consistent with.3. We will choose three pairs of numbers m, M, m [ 4.707, ], M [7.7077,.974] as follows: i m = m L = , M = MR = , then = δ f à = = ii m = m = 4.707, M = M =.974, then = δ f à = = iii m =, M = 0, then = δ f à = = , , New, we obtain the following improvement of. see.: i A f < A f = < C f < = B f < B f, ii A f < A f = < C f < = B f < B f, iii A f < A f = < C f < = B f 3 < B f.

9 EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY 9 Using Theorem. we get the following result. Corollary.3. Let the assumptions of Theorem. hold. Then n Φ i fa i n Φ i fa i + γ δ f Ã Φ i fa i.3 and n Φ i fa i Φ i fa i γ δ f Ã Φ i fa i.4 holds for every γ, γ in the close interval joining and, where δ f defined by.6. and à are Proof. Adding δ f à in.5 and noticing δ f à 0, we obtain n Φ i fa i n Φ i fa i + δ f Ã Φ i fa i. Taking into account the above inequality and the left hand side of.5 we obtain.3. Similarly, subtracting δ f à in.5 we obtain.4. Remark.4. Let the assumptions of Theorem. be valid. We observe that the following inequality f Φ i A i Φ i fa i δ f Ã Φ i fa i, holds for every continuous convex function f : I R provided that the interval I contains all m i, M i, i =,..., n, where δ f is defined by.6, à Ã,A,Φ,n m, M = m + M K Φ i A i K M m and m [m, m L ], M [MR, M], m < M, are arbitrary numbers. Indeed, by the assumptions of Theorem. we have n n m L H Φ i A i M R H and Φ i A i = which implies m L H Φ i A i M R H. Φ i A i Also m L, M R [m i, M i ] = Ø for i = n +,..., n and n Φ i H = K hold. So we can apply Theorem B on operators A n +,..., A n and mappings Φ i. We obtain the desired inequality.

10 0 J. MIĆIĆ, J. PEČARIĆ, J. PERIĆ We denote by m C and M C the bounds of C = n Φ ia i. If m C, M C [m i, M i ] = Ø, i =,..., n, then series inequality.5 can be extended from the left side if we use refined Jensen s operator inequality.4 n f Φ i A i = f Φ i A i n Φ i fa i δ f à n Φ i fa i n Φ i fa i + δ f Ã Φ i fa i Φ i fa i δ f à where δ f and à are defined by.6, à Ã,A,Φ,n m, M = K M m Φ i fa i, Φ i A i m + M K Remark.5. We obtain the equivalent inequalities to the ones in Theorem. in the case when n Φ i H = γ K, for some positive scalar γ. If + = γ and one of two equalities n Φ i A i = Φ i A i = Φ i A i γ is valid, then n Φ i fa i n Φ i fa i + γ δ fã γ Φ i fa i Φ i fa i γ δ fã Φ i fa i, holds for every continuous convex function f : I R provided that the interval I contains all m i, M i, i =,..., n, where δ f and à are defined by.6. With respect to Remark.5, we obtain the following obvious corollary of Theorem. with the convex combination of operators A i, i =,..., n. Corollary.6. Let A,..., A n be an n tuple of self-adjoint operators A i BH with the bounds m i and M i, m i M i, i =,..., n. Let p,..., p n be an n tuple of non-negative numbers such that 0 < n p i = p n < p n = n p i, where n < n. Let m L = min{m,..., m n }, M R = max{m,..., M n } and { ml, if {M m = i : M i m L, i {n +,..., n}} = Ø, { max {M i : M i m L, i {n +,..., n}}, otherwise, MR, if {m M = i : m i M R, i {n +,..., n}} = Ø, min {m i : m i M R, i {n +,..., n}}, otherwise.

11 If EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY m L, M R [m i, M i ] = Ø for i = n +,..., n, m < M, and one of two equalities is valid, then p n n p n n p i fa i p n p n p i A i = p n p n n p i A i = p n p n p i fa i + p i fa i p n p n δ f à p i A i p n δ f à p n p n p n p n p i fa i p i fa i,.5 holds for every continuous convex function f : I R provided that the interval I contains all m i, M i, i =,..., n, where where δ f is defined by.6, à ÃA,p,n m, M = H p n M m n p i A i and m [m, m L ], M [MR, M], m < M, are arbitrary numbers. m + M H If f : I R is concave, then the reverse inequality is valid in.5. As a special case of Corollary.6 we obtain an extension of [7, Corollary 6]. Corollary.7. Let A,..., A n be an n tuple of self-adjoint operators A i BH with the bounds m i and M i, m i M i, i =,..., n. Let p,..., p n be an n tuple of non-negative numbers such that n p i =. Let m A, M A [m i, M i ] = Ø for i =,..., n, and m < M, where m A and M A, m A M A, are the bounds of A = n p ia i and m = max {M i m A, i {,..., n}}, M = min {m i M A, i {,..., n}}. If f : I R is a continuous convex function provided that the interval I contains all m i, M i, then f p i A i f p i A i + δ fã f p i A i + p i fa i p i fa i.6 δ fã p i fa i, n holds, where δ f is defined by.6, à = H M m p ia i m [m, m A ], M [MA, M], m < M, are arbitrary numbers. If f : I R is concave, then the reverse inequality is valid in.6. m+ M H and

12 J. MIĆIĆ, J. PEČARIĆ, J. PERIĆ Proof. We prove only the convex case. We define n+ tuple of operators B,..., B n+, B i BH, by B = A = n p ia i and B i = A i, i =,..., n +. Then m B = m A, M B = M A are the bounds of B and m Bi = m i, M Bi = M i are the ones of B i, i =,..., n +. Also, we define n + tuple of non-negative numbers q,..., q n+ by q = and q i = p i, i =,..., n +. We have that n+ q i = and m B, M B [m Bi, M Bi ] = Ø, for i =,..., n + and m < M.7 holds. Since then n+ n+ q i B i = B + q i B i = i= p i A i + p i A i = B, q B = n+ n+ q i B i = q i B i..8 Taking into account.7 and.8, we can apply Corollary.6 for n = and B i, q i as above, and we get q fb q fb + δ B f n+ n+ q i fb i q i fb i δ B n+ f q i fb i, where B = H M m B i= i= i= m+ M H, which gives the desired inequality Quasi-arithmetic means In this section we study an application of Theorem. to the quasi-arithmetic mean with weight. For a subset {A n,..., A n } of {A,..., A n }, we denote the quasi-arithmetic mean by M ϕ γ, A, Φ, n, n = ϕ γ n i=n Φ i ϕa i, 3. where A n,..., A n are self-adjoint operators in BH with the spectra in I, Φ n,..., Φ n are positive linear mappings Φ i : BH BK such that n i=n Φ i H = γ K, and ϕ : I R is a continuous strictly monotone function. Under the same conditions, for convenience we introduce the following denotations δ ϕ,ψ m, M = ψm + ψm ψ ϕ ϕm+ϕm, Ã ϕ,n,γm, M = K n γm m Φ ϕai i ϕm+ϕm 3. H, where ϕ, ψ : I R are continuous strictly monotone functions and m, M I, m < M. Of course, we include implicitly that Ãϕ,n,γm, M Ãϕ,A,Φ,n,γm, M.

13 EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY 3 The following theorem is an extension of [7, Theorem 7] and a refinement of [6, Theorem 3.]. Theorem 3.. Let A,..., A n be an n tuple of self-adjoint operators A i BH with the bounds m i and M i, m i M i, i =,..., n. Let ϕ, ψ : I R be continuous strictly monotone functions on an interval I which contains all m i, M i. Let Φ,..., Φ n be an n tuple of positive linear mappings Φ i : BH BK, such that n Φ i H = K, n Φ i H = K, where n < n,, > 0 and + =. Let one of two equalities M ϕ, A, Φ,, n = M ϕ, A, Φ,, n = M ϕ, A, Φ, n +, n 3.3 be valid and let m L, M R [m i, M i ] = Ø for i = n +,..., n, m < M, where m L = min{m,..., m n }, M R = max{m,..., M n }, { ml, if {M m = i : M i m L, i {n +,..., n}} = Ø, { max {M i : M i m L, i {n +,..., n}}, otherwise, MR, if {m M = i : m i M R, i {n +,..., n}} = Ø, min {m i : m i M R, i {n +,..., n}}, otherwise. i If ψ ϕ is convex and ψ is operator monotone, then n M ψ, A, Φ,, n ψ Φ i ψa i + δ ϕ,ψ Ã ϕ,n, M ψ, A, Φ,, n ψ Φ i ψa i δ ϕ,ψ Ã ϕ,n, M ψ, A, Φ, n +, n holds, where δ ϕ,ψ 0 and Ãϕ,n, i If ψ ϕ is convex and ψ is operator monotone, then the reverse inequality is valid in 3.4, where δ ϕ,ψ 0 and Ãϕ,n, 0. ii If ψ ϕ is concave and ψ is operator monotone, then 3.4 holds, where δ ϕ,ψ 0 and Ãϕ,n, 0. ii If ψ ϕ is concave and ψ is operator monotone, then the reverse inequality is valid in 3.4, where δ ϕ,ψ 0 and Ãϕ,n, 0. In all the above cases, we assume that δ ϕ,ψ δ ϕ,ψ m, M, Ãϕ,n, Ãϕ,n, m, M are defined by 3. and m [m, m L ], M [MR, M], m < M, are arbitrary numbers. Proof. We only prove the case i. Suppose that ϕ is a strictly increasing function. Then m L, M R [m i, M i ] = Ø for i = n +,..., n implies ϕm L, ϕm R [ϕm i, ϕm i ] = Ø for i = n +,..., n. 3.5

14 4 J. MIĆIĆ, J. PEČARIĆ, J. PERIĆ Also, by using 3.3, we have n Φ i ϕa i = Φ i ϕa i = Φ i ϕa i. Taking into account 3.5 and the above double equality, we obtain by Theorem. n Φ i f ϕa i n Φ i f ϕa i + δ f à ϕ,n, Φ i f ϕa i Φ i f ϕa i δ f à ϕ,n, Φ i f ϕa i, 3.6 for every continuous convex function f : J R on an interval J which contains all [ϕm i, ϕm i ] = ϕ[m i, M i ], i =,..., n, where δ f = fϕm + fϕm f ϕm+ϕm. Also, if ϕ is strictly decreasing, then we check that 3.6 holds for convex f : J R on J which contains all [ϕm i, ϕm i ] = ϕ[m i, M i ]. Putting f = ψ ϕ in 3.6, we obtain n Φ i ψa i n Φ i ψa i + δ ϕ,ψ à ϕ,n, Φ i ψa i Φ i ψa i δ ϕ,ψ à ϕ,n, Φ i ψa i. Applying an operator monotone function ψ on the above double inequality, we obtain the desired inequality 3.4. We now give some particular results of interest that can be derived from Theorem 3., which are an extension of [7, Corollary 8, Corollary 0] and a refinement of [6, Corollary 3.3]. Corollary 3.. Let A,..., A n and Φ,..., Φ n, m i, M i, m, M, m L, M R, and be as in Theorem 3.. Let I be an interval which contains all m i, M i and m L, M R [m i, M i ] = Ø for i = n +,..., n, m < M. I If one of two equalities is valid, then M ϕ, A, Φ,, n = M ϕ, A, Φ,, n = M ϕ, A, Φ, n +, n n Φ i A i n Φ i A i + δ ϕ à ϕ,n, Φ i A i δ ϕ à ϕ,n, Φ i A i Φ i Φ i A i. 3.7

15 EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY 5 holds for every continuous strictly monotone function ϕ : I R such that ϕ is convex on I, where δ ϕ = m + M ϕ ϕ m+ϕ M 0, Ãϕ,n, = K n M m Φ ϕai i ϕ M+ϕ m H and m [m, m L ], M [MR, M], m < M, are arbitrary numbers. But, if ϕ is concave, then the reverse inequality is valid in 3.7 for δ ϕ 0. II is valid, then If one of two equalities n Φ i A i = Φ i A i = Φ i A i n M ϕ, A, Φ,, n ϕ Φ i ϕa i + δ ϕ Ã n M ϕ, A, Φ,, n ϕ Φ i ϕa i δ ϕ Ã n M ϕ, A, Φ, n +, n 3.8 holds for every continuous strictly monotone function ϕ : I R such that one of the following conditions i ϕ is convex and ϕ is operator monotone, i ϕ is concave and ϕ is operator monotone, is satisfied, where δ ϕ = ϕ m + ϕ M m+ ϕ M n Φ Ai i, Ã n = K M m m+ M H and m [m, m L ], M [MR, M], m < M, are arbitrary numbers. But, if one of the following conditions ii ϕ is concave and ϕ is operator monotone, ii ϕ is convex and ϕ is operator monotone, is satisfied, then the reverse inequality is valid in 3.8. Proof. The inequalities 3.7 follows from Theorem 3. by replacing ψ with the identity function, while the inequalities 3.8 follows by replacing ϕ with the identity function and ψ with ϕ. Remark 3.3. Let the assumptions of Theorem 3. be valid. We observe that if one of the following conditions i ψ ϕ is convex and ψ is operator monotone, i ψ ϕ is concave and ψ is operator monotone, is satisfied, then the following obvious inequality see Remark.4. M ϕ, A, Φ, n +, n ψ Φ i ψa i δ ϕ Ã M ψ, A, Φ, n +, n,

16 6 J. MIĆIĆ, J. PEČARIĆ, J. PERIĆ holds, δ ϕ = ϕ m+ϕ M ϕ m+ M, Ã = K M m n Φ ia i m+ M K and m [m, m L ], M [MR, M], m < M, are arbitrary numbers. We denote by m ϕ and M ϕ the bounds of M ϕ, A, Φ,, n. If m ϕ, M ϕ [m i, M i ] = Ø, i =,..., n, and one of two following conditions i ψ ϕ is convex and ψ is operator monotone ii ψ ϕ is concave and ψ is operator monotone is satisfied, then the double inequality 3.4 can be extended from the left side as follows n Φ i fa i δ ϕ,ψ Ã M ϕ, A, Φ,, n = M ϕ, A, Φ,, n ψ M ψ, A, Φ,, n ψ ψ n Φ i ψa i δ ϕ,ψ Ã ϕ,n, Φ i ψa i + δ ϕ,ψ Ã ϕ,n, M ψ, A, Φ, n +, n, M ψ, A, Φ,, n where δ ϕ,ψ and Ãϕ,n, are defined by 3., Ã = K M m Φ i A i m + M K. As a special case of the quasi-arithmetic mean 3. we can study the weighted power mean as follows. For a subset {A p,..., A p } of {A,..., A n } we denote this mean by M [r] γ, A, Φ, p, p = p /r Φ i A r i, r R\{0}, γ i=p p exp Φ i ln A i, r = 0, γ i=p where A p,..., A p are strictly positive operators, Φ p,..., Φ p are positive linear mappings Φ i : BH BK such that p i=p Φ i H = γ K. Under the same conditions, for convenience we introduce denotations as a special case of 3. as follows δ r,s m, M = { m s + M s m r +M r s/r, r 0, m s + M s mm s/, r = 0, Ã r m, M = { K n M r m r Φ ia r i M r +m r r 0, K ln M m n Φ iln A i ln Mm K, r = 0, K, 3.9

17 EXTENSION OF THE REFINED JENSEN S OPERATOR INEQUALITY 7 where m, M R, 0 < m < M and r, s R, r s. Of course, we include implicitly that Ãrm, M Ãr,Am, M, where A = n Φ ia r i for r 0 and A = n Φ iln A i for r = 0. We obtain the following corollary by applying Theorem 3. to the above mean. This is an extension of [7, Corollary 3] and a refinement of [6, Corollary 3.4]. Corollary 3.4. Let A,..., A n be an n tuple of self-adjoint operators A i BH with the bounds m i and M i, m i M i, i =,..., n. Let Φ,..., Φ n be an n tuple of positive linear mappings Φ i : BH BK, such that n Φ i H = K, n Φ i H = K, where n < n,, > 0 and + =. Let m L, M R [m i, M i ] = Ø for i = n +,..., n, m < M, where m L = min{m,..., m n }, M R = max{m,..., M n } and { ml, if {M m = i : M i m L, i {n +,..., n}} = Ø, { max {M i : M i m L, i {n +,..., n}}, otherwise, MR, if {m M = i : m i M R, i {n +,..., n}} = Ø, min {m i : m i M R, i {n +,..., n}}, otherwise. i If either r s, s or r s and also one of two equalities is valid, then M [r], A, Φ,, n = M [r], A, Φ,, n = M [r], A, Φ, n +, n M [s], A, Φ,, n n holds, where δ r,s 0 and Ãs,n, 0. Φ i A s i + δ r,s à s,n, /s M [s], A, Φ,, n Φ i A s i δ r,s à s,n, /s M [s], A, Φ, n +, n In this case, we assume that δ r,s δ r,s m, M, à s,n, Ãs,n, m, M are defined by 3.9 and m [m, m L ], M [MR, M], m < M, are arbitrary numbers. ii If either r s, r or r s and also one of two equalities is valid, then M [s], A, Φ,, n = M [s], A, Φ,, n = M [s], A, Φ, n +, n M [r], A, Φ,, n n holds, where δ s,r 0 and Ãs,n, 0. Φ i A r i + δ s,r à r,n, /r M [r], A, Φ,, n Φ i A r i δ s,r à r,n, /r M [r], A, Φ, n +, n In this case, we assume that δ s,r δ s,r m, M, à r,n, Ãr,n, m, M are defined by 3.9 and m [m, m L ], M [MR, M], m < M, are arbitrary numbers.

18 8 J. MIĆIĆ, J. PEČARIĆ, J. PERIĆ Proof. In the case i we put ψt = t s and ϕt = t r if r 0 or ϕt = ln t if r 0 in Theorem 3.. In the case ii we put ψt = t r and ϕt = t s if s 0 or ϕt = ln t if s 0. We omit the details. References. S. Abramovich, G. Jameson and G. Sinnamon, Refining Jensens inequality, Bull. Math. Soc. Sci. Math. Roumanie N.S , no., S. S. Dragomir, A new refinement of Jensen s inequality in linear spaces with applications, Math. Comput. Modelling 5 00, F. Hansen, J. Pečarić and I. Perić, Jensen s operator inequality and it s converses, Math. Scand , M. Khosravi, J. S. Aujla, S. S. Dragomir and M. S. Moslehian, Refinements of Choi-Davis- Jensen s inequality, Bull. Math. Anal. Appl. 3 0, J. Mićić, Z. Pavić and J. Pečarić, Jensen s inequality for operators without operator convexity, Linear Algebra Appl , J. Mićić, Z. Pavić and J. Pečarić, Extension of Jensen s operator inequality for operators without operator convexity, Abstr. Appl. Anal. 0 0, J. Mićić, J. Pečarić and J. Perić, Refined Jensen s operator inequality with condition on spectra, Oper. Matrices, 0, submitted for publication. 8. D.S. Mitrinović, J.E. Pečarić and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Acad. Publ., Dordrecht-Boston-London, B. Mond and J. Pečarić, Converses of Jensen s inequality for several operators, Revue d analyse numer. et de théorie de l approxim M. S. Moslehian, Operator extensions of Huas inequality, Linear Algebra Appl , J. Rooin, A refinement of Jensens inequality, J. Ineq. Pure and Appl. Math., 6, 005, Art. 38., 4 pp.. H. M. Srivastava, Z.-G. Xia, and Z.-H. Zhang, Some further refinements and extensions of the HermiteHadamard and Jensen inequalities in several variables, Math. Comput. Modelling 54 0, Z.-G. Xiao, H. M. Srivastava and Z.-H. Zhang, Further refinements of the Jensen inequalities based upon samples with repetitions, Math. Comput. Modelling 5 00, L.-C. Wang, X.-F. Ma and L.-H. Liu, A note on some new refinements of Jensens inequality for convex functions, J. Inequal. Pure Appl. Math., 0, 009, Art. 48., 6 pp. Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lučića 5, 0000 Zagreb, Croatia. address: jmicic@fsb.hr Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovića 30, 0000 Zagreb, Croatia. address: pecaric@hazu.hr 3 Faculty of Science, Department of Mathematics, University of Split, Teslina, 000 Split, Croatia. address: jperic@pmfst.hr