Inequalities among quasi-arithmetic means for continuous field of operators

Transcription

1 Filomat 26:5 202, DOI /FIL205977M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Inequalities among quasi-arithmetic means for continuous field of operators Jadranka Mićić a, Kemal Hot b a Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lučića 5, 0000 Zagreb, Croatia b Electrical Engineering Department, Polytechnic of Zagreb, Konavoska 2, 0000 Zagreb, Croatia Abstract. In this paper we study inequalities among quasi-arithmetic means for a continuous field of self-adjoint operators, a field of positive linear mappings and continuous strictly monotone functions which induce means. We present inequalities with operator convexity and without operator convexity of appropriate functions. Also, we present a general formulation of converse inequalities in each of these cases. Furthermore, we obtain refined inequalities without operator convexity. As applications, we obtain inequalities among power means.. Introduction We recall some notations and definitions. Let be a locally compact Hausdorff space and let A be a C -algebra of operators on some Hilbert space H. We say that a field x t t of operators in A is continuous if the function t x t is norm continuous on. If in addition µ is a Radon measure on and the function t x t is integrable, then we can form the Bochner integral x t dµt, which is the unique element in A such that φ x t dµt = φx t dµt for every linear functional φ in the norm dual A. Assume furthermore that there is a field Φ t t of positive linear mappings Φ t : A B from A to another C -algebra B of operators on a Hilbert space K. We recall that a linear mapping Φ t : A B is said to be a positive mapping if Φ t x t 0 for all x t 0. We say that such a field is continuous if the function t Φ t x is continuous for every x A. Let the C -algebras include the identity operators and the field t ϕ t H be integrable with ϕ t H dµt = k K for some positive scalar k. Let BH be the C -algebra of all bounded linear operators on a Hilbert space H. We define bounds of a self-adjoint operator x BH by m x = inf xξ, ξ and M x = sup xξ, ξ ξ = ξ = 200 Mathematics Subject Classification. Primary 47A63; Secondary 47B5, 47A64 Keywords. Quasi-arithmetic mean, power mean, self-adjoint operator, positive linear mapping, convex function Received: 0 February 20; Accepted: 2 March 202 Communicated by Qamrul Hasan Ansari and Ljubiša D.R. Kočinac Research supported by the Croatian Ministry of Science, Education, and Sports, under Research Grants addresses: jmicic@fsb.hr Jadranka Mićić, khot@tvz.hr Kemal Hot

2 J. Mićić, K. Hot / Filomat 26:5 202, for ξ H. If Spx denotes the spectrum of x, then Spx [m x, M x ]. For an operator x BH we define operators x, x +, x by x = x x /2, x + = x + x/2, x = x x/2. Obviously, if x is self-adjoint, then x = x 2 /2 and x +, x 0 called positive and negative parts of x = x + x. We define a generalized quasi-arithmetic operator mean by M φ M φ x, Φ := φ Φ t φxt dµt, 2 k where x t t is a bounded continuous field of self-adjoint operators in a C -algebra BH with spectra in [m, M] for some scalars m < M, Φ t t is a field of positive linear mappings Φ t : BH BK, such that Φ t H dµt = k H for some positive scalar k and φ C[m, M] is a strictly monotone function. As a special case of the quasi-arithmetic mean 2 we can study the operator power mean /r M [r] x, Φ := k Φ t x r t dµt, r R\{0}, 3 exp k Φ t lnx t dµt, r = 0, where x t t is a bounded continuous field of strictly positive operators in a C -algebra BH, Φ t t is a field of positive linear mappings Φ t : BH BK, such that Φ t H dµt = k K for some k > 0. he first result on studying some inequalities such as the Jensen inequality without operator convexity is obtained in [0]. In [2] some techniques are used while one manipulates some inequalities related to continuous fields of operators. here is extensive literature devoted to quasi-arithmetic means, see, e.g. [] [5], [7] [9], [9], [20]. In this paper we present inequalities with operator convexity and without operator convexity of functions which induce means. Also, we present a general formulation of converse inequalities in each of these cases. Furthermore, we obtain refined inequalities without operator convexity. As applications, we obtain inequalities among power means. hese results are generalizations of our previous results obtained in a series of articles [0,, 3, 4, 6, 7]. he interested reader will find in [2, 5] extensions of inequalities among quasi-arithmetic means in the discrete case = {,..., n}. 2. Inequalities with operator convexity We recall some classical results about quasi-arithmetic means. A result with the monotonicity is given in the next theorem. heorem 2.. [7, heorem 2. and heorem 2.3] Let x t t, Φ t t be as in the definition of the quasi-arithmetic mean 2. Let ψ, φ C[m, M] be strictly monotone functions. If one of the following conditions i ψ φ is operator convex and ψ is operator monotone, i ψ φ is operator concave and ψ is operator monotone, ii φ is operator convex and ψ is operator concave, is satisfied then M φ x, Φ M ψ x, Φ. But, if one of the following conditions iii ψ φ is operator concave and ψ is operator monotone, iii ψ φ is operator convex and ψ is operator monotone, iv φ is operator concave and ψ is operator convex, is satisfied then the reverse inequality is valid in 4. 4

3 J. Mićić, K. Hot / Filomat 26:5 202, Proof. We shall give the proof for the convenience of the reader. We will prove only the case i. We put f = ψ φ in the generalized Jensen s inequality [6, heoem 2.] f ϕ t x t dµt ϕ t f x t dµt, k k thereafter replace x t with φx t and finally we apply the operator monotone function ψ. Next we give inequalities complementary to the ones in heorem 2.. In the following theorem we give a general result. We obtain some better bounds than the ones given in [7, heorem 3.]. heorem 2.2. Let x t t, Φ t t, m and M be as in the definition of the quasi-arithmetic mean 2. Let ψ, φ C[m, M] be strictly monotone functions and F : [m, M] [m, M] R be a bounded and operator monotone function in its first variable. Let m φ and and M φ, m φ < M φ, are bounds of the mean M φ x, Φ. 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 F [ M ψ x, Φ, M φ x, Φ ] 5 [ ] φm φz sup F ψ φz φm ψm + m φ z M φ φm φm φm φm ψm, z K [ ] φm φz sup F ψ φz φm ψm + m z M φm φm φm φm ψm, z K. But, if one of the following conditions ii ψ φ is concave and ψ is operator monotone, ii ψ φ is convex and ψ is operator monotone, is satisfied then the reverse inequality is valid in 5 with inf instead of sup. Proof. We will prove only the case i. Replacing z by φz and f by ψ φ in the inequality f z M z z M M m f m + M m f M, z [m, M], and then using the functional calculus and operator monotonicity of ψ we obtain M ψ ψ φmk φm φ φm φm ψm + φm φ φm K φm φm ψm. 6 Now, by using operator monotonicity of F, v and m K m φ K M φ M φ K M K, we obtain F [ ] M ψ, M φ [ φmk φm F ψ φ ψm + φm ] φ φm K ψm, M φ φm φm φm φm [ ] φm φz sup F ψ φz φm ψm + m φ z M φ φm φm φm φm ψm, z K, which give the desired sequence of inequalities 5. If we put Fu, v = u v and Fu, v = v /2 uv /2 v > 0 in heorem 2.2, then we obtain the difference and the ratio type inequalities among quasi-arithmetic means as follows.

4 J. Mićić, K. Hot / Filomat 26:5 202, Corollary 2.3. Let the assumptions be as in heorem 2.2. If one of the conditions i or i in heorem 2.2 is satisfied then { } φm φz M ψ x, Φ M φ x, Φ + max ψ φz φm ψm + m φ z M φ φm φm φm φm ψm z K. 7 If in addition ψ > 0 on [m ψ, M ψ ], then ψ φm φz φz φm φm φm ψm + M ψ x, Φ max m φ z M φ z φm φm ψm M φx, Φ. 8 But, if one of the conditions i or i in heorem 2.2 is satisfied then the reverse inequalities are valid in 7 and 8 with min instead of max. If we put φt = t r and ψt = t s or φt = t s and ψt = t r in heorem 2. we obtain the monotonicity of power means. Corollary 2.4. Let x t t, Φ t t be as in the definition of the power mean 3. It r s, r,, s, or /2 r s or r s /2 then M [r] x, Φ M [s] x, Φ. Further, putting power functions in 8 we obtain the following ratio type inequalities among power means. Corollary 2.5. Let x t t, Φ t t be as in the definition of the power mean 3. Let r, s R, r s and rs 0 and m [r] and and M [r], m [r] < M [r], are bounds of the mean M [r] x, Φ. i If r s, s,, r, or /2 r s or r s /2 then h [r], r, s M [s] x, Φ M [r] x, Φ M [s] x, Φ. ii If s, < r < /2, r 0 or r, /2 < s <, s 0 then h [r], r, s M [s] x, Φ M [r] x, Φ h [r], r, sm [s] x, Φ. iii If s r s, r 0 or r s r/2 < 0 then h [r], r, h [r], r, s M [s] x, Φ M [r] x, Φ h [r], r, M [s] x, Φ. iv If /2 r/2 < s < r, s 0 then h [r], s, h [r], r, s M [s] x, Φ M [r] x, Φ h [r], s, M [s] x, Φ, where h [r] = M [r] /h [r] and h, r, s is a generalized Specht ratio defined by h, r, s := { rh s h r } s s rh r { } sh r h s r r sh s, h > 0. he proof is similar to [4, heorem 4.4] and we omit it. In the same way, we can obtain the difference type inequality for power means. Remark 2.6. he bounds given in Corollary 2.4 are better than the corresponding bounds given in [4, heorem 4.4] for s 0. We give the short proof. Since Kh, p := h p h p h p p h p h p h p = max m t M M t M m Mp + t m M m mp t p, h = M m

5 J. Mićić, K. Hot / Filomat 26:5 202, and m m [r] M [r] M, then Kh [r], p Kh, p. aking into account that Kh, p for p 0, it follows Kh [r], s r /s Kh, s r /s for s 0. Replacing h [r] and h in the above inequality by h [r] r and h r, respectively, and taking into account that h, r, s := Kh, r s /s we obtain h [r], r, s h, r, s. hen for s 0 in the case i and ii of Corollary 2.4 we have h, r, s M [s] x, Φ h [r], r, s M [s] x, Φ M [r] x, Φ M [s] x, Φ and h, r, s M [s] x, Φ h [r], r, s M [s] x, Φ M [r] x, Φ h [r], r, sm [s] x, Φ h, r, sm [s] x, Φ, respectively. Similarly, we can evaluate bounds in the other cases of Corollary Inequalities without operator convexity In this section we give inequalities among quasi-arithmetic operator means with conditions on spectra of the operators. In the next theorem we obtain the monotonicity of among quasi-arithmetic means without operator convexity in heorem 2.. his is a generalization of [0, heorem 3]. heorem 3.. Let x t t, Φ t t be as in the definition of the quasi-arithmetic mean 2. Let m t and M t, m t M t are bounds of x t, t. Let φ, ψ : I R be continuous strictly monotone functions on an interval I which contains all m t, M t. Let mφ, M φ [mt, M t ] = Ø, t, where m φ and and M φ, m φ M φ, are bounds of the mean M φ x, Φ. 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 M φ x, Φ M ψ x, Φ. 9 But, if one of the following conditions ii ψ φ is concave and ψ is operator monotone, ii ψ φ is convex and ψ is operator monotone is satisfied, then the reverse inequality is valid in 9. Proof. We will prove only the case i. Let m φ < M φ. Since Sp Φ tφx t dµt φ[m φ, M φ ] and ψ φ is convex on φ[m φ, M φ ], then by using the functional calculus we obtain = ψ φ φm φ K k Φ t k Φ tφx t dµt k Φ tφx t dµt k Φ tφx t dµt φm φ K φm φ φm φ ψm φ + φmφ φx t φm φ φm φ ψm φ + φx t φm φ φm φ φm φ ψm φ φm φ φm φ ψm φ dµt. 0

6 J. Mićić, K. Hot / Filomat 26:5 202, On the other hand, since φ is monotone, then mφ, M φ [mt, M t ] = Ø = φm φ, M φ φ[m t, M t ] = Ø, t. By using the above condition and that m t H x t M t H, t, then it follows ψ φ φx t φm φ H φx t φm φ φm φ ψ φ φm φ + φx t φm φ H φm φ φm φ f φm φ for t. Applying a positive linear mapping Φ t and integrating, we obtain k Φ tψ φ φx t dµt = k Φ tψx t dµt k Φ φmφ φx t t φm φ φm φ ψm φ + φx t φm φ φm φ φm φ ψm φ dµt. Combining the two inequalities 0 and, and applying operator monotonicity of ψ we obtain the desired inequality 9. In the case m φ = M φ we use supporting line of a convex functions ψ φ in the point z = φm φ and similarly as above we obtain 9. An example of the condition of spectra in the discrete case is shown in Figure.b. m m 2 m M M M 2 m m M m 2 M 2 M a b Figure : Spectral conditions for a convex function and = {, 2} It is interesting to study the case when 9 holds only under the condition placed on the bounds of operators whose means we are considering. We give it in the next corollary. Corollary 3.2. Let x t t, Φ t t be as in the definition of the quasi-arithmetic mean 2. Let m t and M t, m t M t are bounds of x t, t and let φ, ψ : I R be continuous strictly monotone functions on an interval I which contains all m t, M t. Let m x, M x [m t, M t ] = Ø, t, where m x and M x, m x M x, are the bounds of the operator x = Φ tx t dµt. If one of the following conditions i φ is convex, φ is operator monotone, ψ is concave, ψ is operator monotone, ii φ is convex, φ is operator monotone, ψ is convex, ψ is operator monotone, iii φ is concave, φ is operator monotone, ψ is convex, ψ is operator monotone, iv φ is concave, φ is operator monotone, ψ is concave, ψ is operator monotone is satisfied, then M φ x, Φ M ψ x, Φ. 2 But, if one of the following conditions i φ is convex, φ is operator monotone, ψ is concave, ψ is operator monotone,

7 J. Mićić, K. Hot / Filomat 26:5 202, ii φ is convex, φ is operator monotone, ψ is convex, ψ is operator monotone, iii φ is concave, φ is operator monotone, ψ is convex, ψ is operator monotone, iv φ is concave, φ is operator monotone, ψ is concave, ψ is operator monotone is satisfied, then the reverse inequality is valid in 2. Proof. We will prove only the case i. Replacing φ by the identity function I in heorem 3.i and replacing φ by I and ψ by φ in heorem 3.ii, we obtain 2. Using the condition on spectra we obtain the following generalization of heorem 2.2. heorem 3.3. Let x t t, Φ t t be as in the definition of the quasi-arithmetic mean 2. Let m t and M t, m t M t are bounds of x t, t. Let φ, ψ : [m, M] R be continuous strictly monotone functions, where m = inf {m t} and t M = sup{m t }. Let t mφ, M φ [mt, M t ] = Ø, t, where m φ and M φ, m φ < M φ, are bounds of the mean M φ x, Φ. Let : φ[m φ, M φ ] R and F : [m, M] V R be functions such that φ[m φ, M φ ] V and F be bounded and operator monotone function in its first variable. 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 [ inf F ψ φmφ φz m φ z M φ φm φ φm φ ψm φ + φz φm φ φm φ φm φ ψm φ F M ψx, Φ, k Φ tφx t dµt sup F [ ψ φm φz φm φm m φ z M φ ], φz K ψm + φz φm φm φm ψm, φz ] K. But, if one of the following conditions ii ψ φ is concave and ψ is operator monotone, ii ψ φ is convex and ψ is operator monotone is satisfied, then the reverse inequalities are valid in 3 with replace sup and inf by inf and sup, respectively. 3 Proof. We will prove only the case i. By using the inequality see 6 in the proof of heorem 2.2 M ψ ψ φmk φm φ φm φm ψm + φm φ φm K φm φm ψm and operator monotonicity of F, v we obtain RHS of 3. Applying an operator monotone function ψ on the inequality see in the proof of heorem 3. φmφ k Φ φx t t φm φ φm φ ψm φ + φx t φm φ φm φ φm φ ψm φ dµt ψ M ψ, we obtain ψ φmφ K φm φ φm φ φm φ ψm φ + φm φ φm φ K φm φ φm φ By using operator monotonicity of F, v we obtain LHS of 3. ψm φ M ψ. Putting φ and Fu, v = u αv or Fu, v = v /2 uv /2, we obtain the next corollary.

8 J. Mićić, K. Hot / Filomat 26:5 202, Corollary 3.4. Let x t t, Φ t t be as in the definition of the quasi-arithmetic mean 2. Let m t and M t, m t M t are bounds of x t, t and φ, ψ : [m, M] R be continuous strictly monotone functions, where m = inf t {m t } and M = sup t {M t }. Let mφ, M φ [mt, M t ] = Ø, t, where m φ and M φ, m φ < M φ, are bounds of the mean M φ x, Φ. 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 for any real number α the following sequence of inequalities { } min ψ φmφ φz m φ z M φm φ φm φ ψm φ + φz φm φ φm φ φm φ ψm φ αz K φ M ψ x, Φ αm φ x, Φ 4 { max ψ φm φz φz φm m φ z M φm φm ψm + φm φm ψm αz } K. φ If in addition ψ > 0 on [m ψ, M ψ ], then min m φ z M φ M ψ x, Φ ψ φmφ φz φmφ φmφ ψm φ + φz φmφ φmφ φmφ ψm φ max m φ z M φ z { ψ φm φz φm φm ψm+ φz φm z φm φm ψm M φx, Φ 5 } M φ x, Φ. But, if one of the following conditions ii ψ φ is concave and ψ is operator monotone, ii ψ φ is convex and ψ is operator monotone is satisfied, then for any real number α the reverse inequalities are valid in 4 and 5 for ψ > 0 with replace sup and inf by inf and sup, respectively. If we put φt = t r and ψt = t s in heorem 3. and Corollary 3.4 we obtain the order among power means as follows. hese results are a generalization of [0, Corollary 7] and [, heorem ] /2 /2 r=s a WIHOU CONDIION ON SPECRA: =, for r,s in, 2, 3 s /r = Kh,r/s, for r,s in 4, 5 -/r = Kh,r, for r,s in 6 -/s = Kh,s, for r,s in 7 b WIH CONDIION ON SPECRA: =, for r,s in, 2, 4 or, 3, 5 [r] -/s [s] -/r = Kh,s or = Kh,r, for r,s in 6, 7 Figure 2: Regions for the inequality M [r] A, Φ M [s] A, Φ Corollary 3.5. Let x t t, Φ t t be as in the definition of the power mean 3. Let m t and M t, 0 < m t M t be the bounds of x t, t. If one of the following conditions

9 J. Mićić, K. Hot / Filomat 26:5 202, i r s, s or r s Figure 2.b 4 and m [r], M [r] [m t, M t ] = Ø, t, 6 where m [r] and M [r], m [r] M [r] are the bounds of M [r] x, Φ, ii r s, r or r s Figure 2.b 3,5 and m [s], M [s] [m t, M t ] = Ø, t, 7 where m [s] and M [s], m [s] M [s] are the bounds of M [s] x, Φ is satisfied, then M [r] x, Φ M [s] x, Φ. 8 iii If r, s,, r s Figure 2. 6,7 and 6 is valid, then M [r] x, Φ h [r], s M [s] x, Φ, h [r] = M [r] /m [r]. 9 iv If r, s,, r s Figure 2. 6,7 and 7 is valid, then M [r] x, Φ h [s], r M [s] x, Φ, h [s] = M [s] /m [s]. 20 A constant h, p h, p, is defined as follows where h = M/m. h, p := ph h p ph p p h p h p h, h and p 0, h h h e ln h, h and p = 0,, h = and p R, Proof. he proof is quite similar to the ones [0, Corollary 7] and [, heorem ]. We give it for the convenience of the reader. i: We put φt = t r and ψt = t s for t > 0. hen ψ φ t = t s/r is concave for r s, s 0 and r 0. Since ψ t = t /s is operator monotone for s and m [r], M [r] [m t, M t ] = Ø is satisfied, then by applying heorem 3.-i we obtain 8 for r s. But, ψ φ t = t s/r is convex for r s, s 0 and r 0. Since ψ t = t /s is operator monotone for s, then by applying heorem 3.-i we obtain 8 for r s, s, r 0. If r = 0 and s, we put φt = ln t and ψt = t s, t > 0. Since ψ φ t = expst is convex, then similarly as above we obtain the desired inequality. ii: iii a: We put φt = t s and ψt = t r for t > 0 and we use the same technique as in the case i. Let m [r] < M [r]. Suppose that 0 < r s. hen m [r], M [r] [m t, M t ] = Ø = m [r] r, M [r] r [m r t, Mr t ] = Ø, t. Putting f t = t s/r, which is convex, in the two inequalities 0 and, we obtain s/r Φ t x r t dµt Φ t x s t dµt.

10 J. Mićić, K. Hot / Filomat 26:5 202, Now, applying results about the function order see [8, Corollary 6.5] for the power function t t p, p = s, we obtain the desired inequality 9. the remaining the remaining combinations for r, s, we use the same technique as above. iii b: If m [r] = M [r], we put m [r] M [r] in inequalities iii a, and we use that lim h Ch, p = for p and lim h Ch, 0 =. iv: We use the same technique as in the case iii. 4. Refined inequalities In this section we give a refinement of the inequality 9. For convenience we introduce the following denotations: δ φ,ψ m, M = ψm + ψm 2ψ φ φm+φm 2, x φ m, M = 2 K φm φm k Φ t φxt dµt φm+φm 2 K, 2 where x t t is a bounded continuous field of operators in a C -algebra BH with spectra in I, Φ t t is a field of positive linear mappings Φ t : BH BK, such that Φ t K dµt = k K for some positive scalar k, φ, ψ : I R are continuous strictly monotone functions and m, M I, m < M. Of course, we include implicitly that x φ m, M x φ,x m, M, where x = Φ t φxt dµt. o obtain our main result we need the following result. Lemma 4.. [4, Lemma 2] Let x be a self-adjoint element in BH with Spx [m, M], for some scalars m < M. hen f x M H x M m f m + x m H M m f M δ f x 22 holds for every continuous convex function f : [m, M] R, where m + M δ f = f m + f M 2 f 2 If f is concave, then the reverse inequality is valid in 22. and x = 2 H he next theorem is a generalization of [4, heorem 7]. M m x m + M H. 2 heorem 4.2. Let x t t and Φ t t be as in the definition of the quasi-arithmetic mean 2. Let m t and M t, m t M t be the bounds of x t, t. Let φ, ψ : I R be continuous strictly monotone functions on an interval I which contains all m t, M t. Let mφ, M φ [mt, M t ] = Ø, t, and m < M, where m φ and and M φ, m φ M φ, are bounds of the mean M φ x, Φ and m = sup { M t : M t m φ, t }, M = inf { m t : m t M φ, t }. i If ψ φ is convex and ψ is operator monotone, then M φ x, Φ ψ k Φ t ψxt dµt δ φ,ψ x φ M ψ x, Φ, 23 where δ φ,ψ 0 and x φ 0. i If ψ φ is convex and ψ is operator monotone, then the reverse inequality is valid in 23, where δ φ,ψ 0 and x φ 0.

11 J. Mićić, K. Hot / Filomat 26:5 202, ii If ψ φ is concave and ψ is operator monotone, then 23 holds, where δ φ,ψ 0 and x φ 0. ii If ψ φ is concave and ψ is operator monotone, then the reverse inequality is valid in 23, where δ φ,ψ 0 and x φ 0. In all the above cases, we assume that δ φ,ψ δ φ,ψ m, M, x φ x φ m, M are defined by 2 and m [m, m φ ], M [M φ, M], m < M, are arbitrary numbers. Proof. he proof of heorem 4.2 is similar to the ones of heorem 3., but in addition we use Lemma C. We only prove the case i. Putting a convex function f ψ φ on φ[ m, M] and x = k Φ tφx t dµt = φm φ in 22 we obtain ψm φ φ M K φm φ φ M φ m ψ m + φm φ φ m K ψ M δ φ,ψ x φ, 24 φ M φ m where δ φ,ψ and x φ are defined by 2. On the other hand, since mφ, M φ [mt, M t ] = Ø = m, M [m t, M t ] = Ø = = φ m, M φ [m t, M t ] = Ø, t, then ψx t = ψ φ φx t φ M H φx t ψ m + φx t φ m H φ M φ m φ M φ m ψ M. Applying a positive linear mapping Φ t, integrating and adding δ φ,ψ x φ, we obtain k Φ i ψxt dµt δ φ,ψ x φ φ M K φm φ φ M φ m ψ m + φm φ φ m K φ M φ m ψ M δ φ,ψ x φ. 25 Combining the two inequalities 24 and 25, and considering that δ φ,ψ 0 and x φ 0 holds, we obtain ψm φ ψm ψ δ φ,ψ x φ ψm ψ. Applying the operator monotone function ψ we obtain 23. x 2 x 2 x 2 x 2 x x, 2 2 x~, m m M =m m M m 2 =M M 2 M w he re M m, m M 2, 2 x~ K 2 M m M m 2 x 2 x 2 K Figure 3: Refinement for two operators and a convex function ψ φ Example 4.3. We give a small example of a refined inequality among means for the matrix cases and = {, 2} see Figure 3. We put φt = t /3, ψt = t 5 and we define Φ, Φ 2 : M 2 C M 2 C by Φ B = Φ 2 B = 2 B for B M 2C.

12 If x = J. Mićić, K. Hot / Filomat 26:5 202, and x 2 = 25, then 0 M [/3] = 2 3 x x 2 3 = M [5] = 5 2 x5 + 2 x5 2 = , and m = , M = , m 2 = M 2 = 25, m /3 = and M /3 = rounded to five decimal places. It follows that m = , M = 25, m [7.9443, ] and M [ , 25]. Let m = , M = hen we get δ /3,5 = , x /3 = and M [/3] < 5 2 x5 + 2 x5 2 δ /3,5 x /3 = < M [5]. Putting the identity function in heorem 4.2 we obtain another refinement of 9. Corollary 4.4. Let x t t and Φ t t be as in the definition of the quasi-arithmetic mean 2. Let m t and M t, m t M t be the bounds of x t, t. Let φ, ψ : I R be continuous strictly monotone functions on an interval I which contains all m t, M t, such that φ is convex, ψ is concave. We denote I the identity function on I. If mφ, M φ [mt, M t ] = Ø, t, and m [φ] < M [φ], mψ, M ψ [mt, M t ] = Ø, t, and m [ψ] < M [ψ] are valid, where m φ and M φ, m φ M φ are the bounds of M φ x, Φ and m [φ] = sup { M t : M t m φ, t }, M [φ] = inf { m t : m t M φ, t } and analogously for ψ, then the following inequality M φ x, Φ M φ x, Φ + φ,ψ m, M, m, M M ψ x, Φ, 26 for every m [m [φ], m φ ], M [M φ, M [φ] ], m < M and every m [m [ψ],mψ ], M [M ψ, M [ψ] ], m < M, where and δ φ,i, δ ψ,i, x φ and x ψ are defined by 2. φ,ψ m, M, m, M = δ φ,i m, M x φ m, M δ ψ,i m, M x ψ m, M 0 Proof. Putting ψ = I in heorem 4.2 i, we obtain M φ x, Φ M I x, Φ δ φ,i m, M x φ m, M M I x, Φ, 27 where δ φ,i m, M 0. Putting ψ = I and replacing φ by ψ in heorem 4.2 ii, we obtain M ψ x, Φ M I x, Φ δ ψ,i m, M x ψ m, M M I x, Φ, 28 where δ ψ,i m, M 0. Adding in 27 and taking into account 28, we obtain 26. δ φ,i m, M x φ m, M δ ψ,i m, M x ψ m, M

13 J. Mićić, K. Hot / Filomat 26:5 202, As an application of results given in heorem 4.2 we study a refinement of inequality 8. For convenience we introduce denotations as a special case of 2 as follows m δ r,s m, M = s + M s 2 m r +M r s/r 2, r 0, x r m, M = m s + M s 2 mm s/2, r = 0, 2 K M r m r Φ tx r t dµ Mr +m r 2 K, r 0, 2 K ln M m Φ tln x t dµ ln Mm K, r = 0, where m, M R, 0 < m < M and r, s R, r s. Of course, we include implicitly that x r m, M x r,x m, M, where x = Φ tx r t dµ for r 0 and x = Φ tln x t dµ for r = 0. he next corollary is a generalization of [0, Corollary 2]. Corollary 4.5. Let x t t, Φ t t be as in the definition of the power mean 3. Let m t and M t, 0 < m t M t be the bounds of x t, t. i If r s, s or r s, m [r], M [r] [m t, M t ] = Ø, t, and m < M, 29 where m [r] and M [r], m [r] M [r] are the bounds of M [r] x, Φ and m = sup { M t : M t m [r], t }, M = inf { m t : m t M [r], t }, then /s M [r] x, Φ Φ t x s t dµ δr,s x r M [s] x, Φ, 30 where δ r,s 0 for s, δ r,s 0 for s and x r 0. Here we assume that δ r,s δ r,s m, M, x r x r m, M are defined by 29 and m [m, m [r] ], M [M [r], M], m < M, are arbitrary numbers. ii If r s, r or r s, m [s], M [s] [m t, M t ] = Ø, t, and m < M, where m [s] and M [s], m [s] M [s] are the bounds of M [s] x, Φ and m = sup { M t : M t m [s], t }, M = inf { m t : m t M [s], t }, then /r M [r] x, Φ Φ t x r t dµ δs,r x s M [s] x, Φ, where δ s,r 0 for r, δ s,r 0 for r and x s 0. Here we assume that δ s,r δ s,r m, M, x s x s m, M are defined by 29 and m [m, m [s] ], M [M [s], M], m < M, are arbitrary numbers. In the proof we use the same technique as in the proof of Corollary 3.5. We omit it. Figure 4 shows regions,2,4,6,7 in which the monotonicity of the power mean holds true see Corollary 2.4, also Figure 4 shows regions -7 which this holds true with condition on spectra see Corollary 3.5. We show in [0, Example 2] that the order among power means does not hold generally without a condition on spectra in regions 3,5. Now, by using Corollary 4.5 we obtain a refined inequality in the regions 2-6 see Corollary 4.6. Corollary 4.6. Let x t t, Φ t t be as in the definition of the power mean 3. Let m t and M t, 0 < m t M t be the bounds of x t, t. Let m [r], M [r] [m t, M t ] = Ø, t, m [r] < M [r], m [s], M [s] [m t, M t ] = Ø, t, m [s] < M [s],

14 J. Mićić, K. Hot / Filomat 26:5 202, M [r] x, [s] M x, in, 2, 4, 6, 7 WIHOU CONDIION ON SPECRA /2 /2 M [r] x, [r] [s] M x, + r,s, x M A, in 2, 3, 4 or 4, 5, 6 WIH CONDIION ON SPECRA Figure 4: Regions describing inequalities among power means where m [r], M [r], m [r] M [r] and m [s], M [s], m [s] M [s] are the bounds of M [r] x, Φ and M [s] x, Φ, respectively, and m [r] = max { M t : M t m [r], t }, M [r] = min { m t : m t M [r], t }, m [s] = max { M t : M t m [s], t }, M [s] = min { m t : m t M [s], t }. Let m [m [r], m [r] ], M [M [r], M [r] ], m < M, and m [m [s], m [s] ], M [M [s], M [s] ], m < M be arbitrary numbers. If r s, then M [r] x, Φ M [r] x, Φ + δ r, m, M x r m, M δ s, m, M x s m, M M [s] x, Φ. If r s, then M [r] x, Φ M [r] x, Φ + Φ t x t dµ δs, m, M x s m, M Φ t x t dµ δr, m, M x r m, M M [s] x, Φ. If r, s, then M [r] x, Φ M [r] x, Φ + M [] x, Φ δ s, m, M x s m, M Φ t x t dµ δr, m, MÃr m, M M [s] x, Φ. he proof is similar to that of Corollary 4.4 and we omit it. References [] M. Anwar, J. Pečarić, Means of the Cauchy ype, LAP Lambert Academic Publishing, [2] J.I. Fujii, Path of quasi-means as a geodesic, Linear Algebra Appl [3] J.I. Fujii, M. Nakamura, S.-E. akahasi, Coopers approach to chaotic operator means, Sci. Math. Jpn [4]. Furuta, J. Mićić Hot, J. Pečarić, Y. Seo, Mond-Pečarić method in Operator Inequalities, Monographs in Inequalities, Element, Zagreb, [5] J. Haluska, O. Hutnik, Some inequalities involving integral means, atra Mt. Math. Publs [6] F. Hansen, J. Pečarić, I. Perić, Jensen s operator inequality and its converses, Math. Scand [7] A. Kolesárová, Limit properties of quasi-arithmetic means, Fuzzy Sets and Systems [8] F. Kubo,. Ando, Means of positive linear operators, Math. Ann [9] J.-L. Marichal, On an axiomatization of the quasi-arithmetic mean values without the symmetry axiom, Aequationes Math

15 J. Mićić, K. Hot / Filomat 26:5 202, [0] J. Mićić, Z. Pavić, J. Pečarić, Jensen s inequality for operators without operator convexity, Linear Algebra Appl [] J. Mićić, Z. Pavić and J. Pečarić, Order among power operator means with condition on spectra, Math. Inequal. Appl [2] J. Mićić, Z. Pavić, J. Pečarić, Extension of Jensen s operator inequality for operators without operator convexity, Abstr. Appl. Anal [3] J. Mićić, Z. Pavić, J. Pečarić, Some better bounds in converses of the Jensen operator inequality, Oper. Matrices, to appear. [4] J. Mićić, J. Pečarić, J. Perić, Refined Jensen s operator inequality with condition on spectra, Oper. Matrices, to appear. [5] J. Mićić, J. Pečarić, J. Perić, Extension of the refined Jensen s operator inequality with condition on spectra, Ann. Funct. Anal., to appear. [6] J. Mićić, J. Pečarić, Y. Seo, Converses of Jensen s operator inequality, Oper. Matrices [7] J. Mićić, J. Pečarić, Y. Seo, Order among quasi-arithmetic means of positive operators, Math. Reports 3 202, in print. [8] J.Mićić, J.Pečarić, Y.Seo, Function order of positive operators based on the Mond-Pečarić method, Linear Algebra Appl [9] D.S. Mitrinović, J. Pečarić, A M. Fink, Classical and new Inequalities in Analysis, Kluwer Acad. Publ., Dordrecht-Boston-London, 993. [20] D.S. Mitrinović with P.M. Vasić, Analytic Inequalities, Springer-Verlag, Berlin, 970. [2] M.S. Moslehian, An operator extension of the parallelogram law and related norm inequalities, Math. Inequal. Appl