JENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES FOR OPERATORS IN HILBERT SPACE

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1 JENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES FOR OPERATORS IN HILBERT SPACE MARIO KRNIĆ NEDA LOVRIČEVIĆ AND JOSIP PEČARIĆ Abstract. In this paper we consider Jensen s operator which includes bounded self-adjoint operator on Hilbert space and investigate its properties. Due to derived properties we find lower and upper bound for Jensen s operator which are multiples of non-weighted Jensen s operator. On the other hand we also establish some bounds for spectra of Jensen s operator by means of discrete Jensen s functional. The obtained results are then applied to operator means. In such a way we get refinements and conversions of numerous mean inequalities for Hilbert space operators. 1. Introduction Let H be a Hilbert space and let B h H be the semi-space of all bounded selfadjoint operators on H. Besides let B + H and B ++ H respectively denote the sets of all positive and positive invertible operators in B h H. The weighted operator harmonic mean! µ geometric mean µ and arithmetic mean µ for µ 0 1 and A B B ++ H are defined as follows: 1 A! µ B = 1 µa 1 + µb A µ B = A 1 2 A 1 2 BA 1 2 µa A µ B = 1 µa + µb. If µ = 1/2 we write A! B A B A B for brevity respectively. The above definitions and notations will be valid throughout the whole paper. It is well known the arithmetic-geometric-harmonic mean inequality 4 A! µ B A µ B A µ B µ 0 1 with respect to operator order. Such mean inequalities for Hilbert space operators lie in the fields of interest of numerous mathematicians and we refer here to some recent results. Inspired by Furuichi s refinement of arithmetic-geometric mean see 4 5 A µ B A µ B 2 min{µ 1 µ} A B A B µ Mathematics Subject Classification. Primary 47A63 Secondary 26D15 Key words and phrases. Jensen s inequality Jensen s functional Jensen s operator Hilbert space bounded self-adjoint operator positive invertible operator arithmetic operator mean geometric operator mean harmonic operator mean superadditivity monotonicity refinement conversion Kantorovich constant. 1

2 2 MARIO KRNIĆ NEDA LOVRIČEVIĆ AND JOSIP PEČARIĆ Zuo et. al. 15 also obtained refinement of arithmetic-harmonic mean in difference form. More precisely they have bounded the difference between weighted arithmetic and harmonic mean with the difference of non-weighted means that is 6 A µ B A! µ B 2 min{µ 1 µ} A B A! B µ 0 1 where A B B ++ H. Recently Kittaneh et. al. 9 obtained the following refinement and conversion of arithmetic-geometric operator mean inequality 7 2 max{p 1 p 2 } A B C C 1 BC C A p 1 B C C 1 BC 2 min{p 1 p 2 } 1 p1 C A B C C 1 BC C where A B B ++ H C B 1 H A = C C and p = p 1 p 2 R 2 +. Here and throughout the whole paper B 1 H denotes the set of invertible operators on Hilbert space H. Note that operator C C 1 BC 1 µ C represents generalization of geometric mean since for C = A 1 2 it becomes A µ B µ 0 1. In a similar manner Zuo et. al. 15 also improved arithmetic-geometric mean inequality via Kantorovich constant. More precisely they proved that min{µ1 µ} M 8 A µ B K m 2 A µ B µ 0 1 where A B are positive operators satisfying 9 0 < m 1 H A m1 H < M1 H B M 1 H or 0 < m 1 H B m1 H < M1 H A M 1 H Kt 2 is the well known Kantorovich constant i.e. Kt 2 = t+1 2 /4t t > 0 and 1 H is an identity operator on Hilbert space. Note also that inequality 8 improves Furuichi s result from 5 which includes the well known Specht s ratio instead of Kantorovich constant. The similar problem area as presented here was also considered in papers and 14. In addition for a comprehensive inspection of the recent results about inequalities for bounded self-adjoint operators on Hilbert space the reader is referred to 6. On the other hand it is well known that all inequalities between classical means can be established by means of one of the most important inequality in mathematical analysis that is Jensen s inequality. Furthermore Dragomir et al. see 1 investigated the properties of discrete Jensen s functional n n i=1 10 J n f x p = p i fx i P n f p ix i i=1 where f : I R R is a convex function x = x 1 x 2... x n I n n 2 and p = p 1 p 2... p n is positive n tuple of real numbers with P n = n i=1 p i. They obtained that such functional is superadditive on the set of positive real n tuples that is 11 J n f x p + q J n f x p + J n f x q. P n

3 JENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES... 3 Further above functional is also increasing in the same setting that is 12 J n f x p J n f x q 0 where p q i.e. p i q i i = n. Monotonicity property of discrete Jensen s functional was proved few years before see 13 p.717. Above mentioned properties provided improvements of numerous classical inequalities. For more details about such extensions see 1. Guided by latter idea in this paper we shall establish Jensen s operator which includes self-adjoint operator on Hilbert space and investigate its properties. We are going to provide the properties of superadditivity and monotonicity to hold more generally by analyzing such operator in above mentioned setting. Moreover we shall obtain all operator inequalities presented in this Introduction as a simple consequences of more general results. The paper is organized in the following way: After Introduction in Section 2 we present some auxiliary results that would be used in the sequel. Further in Section 3 we establish general Jensen s operator deduce its important properties and incorporate them with the results presented in Introduction. Finally in Section 4 we establish lower and upper bounds for spectra of Jensen s operator by means of one interesting property of discrete Jensen s functional concerning its monotonicity. In such a way we get both refinements and conversions of previously known mean inequalities for operators in Hilbert space. The techniques that will be used in the proofs are mainly based on classical real and functional analysis especially on the well known monotonicity property for operator functions. 2. Preliminaries In this short section we point out two important facts that will help us in establishing our general results. First of them deals with the well known monotonicity property for bounded self-adjoint operators on Hilbert space: If X B h H with a spectra SpX then 13 ft gt t SpX = fx gx provided that f and g are real valued continuous functions. For more details about this property and its consequences the reader is referred to 6. The other fact a recent result from 10 is a kind of monotonicity of Jensen s functional observed as a function in one variable. It is a content of the following lemma. Lemma 1. Let f : a b R be a convex function and let δ a b p 0 1 be fixed parameters. Then the function ϕ : a b R defined by satisfies the following properties: i function ϕ is decreasing on a δ; ii function ϕ is increasing on δ b. ϕt = 1 pfδ + pft f1 pδ + pt For the proof of Lemma 1 the reader is reffered to 10.

4 4 MARIO KRNIĆ NEDA LOVRIČEVIĆ AND JOSIP PEČARIĆ 3. Jensen s operator and its properties The starting point in this section is a general definition of Jensen s operator in Hilbert space. Let f : a b R R be a continuous convex function and let F a b R denotes the set of all continuous convex functions on interval a b. We define Jensen s operator J : F a b R B h H a b R 2 + B + H as p1 D + p 2 δ1 H 14 J f D δ p = p 1 fd + p 2 fδ1 H f where p = p 1 p 2 a1 H D b1 H and 1 H denotes identity operator on Hilbert space H. Note that operator J is well-defined. Namely positivity of underlying operator J f D δ p follows from Jensen s inequality and monotonicity property 13 for operator functions. For the sake of simplicity positive operator J f D δ p will also be referred to as Jensen s operator. Now we are ready to state and prove our first result that shows that the properties of superadditivity and monotonicity hold in a more general manner. Theorem 1. Suppose J is an operator defined by 14. Then it satisfies the following properties: i J f D δ is superadditive on R 2 + that is 15 J f D δ p + q J f D δ p + J f D δ q. ii If p q R 2 + with p q i.e. p 1 q 1 p 2 q 2 then 16 J f D δ p J f D δ q 0 i.e. J f D δ is increasing on R 2 +. Proof. Consider Jensen s functional 10 for n = 2 and x = x δ that is p1 x + p 2 δ 17 jf x δ p = p 1 fx + p 2 fδ f. Clearly superadditivity and monotonicity properties 11 and 12 provide inequalities 18 jf x δ p + q jf x δ p + jf x δ q and 19 jf x δ p jf x δ q p q. On the other hand according to monotonicity property 13 for operator functions inequalities 18 and 19 are also valid in Hilbert space if we replace x with an operator D B h H assuming a1 H D b1 H. The proof is now completed since jf D δ p = J f D δ p. Superadditivity and monotonicity properties of Jensen s operator are very important properties considering the numerous applications that will follow from them. First regarding monotonicity property 16 we give the consequence of Theorem 1 which includes the lower and upper bound for operator J f D δ p by means of non-weighted operator.

5 JENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES... 5 Corollary 1. Suppose J is an operator defined by 14. Then 20 2 max{p 1 p 2 }J N f D δ J f D δ p 2 min{p 1 p 2 }J N f D δ where J N f D δ = fd + fδ1 H 2 D + δ1h f 2 Proof. It is natural to compare ordered pair p = p 1 p 2 R 2 + with the constant ordered pairs p max = max{p 1 p 2 } max{p 1 p 2 } and p min = min{p 1 p 2 } min{p 1 p 2 }. Clearly p max p p min hence yet another use of property 16 yields interpolating series of inequalities:. J f D δ p max J f D δ p J f D δ p min. Finally since J f D δ p max = 2 max{p 1 p 2 }J N f D δ and J f D δ p min = 2 min{p 1 p 2 }J N f D δ we get relation 20. Our next consequence of Theorem 1 provides Jensen s operator deduced from 14 which includes several self-adjoint operators on Hilbert space. More precisely if J is Jensen s operator defined by 14 we consider the operator C J f C 1 BC 1 δ p C = p 1 C fc 1 BC 1 C + p 2 fδa C p1 C 1 BC 1 + p 2 δ1 H 21 f C where A B ++ H B B h H C B 1 H A = C C and aa B ba. For the sake of simplicity we are going to use abbreviation C J f C 1 BC 1 δ p C for the operator defined by 21. Remark 1. We easily conclude that the operator C J f C 1 BC 1 δ p C is well defined under above assumptions. Namely condition aa B ba implies a1 H C 1 BC 1 b1 H that is spectra of operator C 1 BC 1 belongs to domain of function f. Our next two results show that deduced operator C J f C 1 BC 1 δ p C possess the same properties as original Jensen s operator defined by 14. Theorem 2. Let A B ++ H B B h H C B 1 H be the operators satisfying A = C C and aa B ba. If J is an operator defined by 14 then the operator C J f C 1 BC 1 δ p C has the following properties: i C J f C 1 BC 1 δ C is superadditive on R 2 + that is 22 C J f C 1 BC 1 δ p + q C C J f C 1 BC 1 δ p C + C J f C 1 BC 1 δ q C. ii If p q R 2 + with p q i.e. p 1 q 1 p 2 q 2 then 23 C J f C 1 BC 1 δ p C C J f C 1 BC 1 δ q C 0 i.e. C J f C 1 BC 1 δ C is increasing on R 2 +.

6 6 MARIO KRNIĆ NEDA LOVRIČEVIĆ AND JOSIP PEČARIĆ Proof. If operator J is defined by 14 then according to Remark 1 the operator J f C 1 BC 1 δ p is well defined. Due to superadditivity property 15 of operator J we conclude that operator 24 J f C 1 BC 1 δ p + q J f C 1 BC 1 δ p J f C 1 BC 1 δ q is positive i.e. belongs to B + H. Now if we multiply operator 24 by C on the left and by C on the right we again get positive operator. Obviously as a consequence we get superadditivity property 22. Monotonicity property 23 is deduced in the same way. Remark 2. If C is the square root of operator A that is C = A 1 2 then operator 21 takes form A 1 2 J f A 1 2 BA 1 2 δ p A 1 2. Of course that operator is also superadditive and increasing on R 2 +. Our next result yields bounds for operator C J f C 1 BC 1 δ p C expressed in terms of non-weighted operator. Let s also mention that such result will enable us to deduce some interpolating inequalities which include operator means presented in Introduction. Corollary 2. Let f : a b R R be continuous convex function and δ a b. Suppose A B ++ H B B h H C B 1 H A = C C and p = p 1 p 2 R 2 +. If aa B ba then 25 where 2 max{p 1 p 2 }C J N f C 1 BC 1 δc C J f C 1 BC 1 δ p C 2 min{p 1 p 2 }C J N f C 1 BC 1 δc J N f C 1 BC 1 δ = fc 1 BC 1 + fδ1 H 2 C 1 BC 1 + δ1 H f 2 and C J f C 1 BC 1 δ p C is defined by 21. In particular 2 max{p 1 p 2 }A 1 2 JN f A 1 2 BA 1 2 δ A A 2 J f A 1 2 BA 1 2 δ p A min{p 1 p 2 }A 1 2 JN f A 1 2 BA 1 2 δ A 1 2. Proof. The proof follows the same lines as the proof of Corollary 1 except we use monotonicity property 23 from Theorem 2 instead of monotonicity property 16 from Theorem 1. As we have already mentioned Corollary 2 enables us to provide numerous refinements and conversions of means inequalities for operators in Hilbert space. Recall inequality 6 from Introduction provides refinement of arithmetic-harmonic mean inequality. We also get conversion of above mentioned inequality as a special case of relation 26. Remark 3. Consider series of inequalities 26 equipped with continuous convex function f : 0 R fx = 1/x and parameter δ = 1. In that setting A B B ++ H. Furthermore if we replace operators A and B respectively with A 1 and B 1 then the right inequality in 26 becomes inequality 6 that is

7 JENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES... 7 refinement of arithmetic-harmonic mean inequality. On the other hand the left inequality in 26 yields conversion of arithmetic-harmonic mean inequality that is 27 2 max{p 1 p 2 } A B A! B A p 1 B A! p 1 B. Remark 4. Now we consider series of inequalities 25 equipped with convex function f : R R fx = exp x and parameter δ = 0. In addition we consider operator log C 1 BC 1 assuming B B ++ H instead of operator C 1 BC 1. In described setting relation 25 becomes series of inequalities 7 that is refinement and conversion of arithmetic-geometric mean inequality. Clearly the operator log C 1 BC 1 is well defined since C 1 BC 1 B ++ H. In the next section we develop yet another method for bounding of Jensen s operator J f D δ p. Let s mention here that some different Jensen s type inequalities including self-adjoint operators on Hilbert space were recently obtained in paper Bounds for spectra of Jensen s operator and applications to mean inequalities In the previous section we were concerned with the bounding of Jensen s operator J f D δ p with non-weighted operator J N f D δ. As distinguished from Section 3 in this section we use a different method for bounding of Jensen s operator. Namely regarding monotonicity properties of Jensen s functional considered as a function of one variable see Lemma 1 we get under certain conditions bounds for operator J f D δ p that are multiples of identity operator 1 H. Obviously such estimates can be interpreted as the bounds for the spectra of operator J f D δ p. Now follows our main result concerning above discussion. Theorem 3. Suppose J is an operator defined by 14 and let γ δ a b. If 28 a1 H D γ1 H δ1 H or δ1 H γ1 H D b1 H then 29 J f D δ p j f γ δ p1 H 2 min{p 1 p 2 }j N f γ δ1 H where and In addition if p1 γ + p 2 δ j f γ δ p = p 1 fγ + p 2 fδ f j N f γ δ = fγ + fδ 2 γ + δ f 2 30 a1 H γ1 H D δ1 H or δ1 H D γ1 H b1 H then 31 J f D δ p j f γ δ p1 H 2 max{p 1 p 2 }j N f γ δ1 H..

8 8 MARIO KRNIĆ NEDA LOVRIČEVIĆ AND JOSIP PEČARIĆ Proof. We consider Jensen s functional p1 x + p 2 δ j f x δ p = p 1 fx + p 2 fδ f as a function in variable x and use properties i and ii from Lemma 1 concerning monotonicity of j f x δ p on intervals a δ and δ b. More precisely if a x γ δ or δ γ x b then properties i and ii from Lemma 1 together with monotonicity property 12 of Jensen s functional yield the following series of inequalities: 32 j f x δ p j f γ δ p 2 min{p 1 p 2 }j N f γ δ. Now if D B h H satisfies 28 then according to monotonicity property 13 for operator functions we can insert D in series of inequalities 32. As a result we get relation 29 as required. It remains to prove relation 31. We act in the same way as in the first part of the proof. If a γ x δ or δ x γ b then 12 and properties i ii from Lemma 1 provide relation 33 j f x δ p j f γ δ p 2 max{p 1 p 2 }j N f γ δ. Finally if D B h H satisfies one of conditions in 30 then relation 33 implies 31 due to monotonicity property 13 for operator functions. As a first application we give an analogue of Corollary 2 regarding the method developed in Theorem 3. Corollary 3. Let f : a b R R be continuous convex function. Suppose A B ++ H B B h H C B 1 H A = C C p = p 1 p 2 R 2 + and γ δ a b. If 34 aa B γa δa or δa γa B ba then 35 C J f C 1 BC 1 δ p C j f γ δ pa 2 min{p 1 p 2 }j N f γ δa where C J f C 1 BC 1 δ p C is defined by 21 and the functionals jf γ δ p j N f γ δ are defined in Theorem 3. In addition if 36 aa γa B δa or δa B γa ba then 37 C J f C 1 BC 1 δ p C j f γ δ pa 2 max{p 1 p 2 }j N f γ δa. Proof. Follows simply from Theorem 3. Namely we easily see that the conditions 28 and 30 rewritten for operator D = C 1 BC 1 are equivalent respectively to conditions 34 and 36. Hence if we replace D in 29 and 31 with C 1 BC 1 and then multiply obtained series of inequalities by C on the left and by C on the right we get 35 and 37. Remark 5. If C is the square root of operator A i.e. C = A 1 2 then under the same assumptions as in Corollary 3 the series of inequalities 35 and 37

9 JENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES... 9 respectively read 38 and 39 A 1 2 J f A 1 2 BA 1 2 δ p A 1 2 j f γ δ pa 2 min{p1 p 2 }j N f γ δa A 1 2 J f A 1 2 BA 1 2 δ p A 1 2 j f γ δ pa 2 max{p1 p 2 }j N f γ δa. In the sequel we focus to applications of Theorem 3 and Corollary 3 which provide refinements and conversions of above mentioned operator means in so called difference form. Our next result refers to the difference between arithmetic and harmonic operator mean. Corollary 4. Suppose H is a Hilbert space A B B ++ H p = p 1 p 2 R 2 + and γ > 0. If 40 γa A γb or γb A γa then 41 Furthermore if A p 1 B A! p 1 B γ 12 p 1 p 2 γp 1 γ + p 2 A γ 12 min{p 1 p 2 } γγ + 1 A. 42 γa γb A or A γb γa then 43 A p 1 B A! p 1 B γ 12 p 1 p 2 γp 1 γ + p 2 A γ 12 max{p 1 p 2 } γγ + 1 A. Proof. Relations 41 and 43 are immediate consequences of relations 38 and 39. Namely we consider 38 and 39 endowed with convex function f : 0 R fx = 1/x parameter δ = 1 and operators A and B respectively replaced with A 1 and B 1. In that setting the left expression in 38 and 39 becomes A p 1 B A! p 1 B Jensen s functionals j f γ δ p and j N f γ δ respectively read j f γ δ p = γ 12 p 1 p 2 γp 1 γ + p 2 and j N f γ δ = γ 12 2γγ + 1 which yields required relations 41 and 43. It remains to justify the conditions under which the inequalities in 41 and 43 are valid. More precisely inequalities in 41 are obtained from relation 38 which is valid under conditions 34. Taking into account that we have replaced operators A and B respectively with A 1 and B 1 conditions 34 in described setting respectively read B 1 γa 1 A 1 or A 1 γa 1 B 1. Clearly these conditions are equivalent to those in 40 due to operator monotonicity of function gt = 1/t on 0. The similar discussion holds for relation 43 and

10 10 MARIO KRNIĆ NEDA LOVRIČEVIĆ AND JOSIP PEČARIĆ associated conditions 42. Note that in relations 41 and 43 the difference between arithmetic and harmonic mean was bounded by multiple of one of the operator included in definitions of mentioned means. The same form can be established for the difference between arithmetic and geometric operator mean. To get such a result it is more convenient to use Theorem 3 directly than Corollary 3. Corollary 5. Suppose H is a Hilbert space A B B ++ H C B 1 H A = C C p = p 1 p 2 R 2 + and γ > 0. If 44 B γa A or A γa B then 45 Further if 1 p1 C A p 1 B C C 1 BC p 1 γ + p 2 γ p 1 A min{p 1 p 2 } γ 1 2 A. 46 γa B A or A B γa then 47 1 p1 C A p 1 B C C 1 BC p 1 γ + p 2 γ p 1 A max{p 1 p 2 } γ 1 2 A. Proof. Let f : R R fx = exp x let δ = 0 and let D = log C 1 BC 1. Regarding notations from Theorem 3 we have p1 γ j f γ δ p = p 1 exp γ + p 2 exp exp γ j N f γ δ = 2 while Jensen s operator 14 becomes p 1 C 1 BC 1 + p 2 1 H C 1 BC 1 p 1. Further to apply relation 29 operator D must satisfy one of conditions in 28 i.e. log C 1 BC 1 γ1 H 0 or 0 γ1 H log C 1 BC 1 which reduces to B exp γa A or A exp γa B. Since exponential function is injective we can replace exp γ with γ in all expressions assuming γ > 0. At last if we substitute obtained expressions in 29 and then multiply associated inequalities by C on the left and by C on the right we get 45 as required. To obtain series of inequalities in 47 we act in the same way as above considering operators satisfying 46. Note that in previous two corollaries the difference between two operator means was bounded by a multiple of one of the operator included in definitions of considered means. Regarding the method developed in Theorem 3 for various choices of convex functions we can get some other forms of inequalities for operator means which will be clarified in the sequel.

11 JENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES Emphasize once more that inequality 8 from Introduction provided refinement of arithmetic-geometric inequality via Kantorovich constant Kt 2 = t /4t t > 0. More precisely arithmetic mean was bounded by a multiple of geometric mean. That constant factor was certain power of Kantorovich constant. Our next consequence of Theorem 3 yields even better constant factor then actual power of Kantorovich constant as well as conversion of above mentioned operator inequality. Corollary 6. Suppose H is a Hilbert space A B B ++ H C B 1 H A = C C p = p 1 p 2 R 2 + and γ > 0. If 48 B γa A or A γa B then 49 In addition if A p 1 B p 1γ + p 2 p 1 γ C C 1 BC 1 p 1 Kγ 2 min{p 1 p 2 } C C 1 BC 1 p 1 C. 50 γa B A or A B γa then 51 A p 1 B p 1γ + p 2 p 1 γ C C 1 BC 1 p 1 Kγ 2 max{p 1 p 2 } C C 1 BC 1 p 1 C. Proof. The proof is direct use of Theorem 3. We consider Jensen s operator 14 equipped with convex function f : 0 R fx = log x parameter δ = 1 and operator D B ++ H satisfying 52 D γ1 H 1 H or 1 H γ1 H D. Now taking into account notations from Theorem 3 we have p1 D + p 2 1 H J f D δ p = log D p1 γ + p 2 j f γ δ p = log γ p 1 γ + 1 j N f γ δ = log 2 γ so relation 29 takes form p1 D + p 2 1 H log D p 1 p1 γ + p 2 log that is 53 p 1 D + p 2 1 H p 1γ + p 2 γ p 1 γ log Kγ 2 min{p 1 p 2 } 1 H p 1 D p 1 C C p 1 1 H Kγ 2 min{p 1 p 2 } D p 1. On the other hand if operators A = C C B B ++ H satisfy 48 it follows that operator D = C 1 BC 1 satisfy 52. Finally if we replace D in 53 with

12 12 MARIO KRNIĆ NEDA LOVRIČEVIĆ AND JOSIP PEČARIĆ C 1 BC 1 and multiply expressions in 53 by C on the left and by C on the right we get 49. The series of inequalities in 51 is obtained in the same way as 49 considering relation 31 and operators A = C C B B ++ H satisfying one of the conditions in 50. The following two remarks describe connection between our Corollary 6 and inequality 8 in detail. Remark 6. Relations 49 and 51 provide respectively refinement and conversion of arithmetic-geometric mean for Hilbert space operators. Namely if C = A 1 2 relations 49 and 51 respectively read 54 A p 1 B p 1γ + p 2 and 55 A p 1 B p 1γ + p 2 γ γ p 1 A p 1 p 1 A p 1 On the other hand Fujii et. al. obtained in 3 the inequality B Kγ 2 min{p1p2} A p 1 B B Kγ 2 max{p1p2} A p 1 B. 56 A B m M m σ M A σ B where 0 < m1 H A B M1 H and σ is a symmetric operator mean. Now by letting σ to be the geometric mean we see that 55 represents an extension of 56 to non-symmetric case. Remark 7. Let s compare our relation 49 i.e. 54 with inequality 8 from Introduction. We show that inequality 8 can be obtained from 54. More precisely if we denote h = M/m > 1 then the first condition in 9 yields A < ha m M m 1 H = M1 H B i.e. A < ha B. On the other hand since 1/h < 1 the second condition in 9 yields B m1 H = M m M 1 H 1 h A < A i.e. B 1 h A < A. Hence our conditions in 48 are equivalent to those in 9. Now inequality 8 follows from 54 since Kh 2 = K1/h 2 h > 0. Note also that our series of inequalities 54 also refines inequality 8. Namely if we take into consideration 54 we have interpolated inequality 8 with the multiple of geometric mean operator which includes constant factor not less then above mentioned power of Kantorovich constant. Yet another specific example regarding mean inequalities for operators arises directly from Remark 6. That is a content of the following remark with which we conclude this paper. Remark 8. Series of inequalities 54 and 55 enable us to deduce refinement and conversion of geometric-harmonic operator mean inequality. More precisely if we replace operators A and B in 54 and 55 respectively with A 1 and B 1 and

13 JENSEN S OPERATOR AND APPLICATIONS TO MEAN INEQUALITIES then use the fact that gt = 1/t is operator monotone on 0 relations 54 and 55 become respectively 57 A! p 1 B p 1 γ + p 2 γ p1 A p 1 B Kγ 2 min{p1p2} A p 1 B and 58 A! p 1 B p 1 γ + p 2 γ p1 A p 1 B Kγ 2 max{p1p2} A p 1 B. Furthermore if we replace operators A and B respectively with A 1 and B 1 in conditions 48 and 50 we see that relation 57 holds if γa A γb or γb A γa while 58 holds if γa γb A or A γb γa. Acknowledgments: This research was supported by the Croatian Ministry of Science Education and Sports under Research Grants first author second author and third author. References 1 S.S. Dragomir J.E. Pečarić L.E. Persson Properties of some functionals related to Jensen s inequality Acta Math. Hungar S.S. Dragomir Some Jensen s type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces B. Malays. Math. Sci. So. to appear. 3 J. Fujii M. Nakamura J. Pečarić Y. Seo Bounds for the ratio and difference between parallel sum and series via Mond-Pečarić method Math. Inequal. Appl S. Furuichi On refined Young inequalities and reverse inequalities J. Math. Inequal S. Furuichi Refined Young inequalities with Specht s ratio ArXiv: v2. 6 T. Furuta J. Mićić Hot J. Pečarić Y. Seo Mond-Pečarić Method in Operator Inequalities Element Zagreb F. Kittaneh Y. Manasrah Improved Young and Heinz inequalities for matrices J. Math. Anal. Appl F. Kittaneh Y. Manasrah Reverse Young and Heinz inequalities for matrices Linear Multilinear A. to appear. 9 F. Kittaneh M. Krnić N. Lovričević J. Pečarić Improved arithmetic-geometric and Heinz means inequalities for Hilbert space operators submitted for publication. 10 M. Klaričić Bakula M. Matić J. Pečarić On inequalities complementary to Jensen s inequality Mat. Bilten Skopje 32 LVIII J. Mićić J.E. Pečarić Y. Seo Inequalities between operator means based on the Mond-Pečarić method Houston J. Math J. Mićić J. Pečarić V. Šimić Inequalities involving the arithmetic and geometric means Math. Inequal. Appl D.S. Mitrinović J.E. Pečarić A.M. Fink Classical and New Inequalities in Analysis Kluwer Academic Publishers Dordrecht/Boston/London M. Tominaga Spech t ratio in the Young inequality Sci. Math. Jpn H. Zuo G. Shi M. Fujii Refined Young inequality with Kantorovich constant J. Math. Inequal. to appear.

14 14 MARIO KRNIĆ NEDA LOVRIČEVIĆ AND JOSIP PEČARIĆ Faculty of Electrical Engineering and Computing Unska Zagreb Croatia Mario.Krnic@fer.hr Faculty of Civil Engineering and Architecture Matice hrvatske Split Croatia neda.lovricevic@gradst.hr Faculty of Textile Technology University of Zagreb Pierottijeva Zagreb Croatia pecaric@element.hr