THE HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR CONVEX FUNCTIONS

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1 THE HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR CONVEX FUNCTIONS S.S. DRAGOMIR Abstract. Some Hermite-Hadamard s type inequalities or operator convex unctions o seladjoint operators in Hilbert spaces are given. Applications or particular cases o interest are also provided.. Introduction The ollowing inequality holds or any convex unction de ned on R Z a + b b (a) + (b) (.) (b a) < (x)dx < (b a) a b R: a It was rstly discovered by Ch. Hermite in 88 in the journal Mathesis (see []). But this result was nowhere mentioned in the mathematical literature was not widely known as Hermite s result [7]. E.F. Beckenbach, a leading expert on the history the theory o convex unctions, wrote that this inequality was proven by J. Hadamard in 893 []. In 97, D.S. Mitrinović ound Hermite s note in Mathesis []. Since (.) was known as Hadamard s inequality, the inequality is now commonly reerred as the Hermite- Hadamard inequality [7]. Let X be a vector space, x y X x 6 y. De ne the segment [x y] : ( t)x + ty t [ ]g: We consider the unction : [x y]! R the associated unction g(x y) : [ ]! R g(x y)(t) : [( t)x + ty] t [ ]: Note that is convex on [x y] i only i g(x y) is convex on [ ]. For any convex unction de ned on a segment [x y] X, we have the Hermite- Hadamard integral inequality (see [, p. ], [3, p. ]) x + y (.) [( t)x + ty]dt (x) + (y) which can be derived rom the classical Hermite-Hadamard inequality (.) or the convex unction g(x y) : [ ]! R. Date: February 3,. 99 Mathematics Subject Classi cation. 7A63 7A99. Key words phrases. Seladjoint operators, Positive operators, Hermite-Hadamard s inequality, Operator convex unctions, Functions o seladjoint operators.

2 S.S. DRAGOMIR Since (x) kxk p (x X p < ) is a convex unction, then we have the ollowing norm inequality rom (.) (see [6, p. 6]) p Z (.3) x + y k( t)x + tyk p dt kxkp + kyk p or any x y X. Motivated by the above results we investigate in this paper the operator version o the Hermite-Hadamard inequality or operator convex unctions. The operator quasilinearity o some associated unctionals are also provided. In order to do that we need the ollowing preliminary de nitions results. Let A be a seladjoint linear operator on a complex Hilbert space (H h: :i) : The Gel map establishes a -isometrically isomorphism between the set C (Sp (A)) o all continuous unctions de ned on the spectrum o A denoted Sp (A) the C -algebra C (A) generated by A the identity operator H on H as ollows (see or instance [7, p. 3]): For any g C (Sp (A)) any C we have (i) ( + g) () + (g) (ii) (g) () (g) () (iii) k ()k kk : sup tsp(a) j (t)j (iv) ( ) H ( ) A where (t) (t) t or t Sp (A) : With this notation we de ne (A) : () or all C (Sp (A)) we call it the continuous unctional calculus or a seladjoint operator A: I A is a seladjoint operator is a real valued continuous unction on Sp (A), then (t) or any t Sp (A) implies that (A) i:e: (A) is a positive operator on H: Moreover, i both g are real valued unctions on Sp (A) then the ollowing important property holds: (P) (t) g (t) or any t Sp (A) implies that (A) g (A) in the operator order o B (H) : A real valued continuous unction on an interval I is said to be operator convex (operator concave) i (OC) (( ) ) () ( ) (A) + (B) in the operator order, or all [ ] or every seladjoint operator A B on a Hilbert space H whose spectra are contained in I: Notice that a unction is operator concave i is operator convex. A real valued continuous unction on an interval I is said to be operator monotone i it is monotone with respect to the operator order, i.e., A B with Sp (A) Sp (B) I imply (A) (B) : For some undamental results on operator convex (operator concave) operator monotone unctions, see [7] the reerences therein. As examples o such unctions, we note that (t) t r is operator monotone on [ ) i only i r : The unction (t) t r is operator convex on ( ) i either r or r is operator concave on ( ) i r : The logarithmic unction (t) ln t is operator monotone operator concave on ( ): The entropy unction (t) t ln t is operator concave on ( ): The exponential unction (t) e t is neither operator convex nor operator monotone.

3 HERMITE-HADAMARD S TYPE INEQUALITIES 3 For a recent monograph devoted to various inequalities or unctions o seladjoint operators, see [7] the reerences therein. For other results, see [5], [9], [] []. For recent results, see [], [5] [6].. Some Hermite-Hadamard s Type Inequalities We start with the ollowing result: Theorem. Let : I! R be an operator convex unction on the interval I: Then or any seladjoint operators A B with spectra in I we have the inequality 3 A + 3B (.) + (( + t) A + tb) dt (A) + (B) (A) + (B) : Proo. R First o all, since the unction is continous, the operator valued integral (( t) A + tb) dt exists or any seladjoint operators A B with spectra in I: We give here two proos, the rst using only the de nition o operator convex unctions the second using the classical Hermite-Hadamard inequality or real valued unctions.. By the de nition o operator convex unctions we have the double inequality: C + D (.) [ (( t) C + td) + (( t) D + tc)] [ (C) + (D)] or any t [ ] any seladjoint operators C D with the spectra in I: Integrating the inequality (.) over t [ ] taking into account that (( t) C + td) dt (( t) D + tc) dt then we deduce the Hermite-Hadamard inequality or operator convex unctions Z C + D (HHO) (( t) C + td) dt [ (C) + (D)] that holds or any seladjoint operators C D with the spectra in I: Now, on making use o the change o variable u t we have (( t) A + tb) dt ( u) A + u du by the change o variable u t (( t) A + tb) dt we have ( u) + ub du:

4 S.S. DRAGOMIR Utilising the Hermite-Hadamard inequality (HHO) we can write Z 3 ( u) A + u du (A) + A + 3B ( u) + ub du (A) + which by sumation division by two produces the desired result (.).. Consider now x H kxk two seladjoint operators A B with spectra in I. De ne the real-valued unction ' xab : [ ]! R given by ' xab (t) h (( t) A + tb) x xi : Since is operator convex, then or any t t [ ] with + we have ' xab (t + t ) h (( (t + t )) A + (t + t ) B) x xi h ( [( t ) A + t B] + [( t ) A + t B]) x xi h ([( t ) A + t B]) x xi + h ([( t ) A + t B]) x xi ' xab (t ) + ' xab (t ) showing that ' xab is a convex unction on [ ] : Now we use the Hermite-Hadamard inequality or real-valued convex unctions a + b g Z b g (a) + g (b) g (s) ds b a to get that ' xab a ' xab (t) dt ' xab () + ' xab Z 3 ' xab ' xab (t) dt ' xab + 'xab () which, by sumation division by two, produce 3 A + 3B (.3) + x x h (( t) A + tb) x xi dt (A) + (B) + x x : Finally, since by the continuity o the unction we have h (( t) A + tb) x xi dt (( t) A + tb) dtx x

5 HERMITE-HADAMARD S TYPE INEQUALITIES 5 or any x H kxk any two seladjoint operators A B with spectra in I we deduce rom (.3) the desired result (.). A simple consequence o the above theorem is that the integral is closer to the let bound than to the right, namely we can state: Corollary. With the assumptions in Theorem we have the inequality (.) ( ) (( t) A + tb) dt (A) + (B) (( t) A + tb) dt: Remark. Utilising di erent examples o operator convex or concave unctions, we can provide inequalities o interest. I r [ ] [ [ ] then we have the inequalities or powers o operators (.5) r r r 3 A + 3B + (( t) A + tb) r dt r + Ar + B r Ar + B r or any two seladjoint operators A B with spectra in ( ) : I r ( ) the inequalities in (.5) hold with " " instead o " ": We also have the ollowing inequalities or logarithm (.6) ln ln ln (( ln 3 A + 3B + ln + t) A + tb) dt ln (A) + ln (B) or any two seladjoint operators A B with spectra in ( ) : ln (A) + ln (B) 3. Some Operator Quasilinearity Properties Consider an operator convex unction : I R! R de ned on the interval I two distinct seladjoint operators A B with the spectra in I. We denote by [A B] the closed operator segment de ned by the amily o operators ( t) A + tb, t [ ]g : We also de ne the operator-valued unctional (3.) (A B t) : ( t) (A) + t (B) (( t) A + tb) in the operator order, or any t [ ] : The ollowing result concerning an operator quasilinearity property or the unctional ( t) may be stated:

6 6 S.S. DRAGOMIR Theorem. Let : I R! R be an operator convex unction on the interval I. Then or each A B two distinct seladjoint operators with the spectra in I C [A B] we have (3.) ( ) (A C t) + (C B t) (A B t) or each t [ ] i.e., the unctional ( t) is operator superadditive as a unction o interval. I [C D] [A B] then (3.3) ( ) (C D t) (A B t) or each t [ ] i.e., the unctional ( t) is operator nondecreasing as a unction o interval. Proo. Let C ( s) A + sb with s ( ) : For t ( ) we have (C B t) ( t) (( s) A + sb) + t (B) (A C t) ( t) (A) + t (( s) A + sb) giving that (3.) (A C t) + (C B t) (A B t) (( s) A + sb) + (( t) A + tb) (( t) [( s) A + sb] + tb) (( t) A + t [( s) A + sb]) (( t) ( s) A + [( t) s + t] B) (( ts) A + tsb) : Now, or a convex unction ' : I R! R, where I is an interval, any real numbers t t s s rom I with the properties that t s t s we have that (3.5) ' (t ) ' (t ) t t ' (s ) ' (s ) s s : Indeed, since ' is convex on I then or any a I the unction : In ag! R (t) : ' (t) t ' (a) a is monotonic nondecreasing on In ag. Utilising this property repeatedly we have ' (t ) ' (t ) ' (s ) ' (t ) ' (t ) ' (s ) t t s t t s ' (s ) ' (s ) ' (s ) ' (s ) s s s s which proves the inequality (3.5). For a vector x H, with kxk consider the unction ' x : [ ]! R given by ' x (t) : h (( t) A + tb) x xi : Since is operator convex on I it ollows that ' x is convex on [ ] : Now, i we consider, or given t s ( ) t : ts < s : s t : t < t + ( t) s : s

7 HERMITE-HADAMARD S TYPE INEQUALITIES 7 then ' x (t ) h (( ts) A + tsb) x xi ' x (t ) h (( t) A + tb) x xi giving that ' x (t ) ' x (t ) (( ts) A + tsb) (( t) A + tb) x x : t t t (s ) Also giving that ' x (s ) h (( s) A + sb) x xi ' x (s ) h (( t) ( s) A + [( t) s + t] B) x xi ' x (s ) ' x (s ) s s (( s) A + sb) (( t) ( s) A + [( t) s + t] B) x x : t (s ) Utilising the inequality (3.5) multiplying with t (s ollowing inequality in the operator order (3.6) (( ts) A + tsb) (( t) A + tb) ) < we deduce the (( s) A + sb) (( t) ( s) A + [( t) s + t] B) : Finally, by (3.) (3.6) we get the desired result (3.). Applying repeatedly the superadditivity property we have or [C D] [A B] that giving that which proves (3.3). (A C t) + (C D t) + (D B t) (A B t) (A C t) + (D B t) (A B t) (C D t) For t we consider the unctional (A B) : A B (A) + (B) which obviously inherits the superadditivity monotonicity properties o the unctional ( t) : We are able then to state the ollowing Corollary. Let : I R! R be an operator convex unction on the interval I. Then or each A B two distinct seladjoint operators with the spectra in I we have the ollowing bounds in the operator order (3.7) in C[AB] A + C (C) + (D) (3.8) sup CD[AB] + C + B C + D (C) (A) + (B) :

8 8 S.S. DRAGOMIR Proo. By the superadditivity o the unctional ( ) we have or each C [A B] that (A) + (B) (A) + (C) A + C (C) + (B) C + B + which is equivalent with A + C C + B (3.9) + (C) : Since the equality case in (3.9) is realized or either C A or C B we get the desired bound (3.7). The bound (3.8) is obvious by the monotonicity o the unctional ( ) as a unction o interval. Consider now the ollowing unctional (A B t) : (A) + (B) (( t) A + tb) (( t) B + ta) where, as above, : I R! R is a convex unction on the interval I A B are seladjoint operators with the spectra in I while t [ ] : We notice that (A B t) (B A t) (A B t) (A B t) (A B t) + (A B t) or any A B seladjoint operators with the spectra in I t [ ] : Thereore, we can state the ollowing result as well Corollary 3. Let : I R! R be an operator convex unction on the interval I. Then or each A B two distinct seladjoint operators with the spectra in I the unctional ( t) is operator superadditive operator nondecreasing as a unction o interval. In particular, i C [A B] then we have the inequality (3.) [ (( t) A + tb) + (( t) B + ta)] [ (( t) A + tc) + (( t) C + ta)] + [ (( t) C + tb) + (( t) B + tc)] (C) : Also, i C D [A B] then we have the inequality (3.) (A) + (B) (( t) A + tb) (( t) B + ta) (C) + (D) (( t) C + td) (( t) C + td) or any t [ ] : Perhaps the most interesting unctional we can consider is the ollowing one: (3.) (A B) (A) + (B) (( t) A + tb) dt:

9 HERMITE-HADAMARD S TYPE INEQUALITIES 9 Notice that, by the second Hermite-Hadamard inequality or operator convex unctions we have that (A B) in the operator order. We also observe that (3.3) (A B) (A B t) dt (A B t) dt: Utilising this representation, we can state the ollowing result as well: Corollary. Let : I R! R be an operator convex unction on the interval I. Then or each A B two distinct seladjoint operators with the spectra in I the unctional ( ) is operator superadditive operator nondecreasing as a unction o interval. Moreover, we have the bounds in the operator order (3.) in C[AB] [ (( t) A + tc) + (( t) C + tb)] dt (C) (C) + (D) (3.5) sup CD[AB] (( (A) + (B) t) C + td) dt (( (( t) A + tb) dt t) A + tb) dt: Remark. The above inequalities can be applied to various concrete operator convex unction o interest. I we choose, or instance, the inequality (3.5), then we get the ollowing bounds in the operator order (3.6) sup CD[AB] C r + D r (( t) C + td) r dt Ar + B r (( t) A + tb) r dt where r [ ] [ [ ] A B are seladjoint operators with spectra in ( ) : I r ( ) then (3.7) sup (( t) C + td) r C r + D r dt CD[AB] (( t) A + tb) r dt A B are seladjoint operators with spectra in ( ) : We also have the operator bound or the logarithm ln (C) + ln (D) (3.8) sup ln (( t) C + td) dt CD[AB] ln (( t) A + tb) dt where A B are seladjoint operators with spectra in ( ) : A r + B r ln (A) + ln (B)

10 S.S. DRAGOMIR Reerences [] E.F. Beckenbach, Convex unctions, Bull. Amer. Math. Soc. 5(98), [] S.S. Dragomir, An inequality improving the rst Hermite-Hadamard inequality or convex unctions de ned on linear spaces applications or semi-inner products, J. Inequal. Pure Appl. Math. 3 (), No., Article 3. [3] S.S. Dragomir, An inequality improving the second Hermite-Hadamard inequality or convex unctions de ned on linear spaces applications or semi-inner products, J. Inequal. Pure Appl. Math. 3 (), No.3, Article 35. [] S.S. Dragomir, Grüss type inequalities or unctions o seladjoint operators in Hilbert spaces, Preprint, RGMIA Res. Rep. Coll., (e) (8), Art.. [ONLINE: edu.au/rgmia/v(e).asp]. [5] S.S. Dragomir, Some new Grüss type Inequalities or unctions o seladjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll., (e) (8), Art.. [ONLINE: http: // [6] S.S. Dragomir, Some reverses o the Jensen inequality or unctions o seladjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll., (e) (8), Art.. [ONLINE: http: // [7] T. Furuta, J. Mićić Hot, J. Peµcarić Y. Seo, Mond-Peµcarić Method in Operator Inequalities. Inequalities or Bounded Seladjoint Operators on a Hilbert Space, Element, Zagreb, 5. [8] E. Kikianty S. S. Dragomir, Hermite-Hadamard s inequality the p-hh-norm on the Cartesian product o two copies o a normed space, Math. Inequal. Appl. (in press) [9] A. Matković, J. Peµcarić I. Perić, A variant o Jensen s inequality o Mercer s type or operators with applications. Linear Algebra Appl. 8 (6), no. -3, [] C.A. McCarthy, c p Israel J. Math., 5(967), 9-7. [] J. Mićić, Y. Seo, S.-E. Takahasi M. Tominaga, Inequalities o Furuta Mond-Peµcarić, Math. Ineq. Appl., (999), 83-. [] D.S. Mitrinović I.B. Lacković, Hermite convexity, Aequationes Math. 8 (985), 9 3. [3] B. Mond J. Peµcarić, Convex inequalities in Hilbert space, Houston J. Math., 9(993), 5-. [] B. Mond J. Peµcarić, On some operator inequalities, Indian J. Math., 35(993), -3. [5] B. Mond J. Peµcarić, Classical inequalities or matrix unctions, Utilitas Math., 6(99), [6] J.E. Peµcarić S.S. Dragomir, A generalization o Hadamard s inequality or isotonic linear unctionals, Radovi Mat. (Sarajevo) 7 (99), 3 7. [7] J.E. Peµcarić, F. Proschan, Y.L. Tong, Convex Functions, Partial Orderings, Statistical Applications, Academic Press Inc., San Diego, 99. Mathematics, School o Engineering & Science, Victoria University, PO Box 8, Melbourne City, MC 8, Australia. address: sever.dragomir@vu.edu.au URL: