SOME HERMITE HADAMARD TYPE INEQUALITIES FOR THE PRODUCT OF TWO OPERATOR PREINVEX FUNCTIONS

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1 Banach J. Math. Anal. 9 15), no., uncorrected galleyproo DOI: /bjma/9-- ISSN: electronic) SOME HERMITE HADAMARD TYPE INEQUALITIES FOR THE PRODUCT OF TWO OPERATOR PREINVEX FUNCTIONS A. G. GHAZANFARI AND A. BARANI Communicated by F. Kittaneh Abstract. In this paper we introduce operator preinvex unctions and establish a Hermite Hadamard type inequality or such unctions. We give an estimate o the right hand side o a Hermite Hadamard type inequality in which some operator preinvex unctions o seladjoint operators in Hilbert spaces are involved. Also some Hermite Hadamard type inequalities or the product o two operator preinvex unctions are given. 1. introduction The ollowing inequality holds or any convex unction deined on R and a, b R, with a < b ) a + b 1 b a) + b) x)dx 1.1) b a a Both inequalities hold in the reversed direction i is concave. We note that Hermite Hadamard s inequality may be regarded as a reinement o the concept o convexity and it ollows easily rom Jensen s inequality. The classical Hermite Hadamard inequality provides estimates o the mean value o a continuous convex unction : [a, b] R. The Hermite Hadamard inequality has several applications in nonlinear analysis and the geometry o Banach spaces, see [1, 5]. Date: Received: Feb. 16, 1; Accepted: May 9, 1. Corresponding author. 1 Mathematics Subject Classiication. Primary 7A63; Secondary 6D7, 6D15. Key words and phrases. Hermite Hadamard inequality, invex sets, operator preinvex unctions. 1

2 A. GHAZANFARI, A. BARANI In recent years several extensions and generalizations have been considered or classical convexity. We would like to reer the reader to [3, 7, 17] and reerences therein or more inormation. A number o papers have been written on this inequality providing some inequalities analogous to Hadamard s inequality given in 1.1) involving two convex unctions, see [1, 11, 16]. Pachpatte in [1] has proved the ollowing theorem or the product o two convex unctions. Theorem 1.1. Let and g be real-valued, nonnegative and convex unctions on [a, b]. Then b 1 b a a ) ) a + b a + b g x)gx)dx 1 3 Ma, b) + 1 Na, b), 6 1 b a b a x)gx)dx Ma, b) + 1 Na, b), 3 where Ma, b) = a)ga) + b)gb), Na, b) = a)gb) + b)ga). A signiicant generalization o convex unctions is that o invex unctions introduced by Hanson in [9]. In this paper we introduce operator preinvex unctions and give an operator version o the Hermite Hadamard inequality or such unctions. First, we review the operator order in BH) and the continuous unctional calculus or a bounded seladjoint operator. For seladjoint operators A, B BH) we write A Bor B A) i Ax, x Bx, x or every vector x H, we call it the operator order. Now, let A be a bounded seladjoint linear operator on a complex Hilbert space H;.,. ) and CSpA)) the C -algebra o all continuous complex-valued unctions on the spectrum o A. The Geland map establishes a -isometrically isomorphism Φ between CSpA)) and the C -algebra C A) generated by A and the identity operator 1 H on H as ollows see or instance [15, p.3]): For, g CSpA)) and α, β C i) Φα + βg) = αφ) + βφg); ii) Φg) = Φ)Φg) and Φ ) = Φ) ; iii) Φ) = := sup t) ; t SpA) iv) Φ ) = 1 and Φ 1 ) = A, t SpA). where t) = 1 and 1 t) = t, or I is a continuous complex-valued unctions on SpA), the element Φ) o C A) is denoted by A), and we call it the continuous unctional calculus or a bounded seladjoint operator A. I A is a bounded seladjoint operator and is a real-valued continuous unction on SpA), then t) or any t SpA) implies that A), i.e., A) is a positive operator on H. Moreover, i both and g are real-valued unctions on SpA) such that t) gt) or any t spa), then A) B) in the operator order in BH).

3 SOME HERMITE HADAMARD TYPE INEQUALITIES 3 A real valued continuous unction on an interval I is said to be operator convex operator concave) i 1 λ)a + λb) )1 λ)a) + λb) in the operator order in BH), or all λ [, 1] and or every bounded sel-adjoint operators A and B in BH) whose spectra are contained in I. For some undamental results on operator convex operator concave) and operator monotone unctions, see [15, ] and the reerences therein. Dragomir in [6] has proved a Hermite Hadamard type inequality or operator convex unctions: Theorem 1.. Let : I R be an operator convex unction on the interval I. Then or any seladjoint operators A and B with spectra in I we have the inequality ) A + B ) [ 1 ) )] 3A + B A + 3B + 1 t)a + tb))dt 1 [ ) ] A + B A) + B) + Moslehian in [13] generalized the above theorem 1. as ollows: ) A) + B). Theorem 1.3. I A, B are sel-adjoint operators on a Hilbert space H with spectra in an interval J, is an operator convex unction on J and k, p are positive integers, then ) A + B 1k i + 1 p k A + p 1 i= i= [ i + 1 k A + p 1 i + 1 ) ) B 1 t)a + tb))dt 1 i + 1 ) ) i B + k A + 1 i ) )] B p A) + B). Motivated by the above results we investigate in this paper the operator version o the Hermite Hadamard inequality or operator preinvex unctions. We show that Theorem 1.3 holds or operator preinvex unctions and establish an estimate o the right hand side o a Hermite Hadamard type inequality in which some operator preinvex unctions o seladjoint operators in Hilbert spaces are involved. We also give some Hermite Hadamard type inequalities or the product o two operator preinvex unctions.

4 A. GHAZANFARI, A. BARANI. operator preinvex unctions Deinition.1. Let X be a real vector space, a set S X is said to be invex with respect to the map η : S S X, i or every x, y S and t [, 1], y + tηx, y) S..1) It is obvious that every convex set is invex with respect to the map ηx, y) = x y, but there exist invex sets which are not convex see [1]). Let S X be an invex set with respect to η : S S X. For every x, y S the η path P xv joining the points x and v := x + ηy, x) is deined as ollows P xv := {z : z = x + tηy, x) : t [, 1]}. The mapping η is said to be satisies the condition C i or every x, y S and t [, 1], C) ηy, y + tηx, y)) = tηx, y), ηx, y + tηx, y)) = 1 t)ηx, y). Note that or every x, y S and every t 1, t [, 1] rom condition C we have ηy + t ηx, y), y + t 1 ηx, y)) = t t 1 )ηx, y),.) see [1, 18] or details. Let A be a C -algebra, denote by A sa the set o all sel adjoint elements in A. Deinition.. Let I be an interval in R and S BH) sa be an invex set with respect to η : S S BH) sa. A continuous unction : I R is said to be operator preinvex on I with respect to η or operators in S i B + tηa, B)) 1 t)b) + ta)..3) in the operator order in BH), or all t [, 1] and or every A, B S whose spectra are contained in I. Every operator convex unction is an operator preinvex with respect to the map ηa, B) = A B but the converse does not holds see the ollowing example). Now, we give an example o some operator preinvex unctions and invex sets with respect to the maps η which satisy the conditions C). Example.3. a) Suppose that 1 H is the identity operator on a Hilbert space H, and T : = {A BH) sa : A 1 1 H } U : = {A BH) sa : 1 H A} S : = T U BH) sa. Suppose that the unction η 1 : S S BH) sa is deined by A B A, B U, A B A, B T, η 1 A, B) = 1 H B A T, B U, 1 H B A U, B T.

5 SOME HERMITE HADAMARD TYPE INEQUALITIES 5 Clearly η 1 satisies condition C and S is an invex set with respect to η 1. We show that the real unction t) = t is operator preinvex with respect to η 1 on every interval I R, or operators in S. Since is an operator convex unction on I, or the cases which η 1 A, B) = A B the inequality.3) holds. Let η 1 A, B) = 1 H B, in this case we have 1 H A A and B + tη 1 A, B)) = B + t1 H B)) = 1 t)b + t1 H ) 1 t)b + t1 H 1 t)b + ta. Similarly, or the case η 1 = 1 H B we have B + tη 1 A, B)) = B + t 1 H B)) = 1 t) B) + t1 H ) 1 t)b + t1 H 1 t)b + ta, thereore, the inequality.3) holds. But the real unction gt) = a + bt, a, b R is not operator preinvex with respect to η 1 on S. b) Suppose that V := 1 H, ), W :=, 1 H ), S := V W BH) sa and the unction η : S S BH) sa is deined by { A B A, B V or A, B W, η A, B) = otherwise. Clearly η satisies condition C and S is an invex set with respect to η. The constant unctions t) = a, a R is only operator preinvex unctions with respect to η or operators in S. Because or η =, B + tη A, B)) = B) 1 t)b) + ta), implies that A) B). interchanging A,B we get B) A). c) The unction t) = t is not a convex unction, but it is a operator preinvex unction with respect to η 3, where { A B A, B or A, B, η 3 A, B) = B A otherwise. Let X be a vector space, x, y X, x y. Deine the segment [x, y] := 1 t)x + ty; t [, 1]. We consider the unction : [x, y] R and the associated unction gx, y) : [, 1] R, gx, y)t) := 1 t)x + ty), t [, 1]. Note that is convex on [x, y] i and only i gx, y) is convex on [, 1]. For any convex unction deined on a segment [x, y] X, we have the Hermite- Hadamard integral inequality ) x + y 1 t)x + ty)dt x) + y),.)

6 6 A. GHAZANFARI, A. BARANI which can be derived rom the classical Hermite Hadamard inequality 1.1) or the convex unction gx, y) : [, 1] R. Proposition.. Let I be an interval in R, S BH) sa an invex set with respect to η : S S BH) sa and η satisies condition C on S and : I R a continuous unction. Then, or every A, B S with spectra o A, V := A+ηB, A) in I, the unction is operator preinvex on I with respect to η or operators in η path P AV i and only i the unction ϕ x,a,b : [, 1] R deined by ϕ x,a,b t) := A + tηb, A))x, x.5) is convex on [, 1] or every x H with x = 1. Proo. Let the unction be operator preinvex on I with respect to η or operators in η path P AV. Suppose that A, B S with spectra A, V in I, since A + tηb, A) = tv + 1 t)a thereore or all t [, 1], SpA + tηb, A)) I. I t 1, t [, 1] since η satisies condition C on S, we have ϕ x,a,b 1 λ)t 1 + λt ) = A + 1 λ)t 1 + λt )ηb, A))x, x = A + t 1 ηb, A) + ληa + t ηb, A), A + t 1 ηb, A)) x, x λ A + t ηb, A))x, x + 1 λ) A + t 1 ηb, A))x, x = λϕ x,a,b t ) + 1 λ)ϕ x,a,b t 1 ),.6) or every λ [, 1] and x H with x = 1. Thereore, ϕ x,a,b is convex on [, 1]. Conversely, suppose that x H with x = 1 and ϕ x,a,b is convex on [, 1] and C 1 := A + t 1 ηb, A) P AV, C := A + t ηb, A) P AV. Fix λ [, 1]. By.) we have C 1 + ληc, C 1 ))x, x = A + 1 λ)t 1 + λt )ηb, A))x, x = ϕ x,a,b 1 λ)t 1 + λt ) 1 λ)ϕ x,a,b t 1 ) + λϕ x,a,b t ) = 1 λ) C 1 )x, x + λ C )x, x..7) Hence, is operator preinvex with respect to η or operators in η path P AV. 3. Hermite Hadamard type inequalities In this section we generalize Theorem 1.1 and Theorem 1.3 or operator preinvex unctions and establish an estimate or the right-hand side o the Hermite Hadamard operator inequality or such unctions. Some Hermite Hadamard type inequalities or the product o two operator preinvex unctions is also given. The ollowing Theorem is a generalization o Theorem 1.3 or operator preinvex unctions. Theorem 3.1. Let S BH) sa be an invex set with respect to η : S S BH) sa and η satisies condition C. I or every A, B S with spectra o A, V := A + ηb, A) in the interval I, the unction : I R is operator preinvex with

7 SOME HERMITE HADAMARD TYPE INEQUALITIES 7 respect to η or operators in η path P AV ollowing inequalities holds A + 1 ) ηb, A) 1k p 1 i= [ i= and k, p are positive integers, then the A + i + 1 ηb, A) kp A + tηb, A))dt A + i + 1 ηb, A) kp ) + Proo. For x H with x = 1 and t [, 1], we have ) A + i ηb, A) kp )] A) + B). 3.1) A + tηb, A))x, x = 1 t) Ax, x + t V x, x I, 3.) since Ax, x SpA) I and V x, x SpV ) I. Continuity o and 3.) imply that the operator valued integral A + tηb, A))dt exists. Since η satisied condition C, thereore or every t [, 1] we have A + 1 ηb, A) = A + tηb, A) + 1 ηa + 1 t)ηb, A), A + tηb, A)). 3.3) Let x H be a unit vector, deine the real-valued unction ϕ : [, 1] R given by ϕt) = A + tηb, A))x, x. Since is operator preinvex, by the previous proposition., ϕ is a convex unction on [, 1]. Utilizing the classical Hermite Hadamard inequality or real-valued convex unction ϕ on the interval [ i i+1, ], we get i= i + 1 ϕ ) i+1 i ϕt)dt ϕ i ) + ϕ i+1 ) i= 3.) Summation o the above inequalities over i =,, 1 yields ) i ϕ ϕ i ) + ϕ i+1 ) k ϕt)dt p. 3.5) Hence k 1 i= A + i + 1 ) ηb, A) kp 1 i= [ A + tηb, A))dt A + i + 1 ηb, A) kp ) + A + i ηb, A) kp )]. 3.6)

8 8 A. GHAZANFARI, A. BARANI Inequality 3.3) or t = i+1 and operator preinvexity imply that A + 1 ) ηb, A) 1 A + i + 1 ) ηb, A) + 1 k A + 1 i + 1 ) ) ηb, A). p i= 3.7) Summation o the above inequalities over i =,, 1 and the ollowing equality A + i + 1 ) ηb, A) = A + 1 i + 1 ) ) ηb, A) yield A + 1 ) ηb, A) i= i= In the other hand, rom preinvexity we have 1 i= 1 [ i= ) ηb, A) p A + i + 1 k [ i + 1 k B) + p A + i + 1 ) ηb, A). 3.8) + A + i )] ηb, A) kp 1 i + 1 ) A) + i k B) + p From inequalities 3.6), 3.8) and 3.9) we obtain 3.1). 1 i ) ] A) A) + B). 3.9) 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 3.. With the assumptions in Theorem 3.1 we have the inequality A + tηb, A))dt A + 1 ) ηb, A) A) + B) A + tηb, A))dt. Example 3.3. Let S,, η 1 be as in Example.3, then we have A + 1 η 1B, A)) A + tη 1 B, A)) dt A + B, or every A, B S. Let S BH) sa be an invex set with respect to η : S S BH) sa and, g : I R operator preinvex unctions on the interval I with respect to η or operators in η-path P AV. Then or every A, B S with spectra o A, V :=

9 SOME HERMITE HADAMARD TYPE INEQUALITIES 9 A + ηb, A) in the interval I, we deine real unctions MA, B) and NA, B) on H by MA, B)x) = A)x, x ga)x, x + B)x, x gb)x, x NA, B)x) = A)x, x gb)x, x + B)x, x ga)x, x x H), x H). The ollowing Theorem is a generalization o Theorem 1.1 or operator preinvex unctions. Theorem 3.. Let, g : I R + be operator preinvex unctions on the interval I with respect to η and η satisies condition C. Then or any seladjoint operators A and B on a Hilbert space H with spectra A, V in I, the inequality A + tηb, A))x, x ga + tηb, A))x, x dt holds or any x H with x = MA, B)x) + 1 NA, B)x), 3.1) 6 Proo. Continuity o, g and 3.) imply that the ollowing operator valued integrals exist B + tηb, A))dt, ga + tηb, A))dt, g)a + tηb, A))dt. Since and g are operator preinvex, thereore or t in [, 1] and x H we have A + tηb, A))x, x tb) + 1 t)a))x, x, 3.11) ga + tηb, A))x, x tgb) + 1 t)ga))x, x. 3.1) From 3.11) and 3.1) we obtain A + tηb, A))x, x ga + tηb, A))x, x 1 t) A)x, x ga)x, x + t B)x, x gb)x, x + t1 t) [ A)x, x gb)x, x + B)x, x ga)x, x ]. 3.13) Integrating both sides o 3.13) over [, 1] we get the required inequality 3.1). Theorem 3.5. Let, g : I R be operator preinvex unctions on the interval I with respect to η. I η satisies condition C, then or any seladjoint operators A and B on a Hilbert space H with spectra A, V in I, the inequality

10 1 A. GHAZANFARI, A. BARANI A + 1 ) ηb, A) x, x g A + 1 ) ηb, A) x, x 1 holds or any x H with x = 1. A + tηb, A))x, x ga + tηb, A))x, x dt MA, B)x) + 1 NA, B)x), 3.1) 6 Proo. Put D = A + tηb, A) and E = A + 1 t)ηb, A), by 3.3) we have A + 1ηB, A) = D + 1 ηe, D). Since and g are operator preinvex, thereore or any t I and any x H with x = 1 we observe that A + 1 ) ηb, A) x, x g A + 1 ) ηb, A) x, x = D + 1 ) ηe, D) x, x g D + 1 ) ηe, D) x, x ) ) D) + E) gd) + ge) x, x x, x 1 [ D)x, x + E)x, x ][ gd)x, x + ge)x, x ] 1 [ D)x, x gd)x, x + E)x, x ge)x, x ] + 1 [t A)x, x + 1 t) B)x, x ][1 t) ga)x, x + t gb)x, x ] + [1 t) A)x, x + t B)x, x ][t ga)x, x + 1 t) gb)x, x ] = 1 [ D)x, x gd)x, x + E)x, x ge)x, x ] + 1 t1 t)[ A)x, x ga)x, x + B)x, x gb)x, x ] + t + 1 t) )[ A)x, x gb)x, x + B)x, x ga)x, x ]. 3.15) We integrate both sides o 3.15) over [,1] and obtain A + 1 ) ηb, A) x, x g A + 1 ) ηb, A) x, x 1 [ A + tηb, A))x, x ga + tηb, A))x, x + A + 1 t)ηb, A))x, x ga + 1 t)ηb, A))x, x ]dt MA, B)x) + 1 NA, B)x). 6 This implies the required inequality 3.1). The ollowing Theorem is a generalization o Theorem 3.1 in [].

11 SOME HERMITE HADAMARD TYPE INEQUALITIES 11 Theorem 3.6. Let the unction : I R + is continuous, S BH) sa be an open invex set with respect to η : S S BH) sa and η satisies condition C. I or every A, B S and V = A + ηb, A) the unction is operator preinvex with respect to η on η path P AV with spectra o A and V in I. Then, or every a, b, 1) with a < b and every x H with x = 1 the ollowing inequality holds, a A + sηb, A))ds x, x + 1 b b t 1 b a a A + sηb, A))ds x, x A + sηb, A))ds x, x dt b a { A + aηb, A))x, x + A + bηb, A))x, x }. 3.16) 8 Moreover we have a A + sηb, A))ds + 1 b A + sηb, A))ds 1 b t A + sηb, A))dsdt b a a b a A + aηb, A)) + A + bηb, A)) 8 b a [ A + aηb, A)) + A + bηb, A)) ]. 3.17) 8 Proo. Let A, B S and a, b, 1) with a < b. For x H with x = 1 we deine the unction ϕ : [, 1] R + by t ϕt) := A + sηb, A))ds x, x. Utilizing the continuity o the unction, the continuity property o the inner product and the properties o the integral o operator-valued unctions we have t t A + sηb, A))ds x, x = A + sηb, A)) x, x ds. Since A + sηb, A)), thereore ϕt) or all t I. Obviously or every t, 1) we have ϕ t) = A + tηb, A))x, x, hence, ϕ t) = ϕ t). Since is operator preinvex with respect to η on η path P AV, by Proposition. the unction ϕ is convex. Applying Theorem. in [8] to the unction ϕ implies that ϕa) + ϕb) 1 b a b a ϕs)ds b a) ϕ a) + ϕ b)), 8 and we deduce that 3.16) holds. Taking supremum over both side o inequality 3.16) or all x with x = 1, we deduce that the inequality 3.17) holds.

12 1 A. GHAZANFARI, A. BARANI Reerences 1. T. Antczak, Mean value in invexity analysis, Nonlinear Analysis 6 5), A. Barani, A.G. Ghazanari and S.S. Dragomir, Hermite Hadamard inequality or unctions whose derivatives absolute values are preinvex, J. Inequal. Appl. 1), Article ID N.S. Barnett, P. Cerone and S.S. Dragomir, Some new inequalities or Hermite Hadamard divergence in inormation theory, Stochastic analysis and applications. Vol. 3, 7-19, Nova Sci. Publ., Hauppauge, NY, 3.. R. Bhatia, Matrix analysis, Springer-Verlag, New York, C. Conde, A version o the Hermite Hadamard inequality in a nonpositive curvature space, Banach J. Math. Anal. 6 1), no., S.S. Dragomir, The Hermite Hadamard type inequalities or operator convex unctions, Appl. Math. Comput ), no. 3, S.S. Dragomir and C.E.M. Pearce, Selected Topics on Hermite Hadamard Inequalities and applications, RGMIA Monographs rgmia.vu.edu.au/ monographs/ hermite hadamard.html), Victoria University,. 8. S.S. Dragomir and R.P. Agarwal, Two inequalities or dierentiable mappings and applications to special means o real numbers and to trapezoidal ormula, Appl. Math. Lett ), no. 5, M.A. Hanson, On suiciency o the Kuhn-Tucker conditions, J. Math. Anal. Appl ), E. Kikianty, Hermite Hadamard inequality in the geometry o Banach spaces, PhD thesis thesis, Victoria University, M. Klaričić Bakula and J. Pečarić, Note on some Hadamard-type inequalities, J. Inequal. Pure Appl. Math. 5 ), no. 3, Article S.R. Mohan and S.K. Neogy, On invex sets and preinvex unction, J. Math. Anal. Appl ), M.S. Moslehian, Matrix Hermite Hadamard type inequalities, Houston J. math 39 13) no 1, B.G. Pachpatte, On some inequalities or convex unctions, RGMIA Res. Rep. Coll. E) 6, 3). 15. J. Pečarić, T. Furuta, J. Mićić Hot and Y. Seo, Mond-Pečarić Method in Operator Inequalities. Inequalities or Bounded Seladjoint Operators on a Hilbert Space, Element, Zagreb, M. Tunç, On some new inequalities or convex unctions, Turkish J. Math. 36 1), S. Wu, On the weighted generalization o the Hermite Hadamard inequality and its applications, Rocky Mountain J. Math. 39 9), no. 5, X.M. Yang and D. Li, On properties o preinvex unctions, J. Math. Anal. Appl. 56 1), 9-1. Department o Mathematics, Lorestan University, P.O.Box 65, Khoramabad, Iran. address: ghazanari.amir@gmail.com; alibarani@yahoo.com