Some Hermite-Hadamard type integral inequalities for operator AG-preinvex functions

Transcription

1 Acta Univ. Sapientiae, Mathematica, 8, ( DOI: /ausm-16-1 Some Hermite-Hadamard type integral inequalities or operator AG-preinvex unctions Ali Taghavi Department o Mathematics, Faculty o Mathematical Sciences, University o Mazandaran, Iran. taghavi@umz.ac.ir Haji Mohammad Nazari Department o Mathematics, Faculty o Mathematical Sciences, University o Mazandaran, Iran m.nazari@stu.umz.ac.ir Vahid Darvish Department o Mathematics, Faculty o Mathematical Sciences, University o Mazandaran, Iran vahid.darvish@mail.com (vdarvish@wordpress.com Abstract. In this paper, we introduce the concept o operator AGpreinvex unctions and prove some Hermite-Hadamard type inequalities or these unctions. As application, we obtain some unitarily invariant norm inequalities or operators. 1 Introduction and preliminaries The ollowing Hermite-Hadamard inequality holds or any convex unction deined on R ( a + b b (a + (b (b a (xdx (b a, a, b R. (1 a 1 Mathematics Subject Classiication: 7A63, 15A6, 7B5, 7B1, 6D15 Key words and phrases: Hermite-Hadamard inequality, operator AG-preinvex unction, log-convex unction, positive linear operator 31

2 Some inequalities or operator AG-preinvex unctions 313 It was irstly discovered by Hermite in 1881 in the journal Mathesis (see [8]. But this result was nowhere mentioned in the mathematical literature and was not widely known as Hermite s result [1]. Beckenbach, a leading expert on the history and the theory o convex unctions, wrote that this inequality was proven by Hadamard in 1893 []. In 197, Mitrinovič ound Hermites note in Mathesis [8]. Since (1 was known as Hadamards inequality, the inequality is now commonly reerred as the Hermite-Hadamard inequality [1]. Deinition 1 [13] A continuous unction : I R R + is said to be an AG-convex unction (arithmetic-geometrically or log-convex unction on the interval I i (λa + (1 λb (a λ (b 1 λ. ( or a, b I and λ [, 1], i.e., log is convex. Theorem 1 [13] Let be an AG-convex unction deined on [a, b]. Then, we have ( ( ( a + b 3a + b a + 3b ( 1 b exp log((udu b a a ( a + b. (a. (b where u = log t. (a(b, (3 Let B(H stands or the C -algebra o all bounded linear operators on a complex Hilbert space H with inner product,. An operator A B(H is positive and write A i Ax, x or all x H. Let B(H sa stand or the set o all sel-adjoint elements o B(H. Let A be a sel-adjoint operator in B(H. The Geland map establishes a -isometrically isomorphism Φ between the set C(Sp(A o all continuous unctions deined on the spectrum o A, denoted by Sp(A, and the C -algebra C (A generated by A and the identity operator 1 H on H as ollows: or any, g C(Sp(A and any α, β C we have: Φ(α + βg = αφ( + βφ(g;

3 31 A. Taghavi, H. M. Nazari, V. Darvish Φ(g = Φ(Φ(g and Φ( = Φ( ; Φ( = := sup t Sp(A (t ; Φ( = 1 H and Φ( 1 = A, where (t = 1 and 1 (t = t, or t Sp(A. With this notation we deine (A = Φ( or all C(Sp(A and we call it the continuous unctional calculus or a sel-adjoint operator A. I A is a sel-adjoint operator and 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 and g are real valued unctions on Sp(A then the ollowing important property holds: (t g(t or any t Sp(A implies that (A g(a, ( in the operator order o B(H, see [1]. Deinition A real valued continuous unction on an interval I is said to be operator convex unction i (λa + (1 λb λ(a + (1 λ(b, in the operator order, or all λ [, 1] and sel-adjoint operators A and B in B(H whose spectra are contained in I. In [] Dragomir investigated the operator version o the Hermite-Hadamard inequality or operator convex unctions. Let : I R be an operator convex unction on the interval I then, or any sel-adjoint operators A and B with spectra in I, the ollowing inequalities holds ( A + B 3 1 (ta + (1 tbdt 1 [ ( 3A + B 1 + ( ] A + 3B ((1 ta + tb dt 1 [ ( ] A + B (A + (B + (A + (B,

4 Some inequalities or operator AG-preinvex unctions 315 or the irst inequality in above, see [1]. In [5], Ghazanari et al. gave the concept o operator preinvex unction and obtained Hermite-Hadamard type inequality or operator preinvex unction. Deinition 3 [5] 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], x + tη(x, y S. It is obvious that every convex set is invex with respect to the map η(x, 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. For every x, y S. the η-path P xv joining the points x and v := x + η(y, x is deined as ollows P xv := {z : z = x + tη(y, x, t [, 1]}. The mapping η is said to satisy the condition (C i or every x, y S and t [, 1], η(y, y + tη(y, x = tη(x, y, η(x, y + tη(x, y = (1 tη(x, y. Note that or every x, y S and every t 1, t [, 1], rom conditions in (C, we have η(y + t η(x, y, y + t 1 η(x, y = (t t 1 η(x, y, (5 see [9] or details. Deinition Let S B(H sa be an invex set with respect to η : S S B(H sa. Then, the continuous : R R is said to be operator preinvex with respect to η on S, i or every A, B S and t [, 1], in the operator order in B(H. (A + tη(b, A (1 t(a + t(b, (6 Every operator convex unction is operator preinvex with respect to the map η(a, B = A B, but the converse does not hold (see [5]. Theorem [5] Let S B(H sa be an invex set with respect to η : S S B(H sa and η satisies condition (C. I or every A, B S and V = A+η(B, A the unction : I R R is operator preinvex with respect to η on η-path P AV with spectra o A and spectra o V in the interval I. Then we have the inequality ( A + V 1 ((A + tη(b, Adt (A + (B.

5 316 A. Taghavi, H. M. Nazari, V. Darvish Throughout this paper, we introduce the concept o operator AG-preinvex unctions and obtain some Hermite-Hadamard type inequalities or these class o unctions. These results lead us to obtain some inequalities unitarily invariant norm inequalities or operators. Some inequalities or operator AG-preinvex unctions In this section, we prove some Hermite-Hadamard type inequalities or operator AG-preinvex unctions. Deinition 5 [13] A continuous unction : I R R + is said to be operator AG-convex (concave i (λa + (1 λb ( (A λ (B 1 λ or λ 1 and sel-adjoint operators A and B in B(H whose spectra are contained in I. Example 1 [6, Corollary 7.6.8] Let A and B be to positive deinite n n complex matrices. For < α < 1, we have where denotes determinant o a matrix. αa + (1 αb A α B 1 α (7 Let be an operator AG-convex unction, or commutative positive operators A, B B(H whose spectra are contained in I, then we have ( A + B 1 (αa + (1 αb((1 αa + αbdα (see [13] or more inequalities. (A(B, (8 Deinition 6 Let S B(H sa be an invex set with respect to η : S S B(H sa. A continuous unction : I R R + is called operator AG-preinvex with respect to η on S i (A + tη(b, A (A 1 t (B t or t [, 1] such that spectra o A and B are contained in I.

6 Some inequalities or operator AG-preinvex unctions 317 Remark 1 Let be an operator AG-preinvex unction, in a commutative case, we then get (A + tη(b, A (A 1 t (B t (1 t(a + t(b max{(a, (B} It means that is operator quasi preinvex i.e., (A+tη(B, A max{(a, (B}. We need the ollowing lemma or giving Hermite-Hadamard type inequalities or operator preinvex unction. Lemma 1 Let S B(H sa be an invex set with respect to η : S S B(H sa and : I R R + be a continuous unction on the interval I. Suppose that η satisies condition (C. Then or every A, B S and V = A + η(b, A and or some ixed s (, 1] the unction is operator AG-preinvex with respect to η on η-path P AV with spectra o A and V in the interval I i and only i the unction ϕ A,B deined by ϕ A,B (t = (A + tη(b, A (9 is a log-convex unction on [, 1]. Proo. Let ϕ be a log-convex unction on [, 1], we should prove that is operator AG-preinvex with respect to η. For every C 1 := A + t 1 η(b, A P AV, C := A + t η(b, A P AV, ixed λ [, 1], by (9 we have (C 1 + λη(c, C 1 = (A + t 1 η(b, A + λη(a + t η(b, A, A + t 1 η(b, A = (A + t 1 η(b, A + λ(t t 1 η(b, A = (A + (t 1 + λt λt 1 η(b, A = (A + ((1 λt 1 + λt η(b, A = ϕ((1 λt 1 + λt ϕ(t 1 1 λ ϕ(t λ = ((A + t 1 η(b, A 1 λ ((A + t η(b, A λ.

7 318 A. Taghavi, H. M. Nazari, V. Darvish Conversely, let be operator AG-preinvex, then, by (6 ϕ((1 λt 1 + λt = (A + ((1 λt 1 + λt η(b, A = (A + t 1 η(b, A + λ(t t 1 η(b, A = (A + t 1 η(b, A + λη(a + t η(b, A, A + t 1 η(b, A (A + t 1 η(b, A 1 λ (A + t η(b, A λ = ϕ(t 1 1 λ ϕ(t λ. Theorem 3 Let S B(H sa be an invex set with respect to η : S S B(H sa and : I R R + be a continuous unction on the interval I. Suppose that η satisies condition (C. Then or the operator AG-preinvex unction with respect to η on η-path P AV such that spectra o A and V are in I, we have ( A + V ( ( 3A + V A + 3V exp log((a + tη(b, Adt ( A + V (A (V (A(V (A + (V where A, B S and V = A + η(b, A and or some ixed s (, 1] Proo. Since is an operator AG-preinvex unction, so by Lemma 1 we have ϕ(t = (A + tη(b, A is log-convex on [, 1]. On the other hand, in [11] we obtained the ollowing inequalities or logconvex unction ϕ on [, 1]: ϕ ( 1 ( ( 1 3 ϕ ϕ exp log(ϕ(udu

8 By knowing that Some inequalities or operator AG-preinvex unctions 319 ϕ ( 1. ϕ(. ϕ(1 ϕ(ϕ(1. (1 ϕ( = (A ( 1 ϕ = (A + 1 ( 3A + V η(b, A = ( 1 ϕ = (A + 1 ( A + V η(b, A = ϕ(1 = (V, we obtain ( A + V ( ( 3A + V A + 3V exp log((a + tη(b, Adt ( A + V (A (V (A(V. 3 Some unitarily invariant norm inequalities or operator AG-preinvex unctions In this section we prove some unitarily invariant norm inequalities or operators. We consider the wide class o unitarily invariant norms. Each o these norms is deined on an ideal in B(H and it will be implicitly understood that when we talk o T, then the operator T belongs to the norm ideal associated with. Each unitarily invariant norm is characterized by the invariance property UTV = T or all operators T in the norm ideal associated with and or all unitary operators U and V in B(H.

9 3 A. Taghavi, H. M. Nazari, V. Darvish For 1 p <, the Schatten p-norm o a compact operator A is deined by A p = (Tr A p 1/p, where Tr is the usual trace unctional. Note that or compact operator A we have, A = s 1 (A, and i A is a Hilbert-Schmidt operator, then A = ( j=1 s j (A1/. These norms are special examples o the more general class o the Schatten p-norms which are unitarily invariant [3]. Remark The author o [7] proved that i A, B, X B(H such that A, B are positive operators, then or ν 1 we have Let X = I in above inequality, we then get A ν XB 1 ν AX ν XB 1 ν. (11 A ν B 1 ν A ν B 1 ν. (1 Lemma Let be an operator AG-preinvex unction and η satisies the condition (C. Then the unction ϕ A,B : [, 1] R deined as ollows is log-convex. Proo. Let t 1, t [, 1], we have ϕ(t = (A + tη(b, A ϕ((1 λt 1 + λt = (A + ((1 λt 1 + λt η(b, A = (A + t 1 η(b, A + λ(t t 1 η(b, A = (A + t 1 η(b, A + λη(a + t η(b, A, A + t 1 η(b, A (A + t 1 η(b, A 1 λ (A + t η(b, A λ (A + t 1 η(b, A 1 λ (A + t η(b, A λ by (1 = ϕ(t 1 1 λ ϕ(t λ. Theorem Let S B(H sa be an invex set with respect to η : S S B(H sa and : I R R + be a continuous unction on the interval I. Suppose that η satisies condition (C. Then or the operator AG-preinvex unction with

10 Some inequalities or operator AG-preinvex unctions 31 respect to η on η-path P AV such that spectra o A and V are in I, we have ( A + V ( ( 3A + V A + 3V exp log( (A + tη(b, A dt ( A + V (A (V (A (V (A + (V. where A, B S and V = A + η(b, A and or some ixed s (, 1] Proo. Since is an operator AG-preinvex unction, so by Lemma we have ϕ(t = (A + tη(b, A is log-convex on [, 1]. On the other hand, in [11] we obtained the ollowing inequalities or logconvex unction ϕ on [, 1] : By knowing that ϕ ( 1 ( 1 ϕ exp ϕ ( 1 ϕ ( 3 log(ϕ(udu. ϕ(. ϕ(1 ϕ(ϕ(1. (13 ϕ( = (A ( 1 ϕ = (A + 1 ( η(b, A = 3A + V ( 1 ϕ = (A + 1 ( η(b, A = A + V ϕ(1 = (V,

11 3 A. Taghavi, H. M. Nazari, V. Darvish we obtain ( A + V ( ( 3A + V A + 3V exp log( (A + tη(b, A dt ( A + V (A (V (A (V. Let η(b, A = B A in the above theorem, then we obtain the ollowing inequalities: Reerences ( A + B ( ( 3A + B A + 3B exp log( ((1 ta + tb dt ( A + B (A (B (A (B (A + (B. (1 [1] T. Antczak, Mean value in invexity analysis, Nonlinear Anal., 6 (5, [] E. F. Beckenbach, Convex unctions, Bull. Amer. Math. Soc., 5 (198, [3] R. Bhatia, Matrix Analysis, GTM 169, Springer-Verlag, New York, [] S. S. Dragomir, Hermite-Hadamards type inequalities or operator convex unctions, Applied Mathematics and Computation., 18 (11,

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