Alfakhr, MT

Dragomir, Sever S

Moradi, HR

Omidvar, ME

Victoria University Eprints Repository

Journal of Mathematical AnalysisISSN: 2217-3412, URL: www.ilirias.com/jmaVolume 8 Issue 6 (2017), Pages 64-73.AN OPERATOR EXTENSION OF CARTWRIGHT-FIELDINEQUALITYMAHDI TALEB ALFAKHR, MOHSEN ERFANIAN OMIDVAR, HAMID REZA MORADI,SILVESTRU SEVER DRAGOMIRAbstract. We establish an operator extension of the following inequality, dueto Cartwright and Field12p (1− p) (b− a)2max {a, b} ≤ (1− p) a+ pb− a1−pbp ≤ 12p (1− p) (b− a)2min {a, b}(a, b > 0, p ∈ [0, 1]) .Applications of this inequality for self-adjoint operators, continuous fields ofoperators and unital fields of positive linear mapping are also given.1. IntroductionAs is customary, we reserve m,M for scalars. Other capital letters are used todenote general elements of the C∗-algebra B (H) of all bounded linear operatorsacting on a Hilbert space (H, 〈·, ·〉). An operator A ∈ B (H) is called positive if〈Ax, x〉 ≥ 0 for all x ∈ H, and we then write A ≥ 0. For self-adjoint operatorsA,B ∈ B (H) we say that A ≤ B if B − A ≥ 0. The Gelfand map f (t) 7→ f (A)is an isometrical ∗-isomorphism between the C∗-algebra C (sp (A)) of continuousfunctions on the spectrum sp (A) of a self-adjoint operator A and the C∗-algebragenerated by A and I. If f, g ∈ C (sp (A)), then f (t) ≥ g (t) (t ∈ sp (A)) impliesthat f (A) ≥ g (A) (see for instance [12, p. 15]). A linear map φ is positive ifφ (A) ≥ 0 whenever A ≥ 0. It said to be unital if φ (I) = I. Moreover, C ([m,M ])is the set of all real valued continuous functions defined on an interval [m,M ]. Formore studies in this direction, we refer to [1].Let A be a positive operator on H. Then for any x ∈ H and a given positivereal number p〈Apx, x〉 ≤ ‖x‖2(1−p)〈Ax, x〉p, p ∈ (0, 1] (1.1)and〈Apx, x〉 ≥ ‖x‖2(1−p)〈Ax, x〉p, p ≥ 1. (1.2)2000 Mathematics Subject Classification. Primary 47A63, secondary 47A30, 15A60, 26D15,26D10.Key words and phrases. Cartwright-Field inequality, positive linear map, Ho¨lder-McCarthy,inequality.c©2017 Ilirias Research Institute, Prishtine¨, Kosove¨.Submitted April 4, 2017. Published November 5, 2017.Communicated by Guest Editor F. Kittaneh.64AN OPERATOR EXTENSION OF CARTWRIGHT-FIELD INEQUALITY 65McCarthy [9] proved the inequalities (1.1) and (1.2) by using the spectral resolutionof A and the Ho¨lder inequality. There is considerable amount of literature devotingto the study of Ho¨lder-McCarthy inequality and reverses. In [5, Theorem 3] Fujiet al. obtained the following interesting ratio inequality that provides a reverse ofthe Ho¨lder-McCarthy inequality (1.2) for self-adjoint operator A on H satisfying0 < mI ≤ A ≤MI〈Apx, x〉 ≤(1p1p q1qMp −mp(M −m) 1p (mMp −Mmp) 1p)p〈Ax, x〉p,for any x ∈ H with ‖x‖ = 1, where q = pp−1 , p > 1.Moreover, Dragomir in [4, Theorem 3.1] obtained the following reverse inequalityunder the same conditions0 ≤ 〈Apx, x〉 − 〈Ax, x〉p≤ p12(M −m)(∥∥Ap−1x∥∥2 − 〈Ap−1x, x〉2) 1212(Mp−1 −mp−1) (‖Ax‖2 − 〈Ax, x〉2) 12≤ 14p (M −m) (Mp−1 −mp−1) ,for any x ∈ H with ‖x‖ = 1, where p > 1. For other inequalities of Ho¨lder-McCarthytype, we refer to [6, 7, 8] and references therein.In 1978, Cartwright and Field [2] obtained the following inequality12p (1− p) (b− a)2max {a, b} ≤ (1− p) a+ pb− a1−pbp ≤ 12p (1− p) (b− a)2min {a, b} (1.3)for any a, b > 0 and p ∈ [0, 1]. For some recent operator inequalities obtained viaCartwright and Field inequality see [3].Several inequalities involving positive linear maps have been recently given in[10, 11]. Motivated by the above results, in this paper we continue and complementthis research by proving some reverses of Ho¨lder-McCarthy inequality involvingunital positive linear map by use of Cartwright and Field inequality. In Section 2of this paper, we generalize inequality (1.3). More precisely, we show new versionsof reverse Ho¨lder-McCarthy inequality (Theorem 2.1 and Proposition 2.2). Webegin Section 3 by establishing an operator version of inequality (1.3) involvingcontinuous fields of operators (see Theorem 3.1). It seems that the inequalitiesrelated to positive linear maps are useful in operator theory and mathematicalphysics and are interesting in their own right.2. Some Reverses of Operator Ho¨lder-McCarthy InequalityWe start our work with the following theorem.66 M. TALEB ALFAKHR, M.E. OMIDVAR, H.R. MORADI, S.S. DRAGOMIRTheorem 2.1. Let A ∈ B (H) be a self-adjoint with 0 < mI ≤ A ≤ MI and Φ bea unital positive linear map on B (H). Thenp (1− p)M(〈Φ(A2)x, x〉− 〈Φ (A)x, x〉2) ≤ 〈Φ (A)x, x〉 − 〈Φ (A1−p)x, x〉 〈Φ (Ap)x, x〉≤ p (1− p)m(〈Φ(A2)x, x〉− 〈Φ (A)x, x〉2)(2.1)for each x ∈ H with ‖x‖ = 1 and p ∈ [0, 1].Proof. If a, b ∈ [m,M ], then by Cartwright-Field inequality we have12Mp (1− p) (b− a)2 ≤ (1− p) a+ pb− a1−pbp ≤ 12mp (1− p) (b− a)2for any p ∈ [0, 1].Fix a ∈ [m,M ] and by using the operator functional calculus for A we get12Mp (1− p) (A2 − 2aA+ a2I) ≤ (1− p) aI + pA− a1−pAp≤ 12mp (1− p) (A2 − 2aA+ a2I) .Since Φ is unital positive linear map we have12Mp (1− p) (Φ (A2)− 2aΦ (A) + a2I) ≤ (1− p) aI + pΦ (A)− a1−pΦ (Ap)≤ 12mp (1− p) (Φ (A2)− 2aΦ (A) + a2I) .(2.2)Then for any x ∈ H with ‖x‖ = 1 we have from (2.2) that12Mp (1− p) (〈Φ (A2)x, x〉− 2a 〈Φ (A)x, x〉+ a2I)≤ (1− p) aI + p 〈Φ (A)x, x〉 − a1−p 〈Φ (Ap)x, x〉≤ 12mp (1− p) (〈Φ (A2)x, x〉− 2a 〈Φ (A)x, x〉+ a2I) .(2.3)By applying again functional calculus we deduce12Mp (1− p) (〈Φ (A2)x, x〉 I − 2A 〈Φ (A)x, x〉+A2)≤ (1− p)A+ p 〈Φ (A)x, x〉 I −A1−p 〈Φ (Ap)x, x〉≤ 12mp (1− p) (〈Φ (A2)x, x〉 I − 2 〈Φ (A)x, x〉A+A2) .Since Φ is unital positive linear map we have12Mp (1− p) (〈Φ (A2)x, x〉 I − 2Φ (A) 〈Φ (A)x, x〉+ Φ (A2))≤ (1− p) Φ (A) + p 〈Φ (A)x, x〉 I − Φ (A1−p) 〈Φ (Ap)x, x〉≤ 12mp (1− p) (〈Φ (A2)x, x〉 I − 2Φ (A) 〈Φ (A)x, x〉+ Φ (A2))for any x ∈ H with ‖x‖ = 1 and p ∈ [0, 1].AN OPERATOR EXTENSION OF CARTWRIGHT-FIELD INEQUALITY 67Thereforep (1− p)M(〈Φ(A2)x, x〉− 〈Φ (A)x, x〉2) ≤ 〈Φ (A)x, x〉 − 〈Φ (A1−p)x, x〉 〈Φ (Ap)x, x〉≤ p (1− p)m(〈Φ(A2)x, x〉− 〈Φ (A)x, x〉2)for any x ∈ H with ‖x‖ = 1 and p ∈ [0, 1]. Remark. Let Φ be a identity positive linear map in (2.1), then we have the follow-ing reverses of McCarty inequalityp (1− p)M(〈A2x, x〉− 〈Ax, x〉2) ≤ 〈Ax, x〉 − 〈A1−px, x〉 〈Apx, x〉≤ p (1− p)m(〈A2x, x〉− 〈Ax, x〉2) (2.4)for each x ∈ H with ‖x‖ = 1 and p ∈ [0, 1]. Choose p = 12 in (2.4), then14M(〈A2x, x〉− 〈Ax, x〉2) ≤ 〈Ax, x〉 − 〈A 12x, x〉2 ≤ 14m(〈A2x, x〉− 〈Ax, x〉2)for each x ∈ H with ‖x‖ = 1.On the other hand, if A is a self-adjoint operator satisfying 0 < mI ≤ A ≤MI,then for every x ∈ H with ‖x‖ = 1 we have〈A2x, x〉− 〈Ax, x〉2 ≤ (M −m)24.Based on this fact and the inequality (2.4), we deduce〈Ax, x〉 −〈A12x, x〉2≤ (M −m)216mfor each x ∈ H with ‖x‖ = 1.Remark. All assumptions as in Theorem 2.1. Another interesting case for Theo-rem 2.1 is to apply functional calculus for self-adjoint operator B in (2.3). In factthe following inequality holds for each x ∈ H with ‖x‖ = 1 and p ∈ [0, 1].12Mp (1− p) (〈Φ (A2)x, x〉− 2 〈Φ (B)x, x〉 〈Φ (A)x, x〉+ 〈Φ (B2)x, x〉)≤ (1− p) 〈Φ (B)x, x〉+ p 〈Φ (A)x, x〉 − 〈Φ (B1−p)x, x〉 〈Φ (Ap)x, x〉≤ 12mp (1− p) (〈Φ (A2)x, x〉− 2 〈Φ (B)x, x〉 〈Φ (A)x, x〉+ 〈Φ (B2)x, x〉) .In the following, we give a reverse of AM-GM inequality involving operator norm.In fact, when the operators A and B are bounded above and below by constantswe have the following result as well:Remark. Assume that 0 < mI ≤ B ≤ A ≤MI, thenm−M ≤ ‖Φ (A)‖ − ‖Φ (B)‖ ≤M −mtherefore(‖Φ (A)‖ − ‖Φ (B)‖)2 ≤ (M −m)2.68 M. TALEB ALFAKHR, M.E. OMIDVAR, H.R. MORADI, S.S. DRAGOMIRIf we take a = ‖Φ (A)‖ and b = ‖Φ (B)‖ in (1.3), we getp (1− p)2M(‖Φ (B)‖ − ‖Φ (A)‖)2 ≤ (1− p) ‖Φ (A)‖+ p ‖Φ (B)‖ − ‖Φ (A)‖1−p‖Φ (B)‖p≤ p (1− p)2m(‖Φ (B)‖ − ‖Φ (A)‖)2.(2.5)Put p = 12 in (2.5), then18M(‖Φ (B)‖ − ‖Φ (A)‖)2 ≤ ‖Φ (A)‖+ ‖Φ (B)‖2− (‖Φ (A)‖ ‖Φ (B)‖) 12≤ 18m(‖Φ (B)‖ − ‖Φ (A)‖)2.(2.6)By the inequality (2.6) we deduce‖Φ (A)‖+ ‖Φ (B)‖2− (‖Φ (A)‖ ‖Φ (B)‖) 12 ≤ (M −m)28m. (2.7)Remark. There are several examples of normalized positive linear maps. But forour application we consider among them Φ : M2 (C) → M2 (C), which Φ (A) =(12 tr (A))1H for all Hermitian matrices A ∈ M2 (C). From inequality (2.7), wehave|tr (A)|+ |tr (B)|2− (|tr (A)| |tr (B)|) 12 ≤ (M −m)24m. (2.8)Remark. Taking Φ (A) =n∑i=1X∗i AXi and Φ (B) =n∑i=1X∗i BXi with X∗i Xi = I in(2.7), therefore∥∥∥∥ n∑i=1X∗i AXi∥∥∥∥+ ∥∥∥∥ n∑i=1X∗i BXi∥∥∥∥2−(∥∥∥∥∥n∑i=1X∗i A12Xi∥∥∥∥∥∥∥∥∥∥n∑i=1X∗i B12Xi∥∥∥∥∥)≤ (M −m)28m.Now, from a different view point we may state:Proposition 2.2. Let A,B ∈ B (H) be self-adjoint with 0 < mI ≤ A,B ≤MI andΦ be a unital positive linear map on B (H). Thenp (1− p)2M(〈Φ (A)x, x〉2 − 2 〈Φ (A)x, x〉 〈Φ (B)x, x〉+ 〈Φ (B2)x, x〉)≤ (1− p) 〈Φ (B)x, x〉+ p 〈Φ (A)x, x〉 − 〈Φ (B1−p)x, x〉 〈Φ (A)x, x〉p≤ p (1− p)2m(〈Φ (A)x, x〉2 − 2 〈Φ (A)x, x〉 〈Φ (B)x, x〉+ 〈Φ (B2)x, x〉)(2.9)for each x ∈ H with ‖x‖ = 1 and p ∈ [0, 1].Proof. If we take b = 〈Φ (A)x, x〉 in (1.3) and then applying functional calculus forthe operator B we getp (1− p)2M(〈Φ (A)x, x〉2I − 2 〈Φ (A)x, x〉B +B2)≤ (1− p)B + p 〈Φ (A)x, x〉 I −B1−p〈Φ (A)x, x〉p≤ p (1− p)2m(〈Φ (A)x, x〉2I − 2 〈Φ (A)x, x〉B +B2)AN OPERATOR EXTENSION OF CARTWRIGHT-FIELD INEQUALITY 69for each x ∈ H with ‖x‖ = 1.Since Φ is unital positive linear map we havep (1− p)2M(〈Φ (A)x, x〉2I − 2 〈Φ (A)x, x〉Φ (B) + Φ (B2))≤ (1− p) Φ (B) + p 〈Φ (A)x, x〉 I − Φ (B1−p) 〈Φ (A)x, x〉p≤ p (1− p)2m(〈Φ (A)x, x〉2I − 2 〈Φ (A)x, x〉Φ (B) + Φ (B2)) .Whencep (1− p)2M(〈Φ (A)x, x〉2 − 2 〈Φ (A)x, x〉 〈Φ (B)x, x〉+ 〈Φ (B2)x, x〉)≤ (1− p) 〈Φ (B)x, x〉+ p 〈Φ (A)x, x〉 − 〈Φ (B1−p)x, x〉 〈Φ (A)x, x〉p≤ p (1− p)2m(〈Φ (A)x, x〉2 − 2 〈Φ (A)x, x〉 〈Φ (B)x, x〉+ 〈Φ (B2)x, x〉)(2.10)for each x ∈ H with ‖x‖ = 1. Remark. If we take B = A in Proposition 2.2, we getp (1− p)2M(〈Φ(A2)x, x〉− 〈Φ (A)x, x〉2) ≤ 〈Φ (A)x, x〉 − 〈Φ (A1−p)x, x〉 〈Φ (A)x, x〉p≤ p (1− p)2m(〈Φ(A2)x, x〉− 〈Φ (A)x, x〉2) .3. Continuous Field of OperatorsLet T be a locally compact Hausdorff space, and let A be a C∗-algebra of op-erators on a Hilbert space H. We say that a field (At)t∈T of operators in A iscontinuous if the function t → At is a norm continuous on T . If in addition µis a Radon measure on T and the function t → ‖At‖ is integrable, then we canform the Bochner integral∫Txtdµ (t), which is the unique element in A such thatϕ(∫TAtdµ (t))=∫Tϕ (At) dµ (t) for every linear functional ϕ in the norm dual A∗.Let (φt)t∈T be a field of positive linear mappings Φt : A→ B from A to anotherC∗-algebra B of operators on a Hilbert space K. We say that such a field iscontinuous if the function t→ Φt (A) is continuous for every A ∈ A. If in additionthe C∗-algebras are unital and Φt (1) is integrable with integral 1, we say that(Φt)t∈T is unital.Utilizing the above facts we prove our main result of this section.Theorem 3.1. Let A and B be C∗-algebras of operators containing I, T be alocally compact Hausdorff space equipped with a bounded Radon measure µ suchthat µ (T ) = 1. Also let (At)t∈T be a bounded continuous field of positive elementsin A and (Φt)t∈T be a field of positive linear mappings Φt : A → B defined on T .70 M. TALEB ALFAKHR, M.E. OMIDVAR, H.R. MORADI, S.S. DRAGOMIRThen12p (1− p) c (m,M)∫TΦt (I)dµ (t)≤ 12p (1− p)max {M, 1}(∫TΦt(A2t)dµ (t)− 2∫TΦt (At) dµ (t) +∫TΦt (I) dµ (t))≤ (1− p)∫TΦt (I)dµ (t) + p∫TΦt (At) dµ (t)−∫TΦt (Apt ) dµ (t)≤ 12p (1− p)min {m, 1}(∫TΦt(A2t)dµ (t)− 2∫TΦt (At) dµ (t) +∫TΦt (I) dµ (t))≤ 12p (1− p)C (m,M)∫TΦt (I)dµ (t)for every p ∈ [0, 1], wherec (m,M) =(M − 1)2 if M < 10 if m ≤ 1 ≤M(m−1)2M if 1 < mandC (m,M) =(m−1)2m if M < 11m max{(m− 1)2, (M − 1)2}if m ≤ 1 ≤M(M − 1)2 if 1 < m.Proof. If we write the inequality (1.3) for a = 1 and b = z we get12p (1− p) (z − 1)2max {z, 1} ≤ 1− p+ pz − zp ≤ 12p (1− p) (z − 1)2min {z, 1} (3.1)for any z > 0 and for any p ∈ [0, 1].If z ∈ [m,M ] ⊂ (0,∞), then max {z, 1} ≤ max {M, 1} and min {m, 1} ≤min {x, 1} and by (3.1) we get12p (1− p)minz∈[m,M ](z − 1)2max {M, 1} ≤12p (1− p) (z − 1)2max {M, 1}≤ 1− p+ pz − zp≤ 12p (1− p) (z − 1)2min {m, 1}≤ 12p (1− p)maxz∈[m,M ](z − 1)2min {m, 1}(3.2)for any z > 0 and for any p ∈ [0, 1].Observe thatminz∈[m,M ](z − 1)2 = (M − 1)2if M < 10 if m ≤ 1 ≤M(m− 1)2 if 1 < mAN OPERATOR EXTENSION OF CARTWRIGHT-FIELD INEQUALITY 71andmaxz∈[m,M ](z − 1)2 =(m− 1)2 if M < 1max{(m− 1)2, (M − 1)2}if m ≤ 1 ≤M(M − 1)2 if 1 < m.Thenminz∈[m,M ](z − 1)2max {M, 1} =(M − 1)2 if M < 10 if m ≤ 1 ≤M(m−1)2M if 1 < m≡ c (m,M)andmaxz∈[m,M ](z − 1)2min {m, 1} =(m−1)2m if M < 11m max{(m− 1)2, (M − 1)2}if m ≤ 1 ≤M(M − 1)2 if 1 < m≡ C (m,M) .Using the inequality (3.2) we have12p (1− p) c (m,M) ≤ 12p (1− p) (z − 1)2max {M, 1}≤ 1− p+ pz − zp≤ 12p (1− p) (z − 1)2min {m, 1}≤ 12p (1− p)C (m,M)(3.3)for any z > 0 and for any p ∈ [0, 1].Using functional calculus and taking z = At, from (3.3) follows12p (1− p) c (m,M) I ≤ 12p (1− p) (At − I)2max {M, 1}≤ (1− p) I + pAt −Apt≤ 12p (1− p) (At − I)2min {m, 1}≤ 12p (1− p)C (m,M) I.(3.4)Applying the positive linear mapping Φt and integrating, we obtain12p (1− p) c (m,M)∫TΦt (I)dµ (t)≤ 12p (1− p)max {M, 1}(∫TΦt(A2t)dµ (t)− 2∫TΦt (At) dµ (t) +∫TΦt (I) dµ (t))≤ (1− p)∫TΦt (I)dµ (t) + p∫TΦt (At) dµ (t)−∫TΦt (Apt ) dµ (t)≤ 12p (1− p)min {m, 1}(∫TΦt(A2t)dµ (t)− 2∫TΦt (At) dµ (t) +∫TΦt (I) dµ (t))≤ 12p (1− p)C (m,M)∫TΦt (I)dµ (t) .Hence Theorem 3.1 is proved. 72 M. TALEB ALFAKHR, M.E. OMIDVAR, H.R. MORADI, S.S. DRAGOMIRA discrete version of Theorem 3.1 is the following result obtained by takingT = {1, . . . , n}.Corollary 3.2. Let A1, . . . , An be positive operators acting on a Hilbert space Hand Φi (i = 1, . . . , n) be positive linear mappings on B (H). Then12p (1− p) c (m,M)n∑i=1Φi (I) ≤ 12p (1− p)max {M, 1}(n∑i=1Φi(A2i)− 2 n∑i=1Φi (Ai) +n∑i=1Φi (I))≤ (1− p)n∑i=1Φi (I) + pn∑i=1Φi (Ai)−n∑i=1Φi (Api )≤ 12p (1− p)min {m, 1}(n∑i=1Φi(A2i)− 2 n∑i=1Φi (Ai) +n∑i=1Φi (I))≤ 12p (1− p)C (m,M)n∑i=1Φi (I).By setting Φt (A) = X∗i AXi in Corollary 3.2, we find the following result.Corollary 3.3. Let A1, . . . , An be positive operators acting on a Hilbert space Hwith Ai ≥ 0 and X1, . . . , Xn ∈ B (H). Then12p (1− p) c (m,M)n∑i=1X∗i Xi ≤12p (1− p)max {M, 1}(n∑i=1X∗i A2iXi − 2n∑i=1X∗i AiXi +n∑i=1X∗i Xi)≤ (1− p)n∑i=1X∗i Xi + pn∑i=1X∗i AiXi −n∑i=1X∗i ApiXi≤ 12p (1− p)min {m, 1}(n∑i=1X∗i A2iXi − 2n∑i=1X∗i AiXi +n∑i=1X∗i Xi)≤ 12p (1− p)C (m,M)n∑i=1X∗i Xi.Corollary 3.4. If in addition we taken∑i=1X∗i Xi = I, in Corollary 3.3 we deduce12p (1− p) c (m,M) I ≤ 12p (1− p)max {M, 1}(n∑i=1X∗i (Ai − I)2Xi)≤n∑i=1X∗i ((1− p) + pAi −Api )Xi≤ 12p (1− p)min {m, 1}(X∗i (Ai − I)2Xi)≤ 12p (1− p)C (m,M) I.AN OPERATOR EXTENSION OF CARTWRIGHT-FIELD INEQUALITY 73Remark. If we replace At by Φt (At) in (3.4) and then integrating, we obtain12p (1− p) c (m,M)∫TΦt (I) dµ (t)≤ 12p (1− p)max {M, 1}(∫TΦ2t (At) dµ (t)− 2∫TΦt (At) dµ (t) +∫TΦt (I) dµ (t))≤ (1− p)∫TΦt (I) dµ (t) + p∫TΦt (At) dµ (t)−∫TΦpt (At) dµ (t)≤ 12p (1− p)min {m, 1}(∫TΦ2t (At) dµ (t)− 2∫TΦt (At) dµ (t) +∫TΦt (I) dµ (t))≤ 12p (1− p)C (m,M)∫TΦt (I) dµ (t).Acknowledgments. The authors would like to thank the anonymous referee forhis/her comments that helped us improve this article.References[1] R. Bhatia, Positive Definite Matrices, Princeton University Press, 2007.[2] D. I. Cartwright, M. J. Field, A refinement of the arithmetic mean-geometric mean inequality,Proc. Amer. Math. Soc. 71 (1978), 36–38.[3] S.S. Dragomir, Additive refinements and reverses of Young’s operator inequality via a resultof Cartwright and Field, Preprint RGMIA Res. Rep. Coll. 19 (2016), Art. 6.[4] ——–, Some reverses of the Jensen inequality for functions of self-adjoint operators in Hilbertspaces, J. Inequal. Appl., Volume 2010, Article ID 496821, 15 pages.[5] M. Fuji, S. Izumino, R. Nakamoto, Y. Seo, Operator inequalities related to Cauchy-Schwarzand Ho¨lder-McCarthy inequalities, Nihonkai Math. J. 8 (1997), 117-122.[6] M. Fujii, S. Izumino, R. Nakamoto, Classes of operators determined by the Heinz-Kato Furutainequality and the Ho¨lder-McCarthy inequality, Nihonkai Math. J. 5(1) (1994), 61–67.[7] T. Furuta, Extensions of Ho¨lder-McCarthy and Kantorovich inequalities and their applica-tions, Proc. Japan Acad. Ser. A Math. Sci. 73(3) (1997), 38–41.[8] ——–, Operator inequalities associated with Ho¨lder-McCarthy and Kantorovich inequalities,J. Inequal. Appl. 2(2) (1998), 137-148.[9] C.A. McCarthy, cp, Israel J. Math. 5 (1967), 249–271.[10] H.R. Moradi, M.E. Omidvar, S.S. Dragomir, An operator extension of C˘ebys˘ev inequality,An. S¸tiint¸. Univ. “Ovidius” Constant¸a Ser. Mat. 25 (2) (2017), 135–147[11] H.R. Moradi, M.E. Omidvar, S.S. Dragomir, M.K. Anwary, More operator inequalities forpositive linear maps, Bull. Math. Anal. Appl. 9(7) (2017), 65–73.[12] G. J. Murphy, C∗-Algebras and Operator Theory, Academic Press, San Diego, 1990.Mahdi Taleb AlfakhrDepartment of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, IranE-mail address: mehdifakhr11@gmail.comMohsen Erfanian OmidvarDepartment of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, IranE-mail address: erfanian@mshdiau.ac.irHamid Reza MoradirYoung Researchers and Elite Club, Mashhad Branch, Islamic Azad University, Mash-had, IranE-mail address: hrmoradi@mshdiau.ac.irSilvestru Sever DragomirSchool of Computer Science and Mathematics Victoria University of Technology, POBox 14428, MCMC 8001, Victoria, AustraliaE-mail address: sever.dragomir@vu.edu.au

An operator extension of Cartwright-field inequality

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