Some Slater's Type Trace Inequalities for Convex Functions Of

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Toyama Math. J. Vol. 38(2016), 75-99

Some Slater’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces

Silvestru Sever Dragomir1,2

Abstract. Some trace inequalities of Slater type for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also provided.

1. Introduction

Suppose that I is an interval of real numbers with interior ˚I and f : I → R is a convex function on I. Then f is continuous on ˚I and has finite left and right derivatives at each point of ˚I. Moreover, if x, y ∈ ˚I and x < y, ′ ≤ ′ ≤ ′ ≤ ′ ′ ′ then f− (x) f+ (x) f− (y) f+ (y) which shows that both f− and f+ are nondecreasing function on ˚I. It is also known that a convex function must be differentiable except for at most countably many points. For a convex function f : I → R, the subdifferential( of )f denoted by ∂f is the set of all functions φ : I → [−∞, ∞] such that φ ˚I ⊂ R and

f (x) ≥ f (a) + (x − a) φ (a) for any x, a ∈ I.

′ It is also well known that if f is convex on I, then ∂f is nonempty, f−, ′ ∈ ∈ f+ ∂f and if φ ∂f, then ′ ≤ ≤ ′ ∈ ˚ f− (x) φ (x) f+ (x) for any x I. 2010 Mathematics Subject Classiﬁcation. Primary 47A63; Secondary 47A99. Key words and phrases. Trace class operators, Hilbert-Schmidt operators, Trace, Convex functions, Jensen’s inequality, Slater’s inequality, Trace inequalities for matrices, Power series of operators.

75 76 Silvestru Sever Dragomir

In particular, φ is a nondecreasing function. If f is diﬀerentiable and convex on ˚I, then ∂f = {f ′} . The following result is well known in the literature as Slater inequality:

Theorem 1 (Slater, 1981, [34]). If f : I → R is a nonincreasing (nonde- ∑ creasing) convex function, x ∈ I, p ≥ 0 with P := n p > 0 and ∑ i i n i=1 i n ̸ ∈ i=1 piφ (xi) = 0, where φ ∂f, then (∑ ) 1 ∑n n p x φ (x ) p f (x ) ≤ f ∑i=1 i i i . (1) P i i n p φ (x ) n i=1 i=1 i i As pointed out in [8, p. 208], the monotonicity assumption for the derivative φ can be replaced with the condition ∑ n p x φ (x ) ∑i=1 i i i ∈ n I, (2) i=1 piφ (xi) which is more general and can hold for suitable points in I and for not necessarily monotonic functions. Let A be a selfadjoint linear operator on a complex Hilbert space (H; ⟨., .⟩) . The Gelfand map establishes a ∗-isometric isomorphism Φ between the set C (Sp (A)) of all continuous functions deﬁned on the spectrum of A, denoted Sp (A) , and the C∗-algebra C∗ (A) generated by A and the identity operator 1H on H as follows (see for instance [17, p. 3]): For any f, g ∈ C (Sp (A)) and any α, β ∈ C we have (i) Φ (αf + βg) = αΦ(f) + βΦ(g); ( ) ∗ (ii) Φ (fg) = Φ (f)Φ(g) and Φ f¯ = Φ (f) ; ∥ ∥ ∥ ∥ | | (iii) Φ(f) = f := supt∈Sp(A) f (t) ; (iv) Φ (f0) = 1H and Φ (f1) = A, where f0 (t) = 1 and f1 (t) = t, for t ∈ Sp (A) . With this notation we deﬁne

f (A) := Φ (f) for all f ∈ C (Sp (A)) and we call it the continuous functional calculus for a selfadjoint operator A. If A is a selfadjoint operator and f is a real valued continuous function on Sp (A), then f (t) ≥ 0 for any t ∈ Sp (A) implies that f (A) ≥ 0, i.e. Some Slater’s type trace inequalities for convex functions 77 f (A) is a positive operator on H. Moreover, if both f and g are real valued functions on Sp (A) then the following important property holds:

f (t) ≥ g (t) for any t ∈ Sp (A) implies that f (A) ≥ g (A) (P) in the operator order of B (H) . For a recent monograph devoted to various inequalities for functions of selfadjoint operators, see [17] and the references therein. For other results, see [28], [22] and [24]. The following result that provides an operator version for the Jensen inequality and can be found in Mond & Peˇcari´c[26] (see also [17, p. 5]):

Theorem 2 (Jensen’s inequality). Let A be a selfadjoint operator on the Hilbert space H and assume that Sp (A) ⊆ [m, M] for some scalars m, M with m < M. If f is a convex function on [m, M] , then

f (⟨Ax, x⟩) ≤ ⟨f (A) x, x⟩ , (MP) for each x ∈ H with ∥x∥ = 1.

As a special case of Theorem 2 we have the following H¨older-McCarthy inequality:

Theorem 3 (H¨older-McCarthy, 1967, [23]). Let A be a selfadjoint positive operator on a Hilbert space H. Then (i) ⟨Arx, x⟩ ≥ ⟨Ax, x⟩r for all r > 1 and x ∈ H with ∥x∥ = 1; (ii) ⟨Arx, x⟩ ≤ ⟨Ax, x⟩r for all 0 < r < 1 and x ∈ H with ∥x∥ = 1; (iii) If A is invertible, then ⟨Arx, x⟩ ≥ ⟨Ax, x⟩r for all r < 0 and x ∈ H with ∥x∥ = 1.

The following result that provides a reverse of the Jensen inequality has been obtained in [11]:

Theorem 4 (Dragomir, 2008, [11]). Let I be an interval and f : I → R be a convex and diﬀerentiable function on ˚I (the interior of I) whose derivative f ′ is continuous on ˚I. If A is a selfadjoint operators on the Hilbert space H with Sp (A) ⊆ [m, M] ⊂ ˚I, then ⟨ ⟩ ⟨ ⟩ (0 ≤) ⟨f (A) x, x⟩−f (⟨Ax, x⟩) ≤ f ′ (A) Ax, x −⟨Ax, x⟩ f ′ (A) x, x , (3) for any x ∈ H with ∥x∥ = 1. 78 Silvestru Sever Dragomir

Perhaps more convenient reverses of (MP) are the following inequalities that have been obtained in the same paper [11]:

Theorem 5 (Dragomir, 2008, [11]). Let I be an interval and f : I → R be a convex and diﬀerentiable function on ˚I (the interior of I) whose derivative f ′ is continuous on ˚I. If A is a selfadjoint operators on the Hilbert space H with Sp (A) ⊆ [m, M] ⊂ ˚I, then

(0 ≤) ⟨f (A) x, x⟩ − f (⟨Ax, x⟩) (4)  [ ] 1/2  1 − ∥ ′ ∥2 − ⟨ ′ ⟩2  2 (M m) f (A) x f (A) x, x ≤  [ ]  1/2  1 ′ − ′ ∥ ∥2 − ⟨ ⟩2 2 (f (M) f (m)) Ax Ax, x 1 ( ) ≤ (M − m) f ′ (M) − f ′ (m) , 4 for any x ∈ H with ∥x∥ = 1. We also have the inequality

(0 ≤) ⟨f (A) x, x⟩ − f (⟨Ax, x⟩) (5) 1 ( ) ≤ (M − m) f ′ (M) − f ′ (m) 4  ′ ′ ′ ′ 1  [⟨Mx − Ax, Ax − mx⟩⟨f (M) x − f (A) x, f (A) x − f (m) x⟩] 2 , − ¯ ¯  ¯ ¯ ¯ ′ ′ ¯  ¯⟨ ⟩ − M+m ¯ ¯⟨ ′ ⟩ − f (M)+f (m) ¯ Ax, x 2 f (A) x, x 2 1 ( ) ≤ (M − m) f ′ (M) − f ′ (m) , 4 for any x ∈ H with ∥x∥ = 1. Moreover, if m > 0 and f ′ (m) > 0, then we also have

(0 ≤) ⟨f (A) x, x⟩ − f (⟨Ax, x⟩) (6)  (M−m)(f ′(M)−f ′(m)) ′  1 √ ⟨Ax, x⟩⟨f (A) x, x⟩ ,  4 Mmf ′(M)f ′(m) ≤  (√ √ )(√ √ )  ′ ′ ′ 1 M − m f (M) − f (m) [⟨Ax, x⟩⟨f (A) x, x⟩] 2 , for any x ∈ H with ∥x∥ = 1. Some Slater’s type trace inequalities for convex functions 79

In [13] we obtained the following operator version for Slater’s inequality as well as a reverse of it:

Theorem 6 (Dragomir, 2008, [13]). Let I be an interval and f : I → R be a convex and diﬀerentiable function on ˚I (the interior of I) whose derivative f ′ is continuous on ˚I. If A is a selfadjoint operator on the Hilbert space H with Sp (A) ⊆ [m, M] ⊂ ˚I and f ′ (A) is a positive invertible operator on H then ( ) ⟨ ′ ⟩ ≤ Af (A) x, x − ⟨ ⟩ 0 f ⟨ ′ ⟩ f (A) x, x (7) ( f (A) x, x )[ ] ⟨Af ′ (A) x, x⟩ ⟨Af ′ (A) x, x⟩ − ⟨Ax, x⟩⟨f ′ (A) x, x⟩ ≤ f ′ , ⟨f ′ (A) x, x⟩ ⟨f ′ (A) x, x⟩ for any x ∈ H with ∥x∥ = 1.

For other similar results, see [13]. In order to state other new results on Slater type trace inequalities we need some preliminary facts as follows.

2. Some Facts on Trace of Operators ⟨· ·⟩ { } Let (H, , ) be a complex Hilbert space and ei i∈I an orthonormal basis of H. We say that A ∈ B (H) is a Hilbert-Schmidt operator if ∑ 2 ∥Aei∥ < ∞. (8) i∈I { } { } It is well know that, if ei i∈I and fj j∈J are orthonormal bases for H and A ∈ B (H) then ∑ ∑ ∑ 2 2 ∗ 2 ∥Aei∥ = ∥Afj∥ = ∥A fj∥ (9) i∈I j∈I j∈I showing that the deﬁnition (8) is independent of the orthonormal basis and A is a Hilbert-Schmidt operator iﬀ A∗ is a Hilbert-Schmidt operator.

Let B2 (H) the set of Hilbert-Schmidt operators in B (H) . For A ∈ B2 (H) we deﬁne ( ) ∑ 1/2 ∥ ∥ ∥ ∥2 A 2 := Aei (10) i∈I 80 Silvestru Sever Dragomir

{ } for ei i∈I an orthonormal basis of H. This deﬁnition does not depend on the choice of the orthonormal basis. 2 Using the triangle inequality in l (I) , one checks that B2 (H) is a vector ∥·∥ B space and that 2 is a norm on 2 (H) , which is usually called in the literature as the Hilbert-Schmidt norm. Denote the modulus of an operator A ∈ B (H) by |A| := (A∗A)1/2 . Because ∥|A| x∥ = ∥Ax∥ for all x ∈ H,A is Hilbert-Schmidt iﬀ |A| is ∥ ∥ ∥| |∥ ∈ B Hilbert-Schmidt and A 2 = A 2 . From (9) we have that if A 2 (H) , ∗ ∈ B ∥ ∥ ∥ ∗∥ then A 2 (H) and A 2 = A 2 . The following theorem collects some of the most important properties of Hilbert-Schmidt operators:

Theorem 7. We have: B ∥·∥ (i) ( 2 (H) , 2) is a Hilbert space with inner product ∑ ∑ ⟨ ⟩ ⟨ ⟩ ⟨ ∗ ⟩ A, B 2 := Aei, Bei = B Aei, ei (11) i∈I i∈I and the deﬁnition does not depend on the choice of the orthonormal basis { } ei i∈I ; (ii) We have the inequalities

∥ ∥ ≤ ∥ ∥ A A 2 (12) for any A ∈ B2 (H) and

∥ ∥ ∥ ∥ ≤ ∥ ∥ ∥ ∥ AT 2 , TA 2 T A 2 (13) for any A ∈ B2 (H) and T ∈ B (H);

(iii) B2 (H) is an operator ideal in B (H) , i.e.

B (H) B2 (H) B (H) ⊆ B2 (H);

(iv) Bfin (H) , the space of operators of ﬁnite rank, is a dense subspace of B2 (H);

(v) B2 (H) ⊆ K (H) , where K (H) denotes the algebra of compact operators on H. Some Slater’s type trace inequalities for convex functions 81

{ } ∈ B If ei i∈I is an orthonormal basis of H, we say that A (H) is trace class if ∑ ∥ ∥ ⟨| | ⟩ ∞ A 1 := A ei, ei < . (14) i∈I ∥ ∥ The deﬁnition of A 1 does not depend on the choice of the orthonormal { } B B basis ei i∈I . We denote by 1 (H) the set of trace class operators in (H) . The following proposition holds:

Proposition 8. If A ∈ B (H) , then the following are equivalent:

(i) A ∈ B1 (H); 1/2 (ii) |A| ∈ B2 (H);

(ii) A (or |A|) is the product of two elements of B2 (H) .

The following properties are also well known:

Theorem 9. With the above notations: (i) We have ∥ ∥ ∥ ∗∥ ∥ ∥ ≤ ∥ ∥ A 1 = A 1 and A 2 A 1 (15) for any A ∈ B1 (H);

(ii) B1 (H) is an operator ideal in B (H) , i.e.

B (H) B1 (H) B (H) ⊆ B1 (H);

(iii) We have

B2 (H) B2 (H) = B1 (H);

(iv) We have

∥ ∥ {|⟨ ⟩ | | ∈ B ∥ ∥ ≤ } A 1 = sup A, B 2 B 2 (H) , B 1 ;

B ∥·∥ (v) ( 1 (H) , 1) is a Banach space. (iv) We have the following isometric isomorphisms

∼ ∗ ∗ ∼ B1 (H) = K (H) and B1 (H) = B (H) ,

∗ ∗ where K (H) is the dual space of K (H) and B1 (H) is the dual space of

B1 (H) . 82 Silvestru Sever Dragomir

We define the trace of a trace class operator A ∈ B1 (H) to be ∑ tr (A) := ⟨Aei, ei⟩ , (16) i∈I { } where ei i∈I is an orthonormal basis of H. Note that this coincides with the usual definition of the trace if H is finite-dimensional. We observe that the series (16) converges absolutely and it is independent from the choice of basis. The following result collects some properties of the trace:

Theorem 10. We have: ∗ (i) If A ∈ B1 (H) then A ∈ B1 (H) and

tr (A∗) = tr (A); (17)

(ii) If A ∈ B1 (H) and T ∈ B (H) , then AT, T A ∈ B1 (H) and

| | ≤ ∥ ∥ ∥ ∥ tr (AT ) = tr (TA) and tr (AT ) A 1 T ; (18)

(iii) tr (·) is a bounded linear functional on B1 (H) with ∥tr∥ = 1;

(iv) If A, B ∈ B2 (H) then AB, BA ∈ B1 (H) and tr (AB) = tr (BA);

(v) Bfin (H) is a dense subspace of B1 (H) .

Utilising the trace notation we obviously have that ( ) ⟨ ⟩ ∗ ∗ ∥ ∥2 ∗ | |2 A, B 2 = tr (B A) = tr (AB ) and A 2 = tr (A A) = tr A for any A, B ∈ B2 (H) . The following H¨older’stype inequality has been obtained by Ruskai in [30]

[ ( )] [ ( )] − α − 1 α |tr (AB)| ≤ tr (|AB|) ≤ tr |A|1/α tr |B|1/(1 α) (19)

1/α 1/(1−α) where α ∈ (0, 1) and A, B ∈ B (H) with |A| , |B| ∈ B1 (H) . 1 In particular, for α = 2 we get the Schwarz inequality [ ( )] [ ( )] 1/2 1/2 |tr (AB)| ≤ tr (|AB|) ≤ tr |A|2 tr |B|2 (20) with A, B ∈ B2 (H) . Some Slater’s type trace inequalities for convex functions 83

If A and B are selfadjoint operators with A ≤ B and P ∈ B1 (H) with ≥ 1/2 1/2 ≤ 1/2 1/2 P 0, then P AP P BP . Since tr is a positive linear( functional) 1/2 1/2 ≤ and( since tr(XY) ) = tr(YX), it follows that tr (PA) = tr P AP tr P 1/2BP 1/2 = tr (PB) . Therefore, if A and B are selfadjoint operators with A ≤ B and P ∈ B1 (H) with P ≥ 0, then

tr (PA) ≤ tr (PB) . (21)

If A ≥ 0 and P ∈ B1 (H) with P ≥ 0, then

0 ≤ tr (PA) ≤ ∥A∥ tr (P ) . (22)

Indeed, since A ≤ ∥A∥ 1H for A ≥ 0, then (22) follows by (21). Moreover, for any selfadjoint A, − |A| ≤ A ≤ |A|. So it follows by (21) that

− tr(P |A|) ≤ tr(PA) ≤ tr(P |A|) i.e., |tr (PA)| ≤ tr (P |A|) (23) for any A a selfadjoint operator and P ∈ B1 (H) with P ≥ 0. For the theory of trace functionals and their applications the reader is referred to [33]. For some classical trace inequalities see [5], [7], [29] and [38], which are continuations of the work of Bellman [2]. For related works the reader can refer to [1], [3], [5], [18], [19], [20], [21], [31] and [35].

3. Slater Type Trace Inequalities

B+ { ∈ B ≥ } We denote by 1 (H) := P : P 1 (H) and P 0 . The following result holds:

Theorem 11. Let I be an interval and f : I → R be a convex and differentiable function on ˚I (the interior of I) whose derivative f ′ is continuous on ˚I. If A is a selfadjoint operator on the Hilbert space H with 84 Silvestru Sever Dragomir

Sp (A) ⊆ [m, M] ⊂ ˚I and f ′ (A) is a positive invertible operator on H, then ( ) ′ ≤ tr [P Af (A)] − tr [P f (A)] 0 f ′ (24) ( tr [P f (A)] )( tr (P ) ) tr [P Af ′ (A)] tr [P Af ′ (A)] tr (PA) ≤ f ′ − , tr [P f ′ (A)] tr [P f ′ (A)] tr (P )

∈ B+ \{ } for any P 1 (H) 0 .

Proof. Since f is convex and diﬀerentiable on ˚I, then we have

f ′ (s)(t − s) ≤ f (t) − f (s) ≤ f ′ (t)(t − s) (25) for any t, s ∈ [m, M] . Now, if we ﬁx t ∈ [m, M] and apply the property (P) for the operator A, then we have

′ ′ ′ ′ tf (A) − Af (A) ≤ f (t) · 1H − f (A) ≤ f (t) t · 1H − f (t) A (26) for any t ∈ [m, M]. If we apply the property (21) to the inequality (26) then we have [ ] [ ] t tr P f ′ (A) − tr P Af ′ (A) ≤ f (t) tr (P ) − tr [P f (A)] (27) ≤ f ′ (t) t tr (P ) − f ′ (t) tr (PA)

∈ B+ \{ } for any P 1 (H) 0 . ′ Now, since A is selfadjoint with m1H ≤ A ≤ M1H and f (A) is positive, then mf ′ (A) ≤ Af ′ (A) ≤ Mf ′ (A) .

If we apply again the property (21), then we get [ ] [ ] [ ] m tr P f ′ (A) ≤ tr P Af ′ (A) ≤ M tr P f ′ (A) , which shows that tr [P Af ′ (A)] t := ∈ [m, M] . 0 tr [P f ′ (A)] Observe that since f ′ (A) is a positive invertible operator on H, then tr [P f ′ (A)] > ∈ B+ \{ } 0 for any P 1 (H) 0 . Some Slater’s type trace inequalities for convex functions 85

Finally, if we put t = t0 in the equation (27), then we get ′ [ ] [ ] tr [P Af (A)] ′ − ′ ′ tr P f (A) tr P Af (A) (28) tr [P( f (A)] ) ′ ≤ tr [P Af (A)] − f ′ tr (P ) tr [P f (A)] ( tr [P f (A)] ) ′ ′ ≤ ′ tr [P Af (A)] tr [P Af (A)] f ′ ′ tr (P ) ( tr [P f (A)] ) tr [P f (A)] tr [P Af ′ (A)] − f ′ tr (PA) , tr [P f ′ (A)] which is equivalent to the desired result (24).

Remark 1. It is important to observe that, the condition that f ′ (A) is a positive invertible operator on H can be replaced with the more general assumption that

tr [P Af ′ (A)] [ ] ∈ ˚I and tr P f ′ (A) ≠ 0 (29) tr [P f ′ (A)] ∈ B+ \{ } for any P 1 (H) 0 , which may be easily veriﬁed for particular convex functions f in various examples as follows. Also, as pointed out by the referee, if ⟨f ′ (A) x, x⟩ > 0 for any x ∈ H, ̸ ′ ∈ B+ \{ } x = 0, then tr [P f (A)] > 0 for any P 1 (H) 0 and the inequality (24) is valid as well.

Remark 2. Now, if the function is concave on ˚I and the condition (29) holds, then we have the inequalities ( ) ′ ≤ tr [P f (A)] − tr [P Af (A)] 0 f ′ (30) (tr (P ) )(tr [P f (A)] ) tr [P Af ′ (A)] tr (PA) tr [P Af ′ (A)] ≤ f ′ − , tr [P f ′ (A)] tr (P ) tr [P f ′ (A)] ∈ B+ \{ } for any P 1 (H) 0 .

Utilising the inequality (30) for the concave function f : (0, ∞) → R, f (t) = ln t, then we can state that ( ) ( ) tr (P ln A) tr (P ) tr PA−1 tr (PA) 0 ≤ − ln ≤ − 1 (31) tr (P ) tr (PA−1) tr (P ) tr (P ) 86 Silvestru Sever Dragomir

∈ B+ \{ } for any positive invertible operator A and P with P 1 (H) 0 . Utilising the inequality (24) for the convex function f : (0, ∞) → R, f (t) = t−1, then we can state that ( ) ( ) ( ) ( ) tr PA−2 tr PA−1 tr (PA) tr PA−2 tr PA−1 0 ≤ − ≤ − , (32) tr (PA−1) tr (P ) tr (P ) tr (PA−1) tr (PA−2)

∈ B+ \{ } for any positive invertible operator Aand P with P 1 (H) 0 . If we take B = A−1 in (32), then we get the equivalent inequality ( ) ( ) ( ) tr PB2 tr (PB) tr PB2 tr PB−1 tr (PB) 0 ≤ − ≤ − , (33) tr (PB) tr (P ) tr (PB) tr (P ) tr (PB2) for any positive invertible operator B and P with P ∈ B1 (H) \{0} . If we write the inequality (24) for the convex function f (t) = exp (αt) with α ∈ R \{0} , then we get ( ) tr [PA exp (αA)] tr [P exp (αA)] 0 ≤ exp α − (34) ( tr [P exp (αA)] )( tr (P ) ) tr [PA exp (αA)] tr [PA exp (αA)] tr (PA) ≤ α exp α − , tr [P exp (αA)] tr [P exp (αA)] tr (P )

∈ B+ \{ } for any selfadjoint operator A and P 1 (H) 0 .

4. Further Reverses

We use the following Gr¨uss’type inequalities [14]:

Lemma 12. Let S be a selfadjoint operator with m1H ≤ S ≤ M1H and f :[m, M] → C a continuous function of bounded variation on [m, M]. For ∈ B ∈ B+ \{ } any C (H) and P 1 (H) 0 we have the inequality ¯ ¯ ¯ ¯ ¯tr (P f (S) C) tr (P f (S)) tr (PC)¯ ¯ − ¯ (35) tr (P ) tr (P ) tr (P ) (¯( ) ¯) ∨M ¯ ¯ 1 1 ¯ tr (PC) ¯ ≤ (f) tr ¯ C − 1H P ¯ 2 tr (P ) tr (P ) m  ( )  ¯ ¯ 1/2 ∨M tr P |C|2 ¯ ¯2 1  ¯tr (PC)¯  ≤ (f) − ¯ ¯ , 2 tr (P ) tr (P ) m Some Slater’s type trace inequalities for convex functions 87

∨M where (f) is the total variation of f on the interval. m If the function f :[m, M] → C is Lipschitzian with the constant L > 0 on [m, M] , i.e. |f (t) − f (s)| ≤ L |t − s| for any t, s ∈ [m, M] , then ¯ ¯ ¯ ¯ ¯tr (P f (S) C) tr (P f (S)) tr (PC)¯ ¯ − ¯ (36) °tr (P ) tr° (P ) tr( (P¯() ) ¯) ° ° ¯ ¯ ° tr (PS) ° 1 ¯ tr (PC) ¯ ≤ L °S − 1H ° tr ¯ C − 1H P ¯ tr (P ) tr (P ) tr (P )  ( )  ° ° ¯ ¯ 1/2 ° ° tr P |C|2 ¯ ¯2 ° tr (PS) °  ¯tr (PC)¯  ≤ L °S − 1H ° − ¯ ¯ tr (P ) tr (P ) tr (P )

∈ B ∈ B+ \{ } for any C (H) and P 1 (H) 0 . Proof. For the sake of completeness we give here a simple proof. We observe that, for any λ ∈ C we have [ ( )] 1 tr (PC) tr P (A − λ1H ) C − 1H (37) tr (P ) [ ( tr (P)]) 1 tr (PC) = tr PA C − 1H tr (P ) [ ( tr (P ) )] λ tr (PC) − tr P C − 1 tr (P ) tr (P ) H tr (P AC) tr (PA) tr (PC) = − . tr (P ) tr (P ) tr (P ) Taking the modulus in (37) and utilising the properties of the trace, we have ¯ ¯ ¯ ¯ ¯tr (P AC) tr (PA) tr (PC)¯ ¯ − ¯ (38) tr (P ) ¯ [ tr (P ) tr (P() )]¯ ¯ ¯ 1 ¯ tr (PC) ¯ = ¯tr P (A − λ1H ) C − 1H ¯ tr (P ) ¯ [ ( tr (P ) ) ]¯ ¯ ¯ 1 ¯ tr (PC) ¯ = ¯tr (A − λ1H ) C − 1H P ¯ tr (P ) (¯( tr (P ) ) ¯) ¯ ¯ 1 ¯ tr (PC) ¯ ≤ ∥A − λ1H ∥ tr ¯ C − 1H P ¯ tr (P ) tr (P ) 88 Silvestru Sever Dragomir for any λ ∈ C, where for the last inequality we used the inequality (18). From the inequality (38) we have ¯ ¯ ¯ ¯ ¯tr (P f (S) C) tr (P f (S)) tr (PC)¯ ¯ − ¯ (39) tr (P ) tr (P ) (¯(tr (P ) ) ¯) ¯ ¯ 1 ¯ tr (PC) ¯ ≤ ∥f (S) − λ1H ∥ tr ¯ C − 1H P ¯ tr (P ) tr (P ) for any λ ∈ C. From (39) we get ¯ ¯ ¯ ¯ ¯tr (P f (S) C) tr (P f (S)) tr (PC)¯ ¯ − ¯ (40) ° tr (P ) tr (P ) °tr (P ) (¯( ) ¯) ° ° ¯ ¯ ° f (m) + f (M) ° 1 ¯ tr (PC) ¯ ≤ °f (S) − 1H ° tr ¯ C − 1H P ¯ . 2 tr (P ) tr (P )

Since f is of bounded variation on [m, M] , then we have ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ f (m) + f (M)¯ ¯f (t) − f (m) + f (t) − f (M)¯ ¯f (t) − ¯ = ¯ ¯ (41) 2 2 1 1 ∨M ≤ [|f (t) − f (m)| + |f (M) − f (t)|] ≤ (f) 2 2 m for any t ∈ [m, M] . From (41) we get in the order B (H) that ¯ ¯ ¯ ¯ ∨M ¯ f (m) + f (M) ¯ 1 ¯f (S) − 1H ¯ ≤ (f) 1H , 2 2 m which implies that ° ° ° ° ∨M ° f (m) + f (M) ° 1 °f (S) − 1H ° ≤ (f) . (42) 2 2 m Making use of (41) and (42) we get the ﬁrst inequality in (35). The second part is obvious by the Schwarz inequality for traces  (¯( ) ¯ ) (¯( ) ¯) 2 1/2 ¯ ¯ ¯ tr(PC) 1/2¯ ¯ − tr(PC) ¯ tr ¯ C − 1H P ¯ tr C tr(P ) 1H P  tr(P )  ≤   , tr (P )  tr (P )  Some Slater’s type trace inequalities for convex functions 89 and by noticing that (¯( ) ¯ ) ¯ ¯2 ( ) − tr(PC) 1/2 2 ¯ ¯ tr ¯ C 1H P ¯ tr P |C| ¯ ¯2 tr(P ) ¯tr (PC)¯ = − ¯ ¯ (43) tr (P ) tr (P ) tr (P ) ∈ B ∈ B+ \{ } for any C (H) and P 1 (H) 0 . From (39) we also have ¯ ¯ ¯ ¯ ¯tr (P f (S) C) tr (P f (S)) tr (PC)¯ ¯ − ¯ (44) ° tr (P ) ( tr (P)) ° tr (P ) (¯( ) ¯) ° ° ¯ ¯ ° tr (SP ) ° 1 ¯ tr (PC) ¯ ≤ °f (S) − f 1H ° tr ¯ C − 1H P ¯ tr (P ) tr (P ) tr (P ) ∈ B ∈ B+ \{ } any C (H) and P 1 (H) 0 . Since |f (t) − f (s)| ≤ L |t − s| for any t, s ∈ [m, M] , then we have in the order B (H) that

|f (S) − f (s) 1H | ≤ L |S − s1H | for any s ∈ [m, M] . In particular, we have ¯ ( ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ tr (SP ) ¯ ¯ tr (SP ) ¯ ¯f (S) − f 1H ¯ ≤ L ¯S − 1H ¯ , tr (P ) tr (P ) which implies that ° ( ) ° ° ° ° ° ° ° ° tr (SP ) ° ° tr (SP ) ° °f (S) − f 1H ° ≤ L °S − 1H ° tr (P ) tr (P ) and by (44) we get the ﬁrst inequality in (36). The second part is obvious.

We also have the following reverse of Schwarz inequality [14]:

Lemma 13. If C is a selfadjoint operator with k1H ≤ C ≤ K1H for some real numbers k < K, then ( ) ( ) tr PC2 tr (PC) 2 0 ≤ − (45) tr (P ) tr (P()¯( ) ¯) ¯ ¯ 1 1 ¯ tr (PC) ¯ ≤ (K − k) tr ¯ C − 1H P ¯ 2 tr (P ) tr (P ) [ ] ( ) ( ) 1/2 1 tr PC2 tr (PC) 2 1 ≤ (K − k) − ≤ (K − k)2 , 2 tr (P ) tr (P ) 4 90 Silvestru Sever Dragomir

∈ B+ \{ } for any P 1 (H) 0 .

Proof. If we take in (35) f (t) = t and S = C we get ¯ ¯ ¯ ( ) ( ) ¯ ¯tr PC2 tr (PC) 2¯ ¯ − ¯ (46) ¯ tr (P ) tr (P ) ¯ (¯( ) ¯) ¯ ¯ 1 1 ¯ tr (PC) ¯ ≤ (K − k) tr ¯ C − 1H P ¯ 2 tr (P ) tr (P ) [ ] ( ) ( ) 1/2 1 tr PC2 tr (PC) 2 ≤ (K − k) − . 2 tr (P ) tr (P )

Since by (43) we have ( ) ( ) tr PC2 tr (PC) 2 − ≥ 0, tr (P ) tr (P ) then by (46) we get ( ) ( ) tr PC2 tr (PC) 2 0 ≤ − (47) tr (P ) tr (P()¯( ) ¯) ¯ ¯ 1 1 ¯ tr (PC) ¯ ≤ (K − k) tr ¯ C − 1H P ¯ 2 tr (P ) tr (P ) [ ] ( ) ( ) 1/2 1 tr PC2 tr (PC) 2 ≤ (K − k) − . 2 tr (P ) tr (P )

Utilising the inequality between the ﬁrst and last term in (47) we also have [ ] ( ) ( ) 1/2 tr PC2 tr (PC) 2 1 − ≤ (K − k) , tr (P ) tr (P ) 2 which proves the last part of (45).

Theorem 14. Let I be an interval and f : I → R be a convex and differentiable function on ˚I whose derivative f ′ is continuous on ˚I. If A is a selfadjoint operator on the Hilbert space H with Sp (A) ⊆ [m, M] ⊂ ˚I and f ′ (A) is a positive invertible operator on H, or

tr [P Af ′ (A)] [ ] ∈ ˚I, tr P f ′ (A) ≠ 0 tr [P f ′ (A)] Some Slater’s type trace inequalities for convex functions 91

∈ B+ \{ } for any P 1 (H) 0 , then

( ) ′ ≤ tr [P Af (A)] − tr [P f (A)] 0 f ′ (48) tr [P f (A()] tr (P ) tr (P ) tr [P Af ′ (A)] ( ) ≤ f ′ L P, A, f ′ (A) , tr [P f ′ (A)] tr [P f ′ (A)]

∈ B+ \{ } for any P 1 (H) 0 , where

( ) tr [P Af ′ (A)] tr (PA) tr [P f ′ (A)] L P, A, f ′ (A) := − tr (P ) tr (P ) tr( P¯() ) ¯)  ¯ ¯  1 ′ − ′ 1 ¯ − tr(PA) ¯  2 (f (M) f (m)) tr(P ) tr A tr(P ) 1H P ≤  (¯( ) ¯)  ¯ ′ ¯  1 − 1 ¯ ′ − tr(P f (A)) ¯ 2 (M m) tr(P ) tr f (A) tr(P ) 1H P  [ ] ( ) 1/2  tr(PA2) 2  1 (f ′ (M) − f ′ (m)) − tr(PA)  2 tr(P ) tr(P ) ≤  [ ]  ( ) 1/2  ′ 2 ′ 2  1 − tr(P [f (A)] ) − tr(P f (A)) 2 (M m) tr(P ) tr(P ) 1 ( ) ≤ f ′ (M) − f ′ (m) (M − m) . 4

Proof. Utilising Lemma 12 and Lemma 13 we have

tr (P f ′ (A) A) tr (P f ′ (A)) tr (PA) 0 ≤ − (49) tr (P ) tr (P ) (tr¯( (P ) ) ¯) ( ) ¯ ¯ 1 ′ ′ 1 ¯ tr (PA) ¯ ≤ f (M) − f (m) tr ¯ A − 1H P ¯ 2 tr (P ) tr (P ) [ ] ( ) ( ) 1/2 1 ( ) tr PA2 tr (PA) 2 ≤ f ′ (M) − f ′ (m) − 2 tr (P ) tr (P ) 1 ( ) ≤ f ′ (M) − f ′ (m) (M − m) 4 92 Silvestru Sever Dragomir and

tr (P f ′ (A) A) tr (P f ′ (A)) tr (PA) 0 ≤ − (50) tr (P ) tr(¯ ((P ) tr (P ) ) ¯) ¯ ′ ¯ 1 1 ¯ ′ tr (P f (A)) ¯ ≤ (M − m) tr ¯ f (A) − 1H P ¯ 2 tr (P ) tr (P ) ( )  1/2 ′ 2 ( ) 1 tr P [f (A)] tr (P f ′ (A)) 2 ≤ (M − m)  −  2 tr (P ) tr (P ) 1 ( ) ≤ f ′ (M) − f ′ (m) (M − m) 4

∈ B+ \{ } for any P 1 (H) 0 . The positivity of

′ ′ tr (P f (A) A) − tr (P f (A)) tr (PA) tr (P ) tr (P ) tr (P ) follows by Cebyˇsev’straceˇ inequality for synchronous functions of selfadjoint operators, see [15].

The case of convex and monotonic functions is as follows:

Corollary 15. Let I be an interval and f : I → R be a convex and differentiable function on ˚I whose derivative f ′ is continuous on ˚I. If A is a selfadjoint operator on the Hilbert space H with Sp (A) ⊆ [m, M] ⊂ ˚I and f ′ (m) > 0, then ( ) tr [P Af ′ (A)] tr [P f (A)] f ′ (M) ( ) 0 ≤ f − ≤ L P, A, f ′ (A) , (51) tr [P f ′ (A)] tr (P ) f ′ (m)

∈ B+ \{ } for any P 1 (H) 0 .

The proof follows by (48) observing that ( ) tr (P ) tr [P Af ′ (A)] f ′ (M) 0 ≤ f ′ ≤ tr [P f ′ (A)] tr [P f ′ (A)] f ′ (m)

∈ B+ \{ } for any P 1 (H) 0 . Some Slater’s type trace inequalities for convex functions 93

If we consider the monotonic nondecreasing convex function f (t) = tp with p ≥ 1 and t ≥ 0, then by (51) we have the sequence of inequalities

( ) tr (PAp) p tr (PAp) 0 ≤ − (52) tr (PAp−1) tr (P ) ( ( )) ( ) − M p 1 tr (PAp) tr (PA) tr PAp−1 ≤ p − m tr (P ) tr (P ) tr (P ) ( ) − 1 M p 1 ≤ p2 2 m (¯( ) ¯)  ( ) ¯ ¯  M p−1 − mp−1 1 tr ¯ A − tr(PA) 1 P ¯  tr(P ) tr(P ) H × (¯( ) ¯)  ¯ p−1 ¯  − 1 ¯ p−1 − tr(PA ) ¯ (M m) tr(P ) tr ¯ A tr(P ) 1H P ¯ ( ) − 1 M p 1 ≤ p2 2 m  [ ] ( ) 1/2  ( ) 2 2  p−1 p−1 tr(PA ) tr(PA)  M − m −  tr(P ) tr(P ) ×  [ ( ) ]  2 1/2  2(p−1) p−1  − tr(PA ) − tr(PA )  (M m) tr(P ) tr(P )

( ) − 1 M p 1 ( ) ≤ p2 M p−1 − mp−1 (M − m) 4 m

∈ B+ \{ } ⊆ ⊂ ∞ for any P 1 (H) 0 and A with Sp (A) [m, M] (0, ) .

Theorem 16. Let I be an interval and f : I → R be a convex and twice diﬀerentiable function on ˚I whose second derivative f ′′ is bounded on ˚I, i.e. there is a positive constant K such that 0 ≤ f ′′ (t) ≤ K for any t ∈ ˚I. If A is a selfadjoint operator on the Hilbert space H with Sp (A) ⊆ [m, M] ⊂ ˚I and f ′ (A) is a positive invertible operator on H, or

tr [P Af ′ (A)] [ ] ∈ ˚I, tr P f ′ (A) ≠ 0 tr [P f ′ (A)] 94 Silvestru Sever Dragomir

∈ B+ \{ } for any P 1 (H) 0 , then ( ) ′ ≤ tr [P Af (A)] − tr [P f (A)] 0 f ′ (53) ° tr [P f (A)] ° tr (P )(¯( ) ¯) ° ° ¯ ¯ ° tr (PA) ° 1 ¯ tr (PA) ¯ ≤ K °A − 1H ° tr ¯ A − 1H P ¯ tr (P() tr (P ) ) tr (P ) tr (P ) tr [P Af ′ (A)] × f ′ tr [P f ′ (A)] tr [P f ′ (A)] ° ° [ ( ) ( ) ] ° ° 2 2 1/2 ° tr (PA) ° tr PA tr (PA) ≤ K °A − 1H ° − tr (P ) tr (P ) tr (P ) ( ) ′ × tr (P ) ′ tr [P Af (A)] ′ f ′ tr [P f (A)] ° tr [P f (A)] ° ( ) ° ° ′ 1 ° tr (PA) ° tr (P ) ′ tr [P Af (A)] ≤ (M − m) K °A − 1H ° f 2 tr (P ) tr [P f ′ (A)] tr [P f ′ (A)]

∈ B+ \{ } for any P 1 (H) 0 .

Proof. From (48) we have ( ) ′ ≤ tr [P Af (A)] − tr [P f (A)] 0 f ′ (54) tr [P f (A()] tr (P ) tr (P ) tr [P Af ′ (A)] ( ) ≤ f ′ L P, A, f ′ (A) , tr [P f ′ (A)] tr [P f ′ (A)]

∈ B+ \{ } for any P 1 (H) 0 . From (36) we also have

( ) (0 ≤) L P, A, f ′ (A) (55) ° ° (¯( ) ¯) ° ° ¯ ¯ ° tr (PA) ° 1 ¯ tr (PA) ¯ ≤ K °A − 1H ° tr ¯ A − 1H P ¯ tr (P ) tr (P ) tr (P ) ° ° [ ( ) ( ) ] ° ° 2 2 1/2 ° tr (PA) ° tr PA tr (PA) ≤ K °A − 1H ° − tr (P ) tr (P ) tr (P )

∈ B+ \{ } for any P 1 (H) 0 . Some Slater’s type trace inequalities for convex functions 95

Therefore, by (54) and (55) we get ( ) ′ ≤ tr [P Af (A)] − tr [P f (A)] 0 f ′ ° tr [P f (A)] ° tr (P )(¯( ) ¯) ° ° ¯ ¯ ° tr (PA) ° 1 ¯ tr (PA) ¯ ≤ K °A − 1H ° tr ¯ A − 1H P ¯ tr (P() tr (P ) ) tr (P ) tr (P ) tr [P Af ′ (A)] × f ′ tr [P f ′ (A)] tr [P f ′ (A)] ° ° [ ( ) ( ) ] ° ° 2 2 1/2 ° tr (PA) ° tr PA tr (PA) ≤ K °A − 1H ° − tr (P ) tr (P ) tr (P ) ( ) tr (P ) tr [P Af ′ (A)] × f ′ tr [P f ′ (A)] tr [P f ′ (A)] that proves the second and third inequalities in (53). The last part follows by Lemma 13.

The inequality (53) can be also written for the convex function f (t) = tp with p ≥ 1 and t ≥ 0, however the details are not presented here. Acknowledgement. The author would like to thank very much the anonymous referee for many valuable suggestions that have been imple- mented in the ﬁnal version of the manuscript.

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1Mathematics, College of Engineering & Science Victoria University, PO Box 14428 Melbourne City, MC 8001, Australia. 2School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa

(Received May 23, 2016)