Order generalised gradient and operator inequalities

Journal of Inequalities and Applications, Feb 2015

We introduce the notion of order generalised gradient, a generalisation of the notion of subgradient, in the context of operator-valued functions. We state some operator inequalities of Hermite-Hadamard and Jensen types. We discuss the connection between the notion of order generalised gradient and the Gâteaux derivative of operator-valued functions. We state a characterisation of operator convexity via an inequality concerning the order generalised gradient. MSC: 47A63, 46E40.

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Order generalised gradient and operator inequalities

Dragomir and Kikianty Journal of Inequalities and Applications Order generalised gradient and operator inequalities Sever S Dragomir 1 Eder Kikianty 0 0 Department of Pure and Applied Mathematics, University of Johannesburg , PO Box 524, Auckland Park, 2006 , South Africa 1 School of Engineering and Science, Victoria University , PO Box 14428, Melbourne, Victoria 8001 , Australia We introduce the notion of order generalised gradient, a generalisation of the notion of subgradient, in the context of operator-valued functions. We state some operator inequalities of Hermite-Hadamard and Jensen types. We discuss the connection between the notion of order generalised gradient and the Gâteaux derivative of operator-valued functions. We state a characterisation of operator convexity via an inequality concerning the order generalised gradient. MSC: 47A63; 46E40 f n i= © 2015 Dragomir and Kikianty; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. subgradient inequality; operator convex function; operator inequality 1 Background Convex functions play a crucial role in many fields of mathematics, most prominently in optimisation theory. There are two main important inequalities which characterise convex functions, namely Jensen’s and Hermite-Hadamard’s inequalities. In  (), Jensen ai∗xiai ≤ ai∗f (xi)ai for any a, b ∈ I. f (x) dx ≤ for any a, b ∈ I. Inequality () is referred to as Jensen’s inequality. Hermite-Hadamard’s inequality provides a refinement for Jensen’s inequality, namely, for a convex function f : I ⊂ R → R, We refer the reader to Section  for further details regarding these inequalities. Similarly to the case of real-valued functions, the operator convexity can be characterised by some operator inequalities. Hansen and Pedersen [] characterise operator convexity via a non-commutative generalisation of Jensen’s inequality. If f is a real continuous function on an interval I, and A(H) is the set of bounded self-adjoint operators on a Hilbert space H with spectra in I, then f is operator convex if and only if for x, . . . , xn ∈ A(H) and a, . . . , an ∈ B(H) with in= ai∗ai = . We refer the reader to Section  for further details regarding this characterisation. One of the useful differential properties of convex functions is the fact that their onesided directional derivatives exist universally [, p.]. Just as the ordinary two-sided directional derivatives of a differentiable function can be described in terms of gradient vectors, the one-sided directional derivatives can be described in terms of ‘subgradient’ vectors [, p.]. A vector x∗ is said to be a subgradient of a convex function f : K ⊂ Rn → R at point x if f (x) – f (y) ≥ x∗ · (x – y) for all y ∈ K . This condition is referred to as the subgradient inequality [, p.]. If () holds for every x ∈ K , then () characterises the convexity of f (cf. Eisenberg [, Theorem ]). In this paper, we introduce the notion of order generalised gradient (cf. Section ) for operator-valued functions, which is a generalisation of () (without the assumption of convexity) in the settings of bounded self-adjoint operators on a Hilbert space. Furthermore, we state some inequalities of Hermite-Hadamard and Jensen types for the order generalised gradient in Section . Finally, in Section , we state the connection between the order generalised gradient and Gâteaux derivative of operator-valued functions. We state a characterisation of convexity analogues to () in the context of operator-valued functions. 2 Inequalities for convex functions This section serves as a point of reference for known results regarding some inequalities related to convex functions (both real-valued and operator-valued functions). 2.1 Jensen’s inequality Let C be a convex subset of the linear space X and f be a convex function on C. If p = (p, . . . , pn) is a probability sequence and x = (x, . . . , xn) ∈ Cn, then This inequality is referred to as Jensen’s inequality. Recently, Dragomir [] obtained the following refinement of Jensen’s inequality: pixi ≤ pjxj ≤ k∈m{,i.n..,n} + pkf (xk) + pkf (xk) where f , xk and pk are as defined above. For other refinements of Jensen’s inequality, we refer the reader to Pečarić and Dragomir [] and Dragomir []. The above result provides a different approach to the one that Pečarić and Dragomir [] obtained in  pixi ≤ i,...,ik= ≤ · · · ≤ pixi ≤ pi . . . pik f (qxi + · · · + qkxik+ ) where  ≤ k ≤ n and p, x are as defined above. For more refinements and applications related to the generalised triangle inequality, the arithmetic-geometric mean inequality, the f -divergence measures, Ky Fan’s inequality, etc., we refer the readers to [–] and []. 2.2 Hermite-Hadamard’s inequality The following inequality also holds for any convex function f defined on R: It was first discovered by Hermite in  in the journal Mathesis []. However, this result was nowhere mentioned in the mathematical literature and was not widely known as Hermite’s result []. Beckenbach, a leading expert on the history and the theory of convex functions wrote that this inequality was proven by Hadamard in  []. In , Mitrinović found Hermite’s note in Mathesis []. Since () was known as Hadamard’s inequality, the inequality is now commonly referred to as Hermite-Hadamard’s inequality []. Hermite-Hadamard’s inequality has been extended in many different directions. One of the extensions of this inequality is in the vector space settings. Firstly, we start with the following definitions and notation: Let X be a vector space and x, y be two distinct vectors in X. We define the segment generated by x and y to be the set [x, y] := ( – t)x + ty, t ∈ [, ] . gx,y(t) = f ( – t)x + ty . For any real-valued function f defined on the segment [x, y], there exists an associated function gx,y : [, ] → R with We remark that f is convex on [x, y] if and only if g is convex on [, ]. For any convex function defined on a segment [x, y] ⊂ X, we have the Hermite-Hadamard integral inequality (cf. Dragomir [, p.] and Dragomir [, p.]): which can be derived by the classical Hermite-Hadamard inequality () for the convex function gx,y : [, ] → R. Consider the function f (x) = x p (x ∈ X and  ≤ p < ∞), which is convex on X, then we have the following norm inequality (derived from ()) [, p.]: ( – t)x + ty p dt ≤ for any x, y ∈ X. 2.3 Non-commutative generalisation of Jensen’s inequality Hansen [] discussed Jensen’s operator inequality for operator monotone functions. Motivated by Aujla’s work [] on the matrix convexity of functions of two variables, Hansen [] characterised operator convex functions of two variables in terms of a noncommutative generalisation of Jensen’s inequality (cf. [, Theorem .]). A simplified proof of this result formulated for matrices is given in Aujla []. The case for several variables is given in Hansen []. The case for self-adjoint elements in the algebra Mn of n-square matrices is given in Hansen and Pedersen []. Finally, Hansen and Pedersen [] presented a generalisation of the above results for self-adjoint operators defined on a Hilbert space. 2.4 Subgradient inequality Recall the following definition of a subgradient []. f (x) – f (y) ≥ x∗ · (x – y) for all y ∈ K . The following theorem is a useful characterisation of convexity (cf. Eisenberg [, Theorem ]). Theorem  If U is a nonempty open subset of Rn, f : U → R is a differentiable function on U, and K is a convex subset of U, then f is convex on K if and only if f (x) – f (y) ≥ (x – y)T f (y) for all x, y ∈ K where f (y) denotes the gradient of f at y. This theorem has been generalised and employed in obtaining optimality conditions of a non-differentiable minimax programming problem in complex spaces (cf. Lai and Liu []). Note that (x – y)T f (y) can be written as f (y) · (x – y), which can be interpreted as the directional derivative of f at point y in x – y direction. 3 Order generalised gradient Throughout the paper, we use the following notation. We denote by B(H) the Banach algebra of all bounded linear operators on the Hilbert space (H, ·, · ), and by A(H) the linear subspace of all self-adjoint operators on H. We denote by P+(H) ⊂ A(H) the convex cone of all positive definite operators defined on H, that is, P ∈ P+(H) if and only if Px, x ≥ , and for all x ∈ H, Px, x =  implies x = . This gives a partial ordering (we refer to it as the operator order) on A(H), where two elements A, B ∈ A(H) satisfy A ≤ B if and only if B – A ∈ P+(H). Definition  Let C be a convex set in A(H). A function f : C → A(H) has the function ∇f : C × A(H) → A(H) as an order generalised gradient if f (A) – f (B) ≥ ∇f (B, A – B) for any A, B ∈ C in the operator order of A(H). Remark  We remark that in (), if f is a real-valued differentiable function on an open set U ⊂ Rn, and ∇f is the gradient of f , then () becomes (). We also note that there is no assumption of convexity at this point. We discuss the convexity case in Section . ∇f (B, X) := Q(BX + XB)Q is an order generalised gradient for f . Proof Observe that BX + XB ∈ A(H) and if P ∈ A(H) then P(BX + XB)P ∈ A(H). We need to prove that QAQ – QBQ ≥ Q B(A – B) + (A – B)B Q. f (A) – f (B) ≥ ∇f (B, A – B) for any A, B ∈ A(H), that is, hence () is equivalent to which is also equivalent to Q(A – B)Q ≥  which always holds. This completes the proof. ∇f (B, X) := XPB + BPX as an order generalised gradient. APA – BPB ≥ (A – B)PB + BP(A – B) = APB – BPB + BPA – BPB, APA – APB – BPA + BPB ≥ . Recall P+(H) ⊂ A(H) the convex cone of all positive definite operators defined on H, that is, P ∈ P+(H) if and only if Px, x ≥ , and for all x ∈ H, Px, x =  implies x = . Proposition  Let f : P+(H) → A(H) defined by f (A) = QA–Q, ∇f (B, X) = –QB–XB–Q is an order generalised gradient for f . Proof For B ∈ P+(H), B– ∈ P+(H) then B–XB– ∈ P+(H) for any X ∈ P+(H) and thus QB–XB–Q ∈ P+(H) showing that ∇f (B, X) ∈ A(H). We need to prove that or equivalently QA–Q – QB–Q ≥ –QB–(A – B)B–Q, QA–(B – A)B–Q + QB–(A – B)B–Q ≥  QA–(B – A)B–Q – QB–(B – A)B–Q ≥  Q A– – B– (B – A)B–Q ≥ . A– – B– A A– – B– ≥  and Q ∈ A(H). = Q A– – B– A A– – B– Q ≥ , 4 Inequalities involving order generalised gradients We start this section by the following definition. Definition  An order generalised gradient ∇f : C × A(H) → A(H) is (i) operator convex if for any B ∈ C, X, Y ∈ A(H) and α, β ≥  with α + β = ; (ii) operator sub-additive if ∇f (B, X + Y ) ≤ ∇f (B, X) + ∇f (B, Y ) for any B ∈ C and X, Y ∈ A(H); (iii) positive homogeneous if for any B ∈ C, X ∈ A(H) and α ≥ ; (iv) operator linear if R for any B ∈ C, X, Y ∈ A(H) and α, β ∈ . It can be seen that if ∇f (·, ·) is operator linear, then it is positive homogeneous and subadditive. If ∇f (·, ·) is positive homogeneous and sub-additive, then it is operator convex. Theorem  Let f : C → A(H) be operator convex and ∇f : C × A(H) → A(H) be an order generalised gradient for f . Then, for any A, B, ∈ C and t ∈ [, ], we have the inequalities ≥ tf (A) + ( – t)f (B) – f tA + ( – t)B ≥ ∇f tA + ( – t)B,  . which is equivalent to f (B) – f (A) ≥ ∇f (A, B – A), –∇f (A, B – A) ≥ f (A) – f (B). Since C is a convex set, hence by () we have –∇f A, ( – t)(B – A) ≥ f (A) – f tA + ( – t)B ≥ ∇f tA + ( – t)B, –( – t)(B – A) –∇f B, –t(B – A) ≥ f (B) – f tA + ( – t)B ≥ ∇f tA + ( – t)B, t(B – A) ≥ tf (A) + ( – t)f (B) – f tA + ( – t)B ≥ ∇f tA + ( – t)B,  , which completes the proof. Corollary  Under the assumptions of Theorem , () If ∇f (·, ·) is positive homogeneous, then we have () If ∇f (·, ·) is operator linear, then –t( – t) ∇f (B, A – B) + ∇f (A, B – A) ≥ tf (A) + ( – t)f (B) – f tA + ( – t)B ≥ . t( – t) ∇f (B, B – A) – ∇f (A, B – A) ≥ tf (A) + ( – t)f (B) – f tA + ( – t)B ≥ . Corollary  Under the assumptions of Theorem , if ∇f is positive homogeneous, then we have the following inequality: f tA + ( – t)B dt ≥ . Example  . We consider the function f (A) = QAQ with Q ∈ A(H). We note that the order generalised gradient ∇f (B, X) = Q(BX + XB)Q is operator linear. Then = Q (B – A)X + X(B – A) Q. For X = B – A, we then get Applying inequality () we have t( – t)Q(B – A)Q ∇f (B, B – A) – ∇f (A, B – A) = Q(B – A)Q. for any A, B ∈ A(H) and Q ∈ A(H). . We consider the function f (A) = APA with P ∈ P(H). We note that the order generalised gradient ∇f (B, X) = XPB + BPX is operator linear. Then If X = B – A, we then get = XP(B – A) + (B – A)PX. Applying inequality () we have t( – t)(B – A)P(B – A) for any A, B ∈ A(H) and P ∈ P(H). . For f (A) = QA–Q with Q ∈ A(H) and A ∈ P+(H), we note that the order generalised gradient ∇f (B, X) = –QB–XB–Q is operator linear. Then For X = B – A, we get ∇f (B, B – A) – ∇f (A, B – A) = –QB–(B – A)B–Q + QA–(B – A)A–Q = –Q B– – B–AB– Q + Q A–BA– – A– Q By () we have the inequality t( – t) QA–BA–Q + QB–AB–Q – QB–Q – QA–Q for any A, B ∈ P+(H) and Q ∈ A(H). 4.2 Jensen type operator inequalities In this subsection, we will state inequalities of Jensen type for order generalised gradients. Theorem  Let f : C ⊂ A(H) → A(H) be a function that possesses ∇f : C × A(H) → A(H) as an order generalised gradient. Then, for any Ai ∈ C, i ∈ {, . . . , n} and pi ≥  with Pn := in= pi > , we have the inequalities pjf (Aj) – f pj∇f Pn i= Pn i= piAi, Aj – Pn i= Proof From the definition of an order generalised gradient we have –∇f (A, B – A) ≥ f (A) – f (B) ≥ ∇f (B, A – B). in= piAi in (), then we get ≥ f (Aj) – f ≥ ∇f Pn i= Pn i= piAi, Aj – Pn i= Theorem  Under the assumptions of Theorem , we have the following results: Pn j= ≥ f (A) Pn j= Pn j= pjf (Aj) – pj∇f (A, Aj – A) pjf (Aj) + pj∇f (Aj, A – Aj). for any j ∈ {, . . . , n}. We obtain the desired inequalities () by multiplying the inequalities in () by pj ≥  and taking the sum over j from  to n; and divide the resulted inequalities Pn j= Pn j= Pn j= () If ∇f is linear, we have Pn i= Pn i= pjf (Aj) – f piAi ≥ . pjf (Aj) – f piAi ≥ ∇f Pn i= pj∇f Aj, Aj – Pn i= Pn i= pjf (Aj) – f piAi ≥ . Proof From () we also have Pn j= pj∇f (A, Aj – A) ≥ f (A) – Pn j= pj∇f (Aj, A – Aj), pj∇f (Aj, A) – pj∇f (Aj, Aj) pj∇f (A, Aj) + pj∇f (A, A). Pn j= Pn j= which completes the proof. Pn j= Pn j= Therefore, if A ∈ A(H) is such that Pn j= then we have the Slater type inequality (cf. Slater [] and Pečarić []) 5 Connection with Gâteaux derivatives In this section, we consider the connection between order generalised gradients and Gâteaux derivatives. We refer the reader to Dragomir [] for some inequalities of Jensen type, involving Gâteaux derivatives of convex functions in linear spaces. Let C ⊂ A(H) be a convex set. Then f : C → A(H) is said to be operator convex if for all t ∈ [, ] and A, B ∈ C, we have f ( – t)A + tB ≤ ( – t)f (A) + tf (B). We start with the following lemmas. t→li m± F(t) x, y = lim F(t)x, y t→± for all nonzero x, y ∈ H. Proof We provide the proof for the right-sided limit, as the proof for the left-sided limit follows similarly. Let ε >  and for x, y ∈ H, where x, y = , set ε = ε/( x H y H ). Since limt→+ F(t) = L, there exists δ such that F(t)x, y – Lx, y ≤ when  < t < δ. Note that L ∈ B(H) since B(H) is a Banach space, hence F(t) – L is also a bounded linear operator. Now, we have ≤ F(t) – L B(H) x H y H < ε x H y H = ε, which completes the proof. ∇–f (A) (B) = lim f (A + tB) – f (A) t→– t ∇+f (A) (B) = lim f (A + tB) – f (A) t→+ t exist and are bounded self-adjoint operators. Proof Fix an arbitrary B ∈ A(H), and let We want to show that G is nondecreasing. Let  < t < t, then f (A – tB) – f (A) = – f [A + t(–B)] – f (A) –t t ≤ – f [A + t(–tB)] – f (A) = f (A – t–Bt) – f (A) . Note also that which implies that f (A) ≤ f (A + tB) + f (A – tB); f (A + tB) – f (A) ≥ – f (A – tB) – f (A) , which implies that By the above expositions, we conclude that G is nondecreasing on R \ {}. This proves that both (∇–f (A))(B) and (∇+f (A))(B) exist and are bounded linear operators by Lemma . Note that for all t ∈ R, t =  and A, B ∈ A(H), t→li m± f [B + t(A –t B)] – f (B) x, y = t→li m± f [B + t(A –t B)] – f (B) x, y = t→li m± x, f [B + t(A –t B)] – f (B) y = x, t→li m± f [B + t(A –t B)] – f (B) y , which completes the proof. is an order generalised gradient. f [B + t(A –t B)] – f (B) = f [( – t)B +t tA] – f (B) ( – t)f (B) + tf (A) – f (B) ≤ t = f (A) – f (B). This is equivalent to Note that for all x ∈ H, ∈ P+(H). by Lemma . Since K ∈ P+(H), Kx, x ≥ , hence [limt→± K ]x, x ≥ , which implies that lim f (A) – f (B) – f [B + t(A – B)] – f (B) t→± t ∈ P+(H). Lemma  gives us which implies that ∇–f (B) (A – B) ≤ ∇+f (B) (A – B), ∇–f (B) (A – B) ≤ f (A) – f (B). Proposition  Let f : A(H) → A(H) be operator convex and A ∈ A(H). The right Gâteaux derivative of f is sub-additive, i.e. ∇+f (A) (B + C) ≤ ∇+f (A) (B) + ∇+f (A) (C) ∇–f (A) (B + C) ≥ ∇–f (A) (B) + ∇–f (A) (C) for any B, C ∈ A(H). f [A + t(B + C)] – f (A) t = f [  (A + tB) +  (A + tC)] – f (A) t f (A + tBt) – f (A) + f (A + tCt) – f (A) By a similar argument to the proof of Theorem , we conclude that ≤ tl→im+ f (A + tB) – f (A) + lim f (A + tC) – f (A) t t→+ t = ∇+f (A) (B) + ∇+f (A) (C) as desired. The proof for the left Gâteaux derivative of f follows similarly. Remark  We remark that the Gâteaux (lateral) derivative(s) is always positive homogeneous with respect to the second variable, i.e. for any function f : A(H) → A(H) and fixed A ∈ A(H), for all α ≥  and B ∈ A(H). The Gâteaux derivative, on the other hand, is always homogeneous with respect to the second variable, i.e. for any function f : A(H) → A(H) and fixed A ∈ A(H), for all α ∈ C and B ∈ A(H). The following result restates Theorem  in the setting of operator-valued functions. Corollary  Let C ⊂ A(H) be a convex set and f : C → A(H) be a Gâteaux differentiable function. Then f is operator convex if and only if ∇f defined by ∇±f (A) (B) = lim f (A + tB) – f (A) t→± t ∇f (A) (B) = lim f (A + tB) – f (A) t→ t ∇f (B) (A – B) ≤ f (A) – f (B) are order generalised gradients. Since f is assumed to be Gâteaux differentiable, both limits are equal, hence is an order generalised gradient for any A, B ∈ C. Conversely, we have the following inequality: for any A, B ∈ C. Let C, D ∈ C, t ∈ (, ), and choose A = C and B = tC + ( – t)D. Then we have Multiply () by t and () by ( – t), and add the resulting inequalities to obtain f tC + ( – t)D ≤ tf (C) + ( – t)f (D), which completes the proof. The following result follows by Corollary  and employing the fact that the Gâteaux lateral derivatives are positive homogenous. Corollary  (Hermite-Hadamard type inequality) Let f : C ⊂ A(H) → A(H) be operator convex. The following inequality holds: ∇±f (B) (A – B) + ∇±f (A) (B – A) f tA + ( – t)B dt ≥ . ∇ log(A) (B) = (sI + A)–B(sI + A)– ds ≥ – (sI + B)–(A – B)(sI + B)– ds (sI + A)–(B – A)(sI + A)– ds log tA + ( – t)B dt ≥ . Then we have the following inequalities: tA + ( – t)B log tA + ( – t)B dt ≥ . The following results follow by Theorems  and . Corollary  (Jensen type inequality) Let f : C ⊂ A(H) → A(H) be operator convex. Then, for any Ai ∈ C, i ∈ {, . . . , n} and pi ≥  with Pn := in= pi > , we have the inequalities Pn j= pj ∇±f (Aj) piAi – Aj pjf (Aj) – f Pn i= Pn i= pj ∇±f Pn i= We also have Pn j= ≥ f (A) Pn j= Pn j= pjf (Aj) – pj ∇±f (A) (Aj – A) pjf (Aj) + pj ∇±f (Aj) (A – Aj). piAi – Aj (sI + Aj)– ds Pn j= ≥ – ≥ – Pn j= Pn j= Pn i= Pn i= pj log(Aj) + log Pn i= Pn i= Pn i= Pn j= Pn j= pj log(Aj) + Pn j= ≥ – log(A) ≥ – Pn j= ∞(sI + A)–(Aj – A)(sI + A)– ds pj log(Aj) – ∞(sI + Aj)–(A – Aj)(sI + Aj)– ds. Pn j= Pn i= piAi – Aj ds Pn i= Pn i= pjAj log(Aj) – Pn i= Pn i= Pn i= Pn j= ≥ A log(A) pjAj log(Aj) – Pn j= ∞  (sI + A)–(A – I)(Aj – A) ds pjAj log(Aj) + Pn j= 6 Conclusions f (A) – f (B) ≥ ∇f (B, A – B) for any A, B ∈ C in the operator order of A(H). We have the following operator inequalities. () Operator inequalities of Hermite-Hadamard type: f tA + ( – t)B dt ≥  for any A, B ∈ C. () Operator inequalities of Jensen type: Pn j= ≥ Pn j= pj∇f Pn i= Pn i= pjf (Aj) – f piAi, Aj – Pn i= Pn j= ≥ f (A) Pn j= pjf (Aj) – pj∇f (A, Aj – A) Pn j= pjf (Aj) + pj∇f (Aj, A – Aj) for any A ∈ C, Ai ∈ C, i ∈ {, . . . , n} and pi ≥  with Pn := () Operator inequalities of Slater type: if ∇f is linear and A ∈ A(H) is such that in= pi > . Pn j= Pn j= f (A) ≥ Order generalised gradients extend the notion of subgradients, without the assumption of convexity, for operator-valued functions. This notion is also connected to the Gâteaux f (A + tB) – f (A) A, B ∈ C is an order generalised gradient. Furthermore, if f : C → A(H) is a Gâteaux differentiable f (A + tB) – f (A) A, B ∈ C is an order generalised gradient. This characterisation of convexity is a generalised version of Theorem  of Section  (cf. Eisenberg [, Theorem ]). Competing interests The authors declare that they have no competing interests. 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Sever S Dragomir, Eder Kikianty. Order generalised gradient and operator inequalities, Journal of Inequalities and Applications, 2015, 49, DOI: 10.1186/s13660-015-0574-y