Quasilinearity of some composite functionals associated to Schwarz's inequality for inner products

The quasilinearity of certain composite functionals associated to the celebrated Schwarz’s inequality for inner products is investigated . Applications for operators in Hilbert spaces are given as well. 1. Introduction Let X be a linear space over the real or complex number …eld K and let us denote by H (X) the class of all positive semi-de…nite Hermitian forms on X; or, for simplicity, nonnegative forms on X; i.e., the mapping h ; i : X X ! K belongs to H (X) if it satis…es the conditions: (i) hx; xi 0 for all x 2 X; (ii) h x+ y; zi = hx; zi+ hy; zi for all x; y 2 X and ; 2 K; (iii) hy; xi = hx; yi for all x; y 2 X: If h ; i 2 H (X) ; then the functional k k = h ; i is a semi-norm on X and the following version of Schwarz’s inequality holds: (1.1) kxk kyk jhx; yij for each x; y 2 H: In addition, if h ; i is an inner product on X; i.e., satis…es the condition (iv) hx; xi = 0 only if x = 0; then the equality case holds in (1.1) if and only if the vectors x and y are linearly dependent. Now, let us observe that H (X) is a convex cone in the linear space of all mappings de…ned on X with values in K. Also, we can introduce on H (X) the following binary relation [5] (1.2) h ; i2 h ; i1 if and only if kxk2 kxk1 for any x 2 H: This is an order relation on H (X) ; see [5]: For some classical results related to the celebrated Schwarz’s inequality, see [7], [9], [11], [12] and [13]. 1991 Mathematics Subject Classi…cation. 26D15; 46C05. Key words and phrases. Additive, superadditive and subadditive functionals, Inner products, Hermitian forms, Hilber spaces, Schwarz’s inequality, Selfadjoint operators, Bounded linear operators. 1 2 S.S. DRAGOMIR AND ANCA C. GOŞA For recent results, see [1], [2], [3], [4], [8], [10] and [14] and the references therein. 2. Some Functionals Related to Schwarz’s Inequality Consider the following functional [5]: (2.1) : H (X) X ! R+; (h ; i ;x; y) := kxk kyk jhx; yij ; which is closely related to the Schwarz inequality in (1.1). Theorem 1 ([5]). The functional ( ;x; y) is nonnegative, superadditive, monotonic nondecreasing and quadratic positive homogeneous on H (X) : The nonnegativity of ( ;x; y) is in fact the Schwarz inequality (1.1). The superadditivity property is translated in the fact that (2.2) (h ; i1 + h ; i2 ;x; y) := kxk21 + kxk 2 2 kyk21 + kyk 2 2 jhx; yi1 + hx; yi2j 2 kxk21 kyk 2 1 + kxk 2 2 kyk 2 2 jhx; yi1j 2 jhx; yi2j 2 = (h ; i1 ;x; y) + (h ; i2 ;x; y) for any h ; i1 ; h ; i2 2 H (X) and x; y 2 X: If h ; i2 h ; i1 in the sense speci…ed in (1.2), then the monotonicity property mentioned in Theorem 1 becomes the inequality (2.3) (h ; i2 ;x; y) = kxk 2 2 kyk 2 2 jhx; yi2j 2 kxk21 kyk 2 1 jhx; yi1j 2 = (h ; i1 ;x; y) for any x; y 2 X: This inequality is of interest due to the fact that it creates the possibility to provide various re…nements for the Schwarz inequality for inner products as pointed out below. The quadratic positive homogeneous property means that ( h ; i ;x; y) = 2 (h ; i ;x; y) for any 2 R+. As a natural corollary of the above we have the result: Corollary 1. Let h ; i1 ; h ; i2 2 H (X) be such that there exists the constants M > m > 0 with the property that (2.4) M kxk1 kxk2 m kxk1 for any x 2 X; meaning that the seminorms k k2 and k k1 are equivalent. Then we have the inequalities (2.5) M h kxk21 kyk 2 1 jhx; yi1j 2 i kxk22 kyk 2 2 jhx; yi2j 2 m h kxk21 kyk 2 1 jhx; yi1j 2 i for any x; y 2 X: Another functional that can be associated with Schwarz’s inequality is the following one : H (X) X ! R+; (h ; i ;x; y) := kxk kyk jhx; yij 1=2 : The properties of this functional have been established in 1994 by B. Mond and the author: QUASILINEARITY OF SOME FUNCTIONALS 3 Theorem 2 ([6]). The functional ( ;x; y) is nonnegative, superadditive, monotonic nondecreasing and positive homogeneous on H (X) : One can realize that the superadditivity property of ( ;x; y) implies the same property for ( ;x; y) and therefore provides an alternative proof for Theorem 1. A di¤erent functional associated with the order one version of Schwarz’s inequality, namely kxk kyk jhx; yij ; has been also considered in [5]. The de…nition of this functional is : H (X) X ! R+; (h ; i ;x; y) := kxk kyk jhx; yij and its properties are incorporated in Theorem 3 ([5]). The functional ( ;x; y) is nonnegative, superadditive, monotonic nondecreasing and positive homogeneous on H (X) : As a consequence of this result that may be useful for applications we have: Corollary 2. Let h ; i1 ; h ; i2 2 H (X) be such that there exists the constants M > m > 0 with the property (2.4). Then we have the inequalities (2.6) M [kxk1 kyk1 jhx; yi1j] kxk2 kyk2 jhx; yi2j m 2 [kxk1 kyk1 jhx; yi1j] for any x; y 2 X: Motivated by the above results, we investigate in the present paper some composite functionals that are related to the above ones, establish their superadditivity and monotonicity properties and apply them for bounded linear operators in Hilbert spaces. 3. Some Composite Functionals and Their Properties Now, assume that : H (X) X ! R+ is a nonnegative, superadditive and r-positive homogeneous on H (X) ; meaning that ( h ; i ;x; y) = r (h ; i ;x; y) for any 0: For e; x; y 2 X with e 6= 0 and p; q 1 we consider the composite functional e;p;q ( ;x; y) : H (X)! [0;1) given by (3.1) e;p;q (h ; i ;x; y) := kek 2(1 1 p )q q (h ; i ;x; y) ; where is as above. The following result holds: Theorem 4. Assume that : H (X) X ! R+ is a nonnegative, superadditive and r-positive homogeneous on H (X) ; then the functional e;p;q ( ;x; y) de…ned by (3.1) is superadditive, monotonic nondecreasing and q r + 1 1 p positive homogeneous on H (X) : Proof. First of all we observe that the following elementary inequality holds: (3.2) ( + ) ( ) p + p for any ; 0 and p 1 (0 < p < 1) : 4 S.S. DRAGOMIR AND ANCA C. GOŞA Indeed, if we consider the function fp : [0;1) ! R, fp (t) = (t+ 1) t we have f 0 p (t) = p h (t+ 1) p 1 t 1 i : Observe that for p > 1 and t > 0 we have that f 0 p (t) > 0 showing that fp is strictly increasing on the interval [0;1). Now for t = ( > 0; 0) we have fp (t) > fp (0) giving that + 1 p p > 1; i.e., the desired inequality (3.2). For p 2 (0; 1) we have that fp is strictly decreasing on [0;1) which proves the second case in (3.2). We will prove …rst the case q = 1: Let x; y 2 X: Since ( ;x; y) is superadditive and p 1, then we have by (3.2) that (3.3) p (h ; i1 + h ; i2 ;x; y) [ (h ; i1 ;x; y) + (h ; i2 ;x; y)] p p (h ; i1 ;x; y) + p (h ; i2 ;x; y) for any h ; i1 ; h ; i2 2 H (X) : Let h ; i1 ; h ; i2 2 H (X) : If e 2 X; e 6= 0 is such that either he; ei1 = 0 or he; ei2 = 0; then the superadditivity property is trivially satis…ed, so we can assume further that he; ei1 6= 0 and he; ei2 6= 0: Therefore, by (3.3) we have that p (h ; i1 + h ; i2 ;x; y) he; ei1 + he; ei2 p (h ; i1 ;x; y) + p (h ; i2 ;x; y) he; ei1 + he; ei2 (3.4) = he; ei1 (h ; i1;x;y) he;ei1 + he; ei2 (h ; i2;x;y) he;ei2 he; ei1 + he; ei2 = he; ei1 (h ; i1;x;y) he;ei 1 p + he; ei2 (h ; i2;x;y) he;ei 2 p