Grüss Type Inequalities for Complex Functions Defined on Unit Circle with Applications for Unitary Operators in Hilbert Spaces

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1 v49n4-drgomir 5/6/3 : pge 77 # Revist Colombin de Mtemátics Volumen 49(5), págins Grüss Type Inequlities for Complex Functions Defined on Unit Circle with Applictions for Unitry Opertors in Hilbert Spces Desigulddes de tipo Grüss pr funciones complejs definids sobre el círculo unitrio con plicciones pr operdores en espcios Hilbert Silvestru Sever Drgomir Victori University, Melbourne, Austrli University of the Witwtersrnd, Johnnesburg, South Afric Abstrct. Some Grüss type inequlities for the Riemnn-Stieltjes integrl of continuous complex vlued integrnds defined on the complex unit circle C(, ) nd vrious subclsses of integrtors re given. Nturl pplictions for functions of unitry opertors in Hilbert spces re provided. Key words nd phrses. Grüss type inequlities, Riemnn-Stieltjes integrl inequlities, Unitry opertors in Hilbert spces, Spectrl theory, Qudrture rules. Mthemtics Subject Clssifiction. 6D5, 4A5, 47A63. Resumen. Se proporcionn lguns desigulddes tipo Grüss pr l integrl de Riemnn-Stieltjes de integrndos de vlores continuos complejos definidos sobre el circulo unitrio complejo C(, ) y vris subclses de integrdores son ddos. Aplicciones nturles pr funciones de operdores unitrios en espcios de Hilbert son proporcionds. Plbrs y frses clve. Desigulddes de tipo Grüss, desigulddes integrles de Riemnn-Stieltjes, Operdores unitrios en espcios de Hilbert, teori espectrl, regls de cudrtur. 77

2 v49n4-drgomir 5/6/3 : pge 78 # 78 SILVESTRU SEVER DRAGOMIR. Introduction In 4], in order to extend the Grüss inequlity to Riemnn-Stieltjes integrl, the uthor introduced the following Čebyšev functionl: T (f, g; u) := u(b) u() u(b) u() f(t)g(t) du(t) f(t)du(t) u(b) u() g(t) du(t), () where f, g re continuous on, b] nd u is of bounded vrition on, b] with u(b) u(). The following result tht provides shrp bounds for the Čebyšev functionl defined bove ws obtined in 4]. Theorem.. Let f :, b] R, g :, b] C be continuous functions on, b] nd u :, b] C with u() u(b). Assume lso tht there exists the rel constnts γ, Γ such tht γ f(t) Γ for ech t, b]. () ) If u is of bounded vrition on, b], then we hve the inequlity: T (f, g; u) Γ γ u(b) u() g b b g(s) du(s) (u), (3) u(b) u() where b (u) denotes the totl vrition of u in, b]. The constnt is shrp, in the sense tht it cnnot be replced by smller quntity. b) If u :, b] R is monotonic nondecresing on, b], then one hs the inequlity: T (f, g; u) The constnt Γ γ u(b) u() is shrp. g(t) u(b) u() g(s) du(s) du(t). (4) c) Assume tht f, g re Riemnn integrble functions on, b] nd f stisfies the condition (). If u :, b] C is Lipschitzin with the constnt L, then we hve the inequlity T (f, g; u) L(Γ γ) u(b) u() g(t) u(b) u() g(s) du(s) dt. (5) Volumen 49, Número, Año 5

3 v49n4-drgomir 5/6/3 : pge 79 #3 GRÜSS TYPE INEQUALITIES FOR COMPLEX FUNCTIONS 79 The constnt is best possible in (5). For some recent inequlities for Riemnn-Stieltjes integrl see ]- ], 3]-5] nd 8]. For continuous functions f, g : C(, ) C, where C(, ) is the unit circle from C centered in nd u :, b], π] C is function of bounded vrition on, b] with u() u(b), we cn define the following functionl s well: ( ) S C f, g; u,, b := u(b) u() u(b) u() f ( e it) g ( e it) du(t) f ( e it) du(t) u(b) u() g ( e it) du(t). (6) In this pper we estblish some bounds for the mgnitude of S C ( f, g; u,, b ) when the integrnds f, g : C(, ) C stisfy some Hölder s type conditions on the circle C(, ) while the integrtor u is of bounded vrition. It is lso shown tht this functionl cn be nturlly connected with continuous functions of unitry opertors on Hilbert spces to obtin some Grüss type inequlities for two functions of such opertors. We recll here some bsic fcts on unitry opertors nd spectrl fmilies tht will be used in the sequel. We sy tht the bounded liner opertor U : H H on the Hilbert spce H is unitry iff U = U. It is well known tht (see for instnce 7, p. 75-p. 76]), if U is unitry opertor, then there exists fmily of projections { E λ, clled the }λ,π] spectrl fmily of U with the following properties: ) E λ E µ for λ µ π; b) E = nd E π = H (the identity opertor on H); c) E λ+ = E λ for λ < π; d) U = π e iλ de λ, where the integrl is of Riemnn-Stieltjes type. Moreover, if { F λ is fmily of projections stisfying the requirements )-d) bove for the opertor U, then F λ = E λ for ll λ, }λ,π] π]. Also, for every continuous complex vlued function f : C(, ) C on the complex unit circle C(, ), we hve f(u) = π f ( e iλ) de λ (7) Revist Colombin de Mtemátics

4 v49n4-drgomir 5/6/3 : pge 8 #4 8 SILVESTRU SEVER DRAGOMIR where the integrl is tken in the Riemnn-Stieltjes sense. nd In prticulr, we hve the equlities π f(u)x, y = f ( e iλ) d E λ x, y (8) π f(u)x ( = f e iλ ) d Eλ x = for ny x, y H. nd π Exmples of such functions of unitry opertors re for n n integer. exp(u) = U n = π π exp ( e iλ) de λ e inλ de λ ( f e iλ ) d E λ x, x, (9) We cn lso define the trigonometric functions for unitry opertor U by: sin(u) = π nd the hyperbolic functions by sinh(u) = where sinh(z) := π sin ( e iλ) de λ nd cos(u) = sinh ( e iλ) de λ nd cosh(u) = exp z exp( z) ] π π cos ( e iλ) de λ cosh ( e iλ) de λ nd cosh(z) := exp z+exp( z) ], z C.. Inequlities for Riemnn-Stieltjes Integrl We sy tht the complex function f : C(, ) C stisfies n H-r-Hölder s type condition on the circle C(, ), where H > nd r (, ] re given, if f(z) f(w) H z w r () for ny w, z C(, ). If r = nd L = H then we cll it of L-Lipschitz type. Consider the power function f : C {} C, f(z) = z m where m is nonzero integer. Then, obviously, for ny z, w belonging to the unit circle C(, ) we hve the inequlity f(z) f(w) m z w Volumen 49, Número, Año 5

5 v49n4-drgomir 5/6/3 : pge 8 #5 GRÜSS TYPE INEQUALITIES FOR COMPLEX FUNCTIONS 8 which shows tht f is Lipschitzin with constnt L = m on the circle C(, ). For ±, rel numbers, consider the function f : C(, ) C, f (z) = z. Observe tht f(z) f(w) z w = () z w for ny z, w C(, ). If z = e it with t, π], then we hve Therefore, for ny z, w C(, ). z = Re(z) + z = cos t + z + = ( ). Utilizing () nd () we deduce nd w () f (z) f (w) ( ) z w (3) for ny z, w C(, ), showing tht the function f is Lipschitzin with constnt L = ( ) on the circle C(, ). For other exmples of Lipschitzin functions tht cn be constructed for power series on Bnch lgebrs, see 6]. The following result holds: Theorem.. Assume tht f : C(, ) C is of H-r-Hölder s type nd g : C(, ) C is of K-q-Hölder s type. If u :, b], π] C is function of bounded vrition with u() u(b), then where S C (f, g; u,, b) HKB r,q (, b) B r,q (, b) := r+q Proof. We hve the following identity mx (s,t),b] u(b) u() b (u)] (4) ( ) s t r+q sin. (5) S C (f, g; u,, b) = u(b) u() ] ( b f ( e it) f ( e is)] g ( e it) g ( ) e is)] du(s) du(t). (6) Revist Colombin de Mtemátics

6 v49n4-drgomir 5/6/3 : pge 8 #6 8 SILVESTRU SEVER DRAGOMIR It is known tht if p : c, d] C is continuous function nd v : c, d] C is of bounded vrition, then the Riemnn-Stieltjes integrl d p(t) dv(t) exists c nd the following inequlity holds d p(t) dv(t) mx d p(t) (v). (7) c t c,d] c Applying this property twice, we hve SC (f, g; u,, b) = u(b) u() ( f ( e it) f ( e is)] g ( e it) g ( e is)] ) du(s) du(t) u(b) u() mx f ( e it) f ( e is)] g ( e it) g ( e is)] b du(s) (u) t,b] u(b) u() mx (t,s),b] f ( e it) f ( e is)] g ( e it) g ( e is)] b (u)]. (8) Utilizing the properties of f nd g we hve f ( e it) f ( e is)] g ( e it) g ( e is)] HK e is e it r+q (9) for ny s, t, b]. Since e is e it = e is ( Re e i(s t)) + e it = cos(s t) = 4 sin ( s t for ny t, s R, then e is e it ( ) r+q s t r+q = r+q sin for ny t, s R. Utilizing (8) nd () we deduce the desired result (4). ) () Volumen 49, Número, Año 5

7 v49n4-drgomir 5/6/3 : pge 83 #7 GRÜSS TYPE INEQUALITIES FOR COMPLEX FUNCTIONS 83 Remrk.. If b = π nd = then obviously there re s, t, π] such tht s t = π showing tht mx (s,t),π] ( ) s t r+q sin =. In this sitution we hve SC (f, g; u,, π) r+q HK u(π) u() π (u)]. () Moreover, if f nd g re Lipschitzin with constnts L nd N, respectively the inequlity () becomes S C (f, g; u,, π) π LN u(π) u() (u)]. () Remrk.3. For intervls smller thn π, i.e. < b π then for ll t, s, b], π] we hve t s (b ) π. Since the function sin is incresing on ], π, then we hve successively tht mx (t,s),b] ( ) ( ) s t sin = sin mx t s = sin (t,s),b] ( b ). (3) In this cse we get the inequlity SC (f, g; u,, b) HKBr,q (, b) u(b) u() b (u)] (4) where ( ) b r+q B r,q (, b) := r+q sin. (5) Moreover, if f nd g re Lipschitzin with constnts L nd N, respectively then ( ) b B(, b) := B, (, b) = sin nd the inequlity (4) becomes SC (f, g; u,, b) LN sin We lso hve: ( ) b b u(b) u() (u)]. (6) Revist Colombin de Mtemátics

8 v49n4-drgomir 5/6/3 : pge 84 #8 84 SILVESTRU SEVER DRAGOMIR Theorem.4. Assume tht f : C(, ) C is of H-r-Hölder s type nd g : C(, ) C is of K-q-Hölder s type. If u :, b], π] C is M- Lipschitzin function with u() u(b), then SC (f, g; u,, b) p+q M HK u(b) u() C p,q(, b) (7) where ( ) C r,q (, b) := s t r+q sin ds dt r+q (r + q + )(r + q + ) (b )r+q+. (8) Proof. It is well known tht if p :, b] C is Riemnn integrble function nd v :, b] C is Lipschitzin with the constnt M >, then the Riemnn- Stieltjes integrl p(t) dv(t) exists nd the following inequlity holds: p(t) dv(t) M p(t) dt. (9) Utilizing this property nd the identity (6) we hve SC (f, g; u,, b) = u(b) u() ( f ( e it) f ( e is)] g ( e it) g ( e is)] ) du(s) du(t) M u(b) u() f ( e it) f ( e is)] g ( e it) g ( e is)] du(s) dt M u(b) u() f ( e it) f ( e is)] g ( e it) g ( e is)] ds dt. Utilizing the properties of f nd g we hve f ( e it) f ( e is)] g ( e it) g ( e is)] HK e is e it r+q (3) Volumen 49, Número, Año 5

9 v49n4-drgomir 5/6/3 : pge 85 #9 GRÜSS TYPE INEQUALITIES FOR COMPLEX FUNCTIONS 85 for ny s, t, b], which implies tht HK f ( e it) f ( e is)] g ( e it) g ( e is)] ds dt e is e it r+q ds dt = r+q HK Utilizing the well known inequlity we hve sin x x, for ny x R ( s t sin ) r+q ds dt. ( s t sin ) r+q ds dt r+q = r+q s t r+q ds dt t (t s) r+q ds + t ] (s t) r+q ds dt = (t ) r+q+ + (b t) r+q+ r+q dt r + q + (b ) r+q+ = r+q (r + q + )(r + q + ) = r+q (b )r+q+ (r + q + )(r + q + ) nd the bound in (8) is proved. The cse of Lipschitzin integrtors is of importnce nd cn be stted s follows: Corollry.5. Assume tht u :, b], π] C is M-Lipschitzin function with u() u(b). If f nd g re Lipschitzin with constnts L nd N, respectively then SC (f, g; u,, b) 4M ( NL b u(b) u() ) ( )] b sin. (3) Revist Colombin de Mtemátics

10 v49n4-drgomir 5/6/3 : pge 86 # 86 SILVESTRU SEVER DRAGOMIR Proof. We hve to clculte C, (, b) = = = ( ) s t sin ds dt cos(s t) ds dt (b ) sin(b t) sin( t) ] dt ]. Since nd then C, (, b) = sin(b t) dt = cos(b ) sin( t) dt = cos(b ) (b ) ( cos(b ) )] ( b (b ) 4 sin = ( b = ) sin ( b )] )] nd the inequlity (3) then follows from (7). The cse of monotonic nondecresing integrtors is s follows: Theorem.6. Assume tht f : C(, ) C is of H-r-Hölder s type nd g : C(, ) C is of K-q-Hölder s type. If u :, b], π] R is monotonic nondecresing function with u() < u(b), then S C (f, g; u,, b) p+q HK ] D r,q (, b) (3) u(b) u() where D r,q (, b) := mx s,t,b] ( ) s t r+q sin du(s) du(t) ( ) s t r+q (33) sin ]. u(b) u() Volumen 49, Número, Año 5

11 v49n4-drgomir 5/6/3 : pge 87 # GRÜSS TYPE INEQUALITIES FOR COMPLEX FUNCTIONS 87 Proof. It is well known tht if p :, b] C is continuous function nd v :, b] R is monotonic nondecresing on, b], then the Riemnn-Stieltjes integrl p(t) dv(t) exists nd the following inequlity holds: p(t) dv(t) p(t) dv(t). (34) Utilizing this property nd the identity (6) we hve SC (f, g; u,, b) = u(b) u() ] ( f ( e it) f ( e is)] g ( e it) g ( e is)] ) du(s) du(t) u(b) u() ] f ( e it) f ( e is)] g ( e it) g ( e is)] du(s) du(t) u(b) u() ] f ( e it) f ( e is)] g ( e it) g ( e is)] du(s) du(t). Since f ( e it) f ( e is)] g ( e it) g ( e is)] HK e is e it r+q for ny s, t, b], then f ( e it) f ( e is)] g ( e it) g ( e is)] du(s) du(t) HK nd the inequlity (3) is proved. e is e it r+q du(s) du(t) = r+q HK The bound (33) for D r,q (, b) is obvious. ( s t sin (35) ) r+q du(s) du(t) The Lipschitzin cse is of interest due to mny exmples tht cn be provided s follows: Revist Colombin de Mtemátics

12 v49n4-drgomir 5/6/3 : pge 88 # 88 SILVESTRU SEVER DRAGOMIR Corollry.7. If f nd g re Lipschitzin with constnts L nd N, respectively nd u :, b], π] R is monotonic nondecresing function with u() < u(b), then SC (f, g; u,, b) LN ] u(b) u() u(b) u() ] ( Proof. We hve to clculte D, (, b) = = = ( ) ] cos s du(s)) sin s du(s). (36) ( ) s t sin du(s) du(t) cos(s t) du(s) du(t) (u(b) ) ] u() J(, b) where Since then J(, b) := ( J(, b) = cos(s t) du(s) du(t). cos(s t) = cos s cos t + sin s sin t ( cos s du(s)) + sin s du(s)). Utilizing (3) we deduce the desired result (36). Remrk.8. Utilizing the integrtion by prts formul for the Riemnn- Stieltjes integrl, we hve nd cos s du(s) = u(b) cos b u() cos + sin s du(s) = u(b) sin b u() sin u(s) sin s ds u(s) cos s ds. If we insert these vlues in the right hnd side of (36) we cn get some expressions contining only Riemnn integrls. However they re complicted nd will not be presented here. Volumen 49, Número, Año 5

13 v49n4-drgomir 5/6/3 : pge 89 #3 GRÜSS TYPE INEQUALITIES FOR COMPLEX FUNCTIONS Applictions for Functions of Unitry Opertors We hve the following vector inequlity for functions of unitry opertors. Theorem 3.. Assume tht f : C(, ) C is of H-r-Hölder s type nd g : C(, ) C is of K-q-Hölder s type. If the opertor U : H H on the Hilbert spce H is unitry, then x, y f(u)g(u)x, y f(u)x, y g(u)x, y r+q HK π E( )x, y ] r+q HK x y (37) for ny x, y H. In prticulr, if f nd g re Lipschitzin with constnts L nd N, respectively then x, y f(u)g(u)x, y f(u)x, y g(u)x, y for ny x, y H. LN π E( )x, y ] LN x y (38) Proof. For given x, y H, define the function u(λ) := E λ x, y, λ, π]. We will show tht u is of bounded vrition nd π (u) := π ( E( )x, y ) x y. (39) It is well known tht, if P is nonnegtive selfdjoint opertor on H, i.e., P x, x for ny x H, then, the following inequlity is generliztion of the Schwrz inequlity in H: P x, y P x, x P y, y, (4) for ny x, y H. Now, if d : = t < t < < t n < t n = π is n rbitrry prtition of the intervl, π], then we hve by Schwrz s inequlity for nonnegtive opertors (4)) tht π ( E( )x, y ) = sup d sup d { n { n (Eti+ E ) } t i x, y i= ) / ) ] } / (Eti+ E ti x, x (Eti+ E ti y, y := I. (4) i= Revist Colombin de Mtemátics

14 v49n4-drgomir 5/6/3 : pge 9 #4 9 SILVESTRU SEVER DRAGOMIR By the Cuchy-Bunikovski-Schwrz inequlity for sequences of rel numbers we lso hve tht I sup d { n ) ] / n ) ] /} (Eti+ E ti x, x (Eti+ E ti y, y i= π ( E( )x, x )] / π i= ( E( )y, y )] / = x y (4) for ny x, y H. Now, from the inequlity () we hve ( Eπ x, y E x, y ) π f ( e it) g ( e it) de t x, y π f ( π e it ) de t x, y g ( e it) de t x, y r+q HK for ny x, y H. The proof is complete. π ( E( )x, y )] (43) Remrk 3.. If U : H H is n unitry opertor on the Hilbert spce H, then for ny integers m, n we hve from (38) the power inequlities x, y U m+n x, y U m x, y U n x, y mn π for ny x, y H. In prticulr, we hve x, y U x, y Ux, y π E( )x, y ] E( )x, y ] mn x y (44) x y (45) nd x, y Ux, y x, Uy π E( )x, y ] x y (46) for ny x, y H. Volumen 49, Número, Año 5

15 v49n4-drgomir 5/6/3 : pge 9 #5 GRÜSS TYPE INEQUALITIES FOR COMPLEX FUNCTIONS 9 For ±, rel numbers, consider the function f : C(, ) C, f (z) = z. This function is Lipschitzin with the constnt L = ( ) on the circle C(, ). Now, if we tke, b ±, nd use the inequlity (38) then we hve x, y ( H U) ( H bu) x, y (H U) x, y ( H bu) x, y b ( ) ( b ) π E( )x, y ] b ( ) ( b ) x y (47) for ny x, y H. In prticulr, we hve x, y ( H U) x, y ( H U) x, y ( ) 4 π E( )x, y ] ( ) 4 x y (48) for ny x, y H. Theorem 3.3. If f nd g re Lipschitzin with constnts L nd N, respectively nd U : H H is n unitry opertor on the Hilbert spce H, then x f(u)g(u)x, x f(u)x, x g(u)x, x x 4 Re(U)x, x ] Im(U)x, x LN = LN x 4 Ux, x ] (49) for ny x H, where Re(U) := U + U nd Im(U) := U U. i Revist Colombin de Mtemátics

16 v49n4-drgomir 5/6/3 : pge 9 #6 9 SILVESTRU SEVER DRAGOMIR Proof. From the inequlity (36) we hve ( Eπ x, x E x, x ) π f ( e it) g ( e it) de t x, x π f ( e it) π de t x, x g ( e it) de t x, x LN (Eπ x, x E x, x ) ( π ( π ) ] cos t de t x, x ) sin t de t x, x (5) for ny x, y H. Since Re ( e it) = cos t nd Im ( e it) = sin t, then we hve from the representtion (7) tht ( π ( π nd due to the fct tht Ux, x = Re(U) + i Im(U)]x, x ) cos t de t x, x = Re(U)x, x ) sin t de t x, x = Im(U)x, x = Re(U)x, x + i Im(U)x, x = Re(U)x, x + Im(U)x, x we deduce from (5) the desired inequlity (49). Remrk 3.4. If U : H H is n unitry opertor on the Hilbert spce H, then for ny integers m, n we hve from (49) the power inequlities x U n+m x, x U n x, x U m x, x mn x 4 ] Ux, x (5) for ny x H. In prticulr, we hve for n = m = x U x, x Ux, x x 4 Ux, x (5) for ny x H. If we tke n = nd m = n nd tke into ccount tht U n x, x = (U n ) x, x = x, U n x = U n x, x Volumen 49, Número, Año 5

17 v49n4-drgomir 5/6/3 : pge 93 #7 GRÜSS TYPE INEQUALITIES FOR COMPLEX FUNCTIONS 93 for ny x H, then we get from (5) tht x 4 U n x, x n x 4 ] Ux, x (53) for ny x H. Now, if we tke, b ±, nd use the inequlity (49), then we get x ( U) ( bu) x, x ( U) x, x ( bu) x, x b ( ) ( b ) x 4 Ux, x ] (54) for ny x H. In prticulr, we hve x ( U) x, x ( U) x, x ( ) 4 x 4 Ux, x ] (55) for ny x H. Acknowledgement. The uthor would like to thnk the nonymous referees for their comments tht hve been implemented in the finl version of the pper. References ] M. W. Alomri, A Compnion of Ostrowski s Inequlity for the Riemnn Stieltjes Integrl f(t) du(t), where f is of Bounded Vrition nd u is of r H-Hölder Type nd Applictions, Appl. Mth. Comput. 9 (3), no. 9, ] G. A. Anstssiou, A New Expnsion Formul, Cubo Mt. Educ. 5 (3), no., ] S. S. Drgomir, On the Ostrowski s Inequlity for Riemnn-Stieltjes Integrl nd Applictions, Koren J. Comput. & Appl. Mth 7 (), no. 3, ], Shrp Bounds of Čebyšev Functionl for Stieltjes Integrls nd Applictions, Bull. Austrl. Mth. Soc. 67 (3), no., Revist Colombin de Mtemátics

18 v49n4-drgomir 5/6/3 : pge 94 #8 94 SILVESTRU SEVER DRAGOMIR 5], Approximting the Riemnn Stieltjes Integrl by Trpezoidl Qudrture Rule With Applictions, Mthemticl nd Computer Modelling 54 (), ], Inequlities of Lipschitz Type for Power Series in Bnch Algebrs, RGMIA Reserch Report Collection 5 (), no. 77, 7 pp., Preprint, Online 7] G. Helmberg, Introduction to Spectrl Theory in Hilbert Spce, John Wiley, New York, USA, ] Z. Liu, Refinement of n Inequlity of Grüss Type for Riemnn-Stieltjes Integrl, Soochow J. Mth. 3 (4), no. 4, (Recibido en febrero de 4. Aceptdo en diciembre de 4) Mthemtics, College of Engineering & Science Victori University, PO Box 448 Melbourne City, MC 8 Austrli School of Computtionl & Applied Mthemtics University of the Witwtersrnd Privte Bg 3, Johnnesburg 5 South Afric e-mil: sever.drgomir@vu.edu.u Volumen 49, Número, Año 5