OSTROWSKI AND TRAPEZOID TYPE INEQUALITIES RELATED TO POMPEIU S MEAN VALUE THEOREM WITH COMPLEX EXPONENTIAL WEIGHT

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1 Journl of Mthemticl Ineulities Volume, Numer 4 (07), doi:0.753/jmi OSTROWSKI AND TRAPEZOID TYPE INEQUALITIES RELATED TO POMPEIU S MEAN VALUE THEOREM WITH COMPLEX EXPONENTIAL WEIGHT PIETRO CERONE, SEVER S. DRAGOMIR AND EDER KIKIANTY (Communicted y A. Aglić Aljinović) Astrct. We present some ineulities of Ostrowski nd trpezoid type with complex exponentil weight, for complex-vlued solutely continuous functions. These ineulities re relted to Pompeiu s men vlue theorem. Specil cses of these ineulities re pplied to otin (i) some pproximtion results for the finite Fourier nd Lplce trnsforms; (ii) refinements of the Ostrowski nd trpezoid ineulities; nd (iii) new Ostrowski nd trpezoid type ineulities.. Introduction In 938, Ostrowski 7 proved the following estimte of the integrl men: THEOREM. Let f :, R e continuous on, nd differentile on (,) with f (t) M < for ll t (,). Then, for ny x,, we hve f (x) ( ) dt x M ( ). () The constnt 4 is est possile, in the sense tht it cnnot e replced y smller untity. Ineulity () is referred to s Ostrowski s ineulity. For its generlistions nd relted results we refer the reders to Drgomir nd Rssis 5. Another estimte of the integrl men is given y the trpezoid rule s follows. THEOREM. (Cerone nd Drgomir 7) Under the ssumptions of Theorem, we hve (x ) f ()+( x) f () ( ) dt x M ( ), () for ny x,. The constnt 4 is est possile. Mthemtics suject clssifiction (00): Primry: 6D0, 6D5. Keywords nd phrses: Ostrowski ineulity, trpezoid ineulity, complex exponentil weight, Lplce trnsform, Fourier trnsform. c D l,zgre Pper JMI

2 948 P. CERONE, S.S.DRAGOMIR AND E. KIKIANTY Ineulity () is known s the trpezoid ineulity. For its generlistions nd relted results, we refer the reders to Cerone nd Drgomir 7. It is importnt to note tht the ounds in ineulities () nd() re the sme. Cerone 6, Remrk stted tht there is strong reltionship etween the Ostrowski nd the trpezoidl functionls which is highlighted y the symmetric trnsformtions mongst their kernels. In 946, Pompeiu 9 derived vrint of Lgrnge s men vlue theorem, known s Pompeiu s men vlue theorem (cf. Shoo nd Riedel, p. 83), s given elow: THEOREM 3. For every rel-vlued function f differentile on n intervl, not contining 0 nd for ll pirs x x in,, there exists point ξ etween x nd x such tht x f (x ) x f (x ) = f (ξ ) ξ f (ξ ). (3) x x Pompeiu s men vlue theorem is utilised in order to provide nother pproximtion of the integrl men, s given elow. THEOREM 4. (Drgomir, 005 9) Let f :, R e continuous on, nd differentile on (,) with, not contining 0. Then for ny x,, we hve the ineulity + f (x) dt x x ( x ) f l f. (4) where l(t)=t, t,. The constnt 4 is est possile. We refer the reders to Pop 0, Pečrić nd Ungr 8, Acu nd Sofone, nd Acu et l. for the generlistions nd extensions of Theorem 4. Ineulities of Ostrowski type which re relted to the Pompeiu s men vlue theorem re given in the ppers y Drgomir 0,. Further ineulities of Ostrowski nd trpezoid types which re relted to the Pompeiu s men vlue theorem cn e found in Cerone, Drgomir, nd Kikinty 8. Some exponentil Pompeiu type ineulities for complex-vlued solutely continuous functions re given in Drgomir, with pplictions to otin some new Ostrowskitype ineulities. We recllthe results on the Ostrowski type ineulities, in the next theorem. THEOREM 5. Let f :, C e n solutely continuous function on the intervl, nd α = β + iγ C with β > 0. Then, for ny x, we hve exp(α) f (x)exp(α) exp(αx) dt α (5) β B (,,x,α) f α f, f α f L,, / β / ( ) /p B (,,x,α) / f α f p, f α f L p,, p >, p + =, B (,,x,α) f α f.

3 INEQUALITIES WITH COMPLEX EXPONENTIAL WEIGHT 949 where B (,,x,α)= e xβ ( x + ) + ( e β + e β e ), xβ β for nd B (,,x,α) := exp(xβ )(x )+β exp(β ) exp(xβ ). If β = 0, then for ny x, we hve exp(iγ) f (x)exp(iγ) exp(iγx) dt iγ (6) 4 + ( x + ) ( ) f iγ f, f iγ f L,, ( x ) + + ( ) + + x + ( ) f iγ f p, f iγ f L p,, p >, p + =, ( ) f iγ f. In this pper, we give refinements of the ineulities in Theorem 5. Welsopresent similr results for trpezoid type ineulities with complex exponentil weights. The pper is orgnised s follows. We present the min theorems concerning ineulities with complex weights in Section. We consider specil cses of the min theorems, y choosing x =( + )/, in Section 3. The ineulities involving the p-norms (with imginry weights) where p re proven to e shrp. The Fourier trnsform hs een principl nlyticl tool in mny fields of reserch, such s proility theory, untum physics, nd oundry-vlue prolems 3. The pproximtions of the finite Fourier trnsform of different clsses of functions hve een considered y employing integrl ineulities of Ostrowski type. We refer to Brnett nd Drgomir 4 for the pproximtions of the Fourier trnsform of solutely continuous functions; to Brnett, Drgomir, nd Hnn 5 for functions of ounded vrition; nd to Drgomir, Cho, nd Kim 3 for Leesgue integrle mppings. Using pre-grüss type ineulity, Drgomir, Hnn, nd Roumeliotis 4 otined some pproximtions of the finite Fourier trnsform for complex-vlued functions. We pply the ineulities in Section 3 to otin some pproximtion results for the finite Fourier nd Lplce trnsforms. Finlly, we providesome refinements of the Ostrowski nd trpezoid ineulities in Section 4 y considering specil cses of the min theorems in Section. Welso otin some new Ostrowski nd trpezoid type ineulities.. Min theorems The min results concerning the Ostrowski nd trpezoid type ineulities with complex exponentil weights re given in this section... Ostrowski type ineulities We recll the definition of the Gmm nd incomplete Gmm functions: Γ(t)= 0 x t e x dx, nd Γ(s,x)= x t s e t dt.

4 950 P. CERONE, S.S.DRAGOMIR AND E. KIKIANTY Throughout the text, for α = β + iγ C nd <, we use the following nottion: Ψ +,α (s,t)=e sβ (β ) + Γ( + ) Γ( +,β (t s)) ; nd Ψ,α (s,t)=e tβ ( β ) + Γ( + ) Γ( +, β(t s)). THEOREM 6. Let f :, C e n solutely continuous function on,, α = β +iγ C nd < p. Let < e the Hölder conjugte of p. If β 0, then f (x) e αx ( ) e αt dt Ψ +,α(,x) f α f,x,p + Ψ,α(x,) f α f x,,p (7) Ψ +,α (,x)+ψ,α (x,) f α f,,p, for x,. Ifβ = 0,then f (x) e ixγ ( ) + (x ) ( + ) e itγ dt f iγ f,x,p + + (x ) +( x) + ( + ) + ( x) ( + ) f iγ f,,p, f iγ f x,,p (8) for x,. The ineulities in (8) re shrp. Proof. We use the Montgomery identity for the solutely continuous function g :, C (cf. Mitrinović, Pečrić, nd Fink 6, p. 565): x g(x)( ) g(t)dt = (t )g (t)dt + (t )g (t)dt, (9) x where x,. If g(t) = /e αt,theng (t) =(f (t) α )/e αt ; nd with this choice of g, (9) ecomes: f (x) e αx ( ) x = e αt dt (t ) f (t) α e αt dt + x (t ) f (t) α e αt dt. (0)

5 INEQUALITIES WITH COMPLEX EXPONENTIAL WEIGHT 95 Tke the modulus of (0) nd mke use of the Hölder s ineulity to otin the following ineulities for < p nd its Hölder s conjugte : f (x) e αx ( ) e αt dt ( x ( (t ) e αt ) ) dt f ( α f,x,p +( ( t) e αt ) ) dt f α f x,,p ( x = (t ) e dt) tβ ( f α f,x,p + ( t) e dt) tβ f α f x,,p. We evlute the integrl x β (x ) (t ) e tβ dt = e β (β ) z e z dz y letting z =(t )β, nd thus ( x (t ) e tβ dt x 0 x x = e β (β ) Γ( + ) Γ( +,β (x )), ) = e β (β ) + Γ( + ) Γ( +,β (x )) = Ψ +,α(,x). Now, we evlute the integrl ( β )( x) ( t) e tβ dt = e β ( β ) z e z dz y letting z = ( t)β, nd thus ( ) ( t) e tβ dt x = e β ( β ) + 0 = e β ( β ) Γ( + ) Γ( +, β( x)), Γ( + ) Γ( +, β( x)) = Ψ,α(x,), nd this proves (7). If β = 0, then f (x) e ixγ ( ) dt eitγ ( x (t ) dt) ( f iγ f,x,p + ( t) dt) f iγ f x,,p = + (x ) ( + ) f iγ f,x,p + + (x ) +( x) + ( + ) + ( x) ( + ) f iγ f,,p, x f iγ f x,,p

6 95 P. CERONE, S.S.DRAGOMIR AND E. KIKIANTY for ny x,. This completes the proof. The shrpness of the ineulities in (8) is given y Proposition. In prticulr, when p = ( = ) in Theorem 6, ineulities (7) nd(8) tke simpler forms s follows: COROLLARY. Let f :, C e n solutely continuous function on, nd α = β + iγ C. Ifβ 0, then we hve f (x)( ) e αx e αt dt e β (x )β + e xβ β f α f,x, + e β +( x)β e xβ β f α f x,, () (( ) + β e β + e β + x β )e xβ f α f,,, for ny x,. Ifβ = 0, then we hve f (x)( ) e ixγ dt eitγ (x ) f iγ f,x, +( x) f iγ f x,, ( 4 ( ) + x + ) f iγ f,,, () for ny x,. The constnts nd 4 in () re shrp. The cse for the -norm is s follows: THEOREM 7. Let f :, C e n solutely continuous function on,, α = β + iγ C. Ifβ 0,then f (x) e αx ( ) e αt dt (3) β e (β +) f α f,x, +( x)e xβ f α f x,,, β > 0& + β x, (x )e xβ f α f,x, + β e (β +) f α f x,,, β < 0& + β x, (x )e xβ f α f,x, +( x)e xβ f α f x,,, otherwise, β e (β +) +( x)e xβ f α f,,, β > 0& + β x, (x )e xβ + β e (β +) f α f,,, β < 0& + β x, ( )e xβ f α f x,,, otherwise,

7 INEQUALITIES WITH COMPLEX EXPONENTIAL WEIGHT 953 for ny x,. Ifβ = 0,then f (x) e ixγ ( ) dt eitγ (x ) f iγ f,x, +( x) f iγ f x,, ( ) f iγ f,,, (4) for ny x,. Proof. Tke the modulusof (0) nd mkeuse of the Hölder s ineulity to otin: f (x) e αx ( ) e αt dt sup (t )e tβ f α f,x, + sup ( t)e tβ f α f x,,. t,x t x, Define the functions: A(t)=(t )e tβ for t,x nd B(t)=( t)e tβ for t x,. Cse : β > 0. We hve B (t)= e βt ( + β ( t)). If β > 0, then B (t) 0 for ll t x,, thus, the supremum is ttined t t = x. We hve A (t) =e βt ( β (t )). The sttionry point of A is t =(β + )/β. Cse : t x ( + β x),wehvea (t )= βe (β +) 0; thus, the supremum is ttined t t = t. Cse : t > x ( + β > x),wehve β (t ) > β (x ) > 0, which implies tht A (t) > 0forllt,x. Thus, the supremum is ttined t t = x. Cse : β < 0. We hve A (t) 0forllt,x, thus, the supremum is ttined t t = x. The sttionry point of B is t =(β + )/β. Cse : t x ( + β x),wehveb (t )=βe (β +) 0; thus, the supremum is ttined t t = t. Cse : t < x ( + β < x),wehve0< + β ( x) < + β (t x), which implies tht B (t) < 0forllt x,. Thus, the supremum is ttined t t = x. This completes the proof of (3). If β = 0, then f (x) e ixγ ( ) e itγ dt sup (t ) f iγ f,x, + sup ( t) f iγ f x,, t,x t x, (x ) f iγ f,x, +( x) f iγ f x,, ( ) f iγ f,,. This completes the proof... Trpezoid type ineulities The proofs for Theorems 8, 9, nd Corollry elow re similr to Theorems 6, 7, nd Corollry. We utilise the trpezoid identity for solutely continuous function g :, C g()( x)+g()(x ) g(t)dt = (t x)g (t)dt, x,, (5)

8 954 P. CERONE, S.S.DRAGOMIR AND E. KIKIANTY insted of the Montgomery identity. We omit the proofs. THEOREM 8. Let f :, C e n solutely continuous function on,, α = β + iγ C nd < p. Let < e rel numer such tht p + =. If β 0,then f () f () ( x)+ eα e α (x ) e αt dt Ψ,α (,x) f α f,x,p + Ψ +,α (x,) f α f x,,p (6) Ψ,α (,x)+ψ+,α (x,) f α f,,p, for ny x,. Ifβ = 0,then f () f () ( x)+ eiγ e iγ (x ) + (x ) ( + ) f iγ f,x,p + + (x ) +( x) + ( + ) e itγ dt + ( x) ( + ) f iγ f,,p, for ny x,. The ineulities in (7) re shrp. f iγ f x,,p (7) COROLLARY. Let f :, C e n solutely continuous function on, nd α = β + iγ C. Ifβ 0, then we hve f () f () ( x)+ eα e α (x ) e αt dt e xβ +(x )β e β β f α f,x, + e xβ ( x)β + e β β f α f x,, (8) β e xβ +(x )β e β ( x)β + e β f α f,,, for ny x,. Ifβ = 0, then we hve f () f () ( x)+ eiγ e iγ (x ) dt eitγ (x ) f iγ f,x, +( x) f iγ f x,, ( 4 ( ) + x + ) f iγ f,,, for ny x,. The constnts nd 4 in (9) re shrp. (9)

9 INEQUALITIES WITH COMPLEX EXPONENTIAL WEIGHT 955 THEOREM 9. Let f :, C e n solutely continuous function on,, α = β + iγ C. Ifβ 0,then f () f () ( x)+ eα e α (x ) e αt dt (0) (x )e β f α f,x, + β e (xβ +) f α f x,,, β > 0&x + β, β e (xβ +) f α f,x, +( x)e β f α f x,,, β < 0&x + β, (x )e β f α f,x, +( x)e β f α f x,,, otherwise, (x )e β + β e (xβ +) f α f,,, β > 0&x + β, β e (xβ +) +( x)e β f α f,,, β < 0&x + β, (x )e β +( x)e β f α f,,, otherwise, for x,. Ifβ = 0,then f () f () ( x)+ eiγ e iγ (x ) for x,. e itγ dt (x ) f iγ f,x, +( x) f iγ f x,, ( ) f iγ f,,, () REMARK. Note the similrity of the ounds in Corollries nd lso, Theorems 6 nd 8, nd similrly, Theorems 7 nd 9. The first set of upper ounds in () (7)nd(3), respectively nd () (8)nd(4), respectively cn e otined y letting = x, x = in the first term, nd x =, = x in the second term in (8) (6) nd (0), respectively nd (9)(7)nd(), respectively. 3. Specil cses of the min theorems nd pproximtions of the Lplce nd Fourier trnsforms One my otin simpler ineulities from the min theorems, y choosing x = ( + )/. These specil cses re pplied to pproximte the Lplce nd Fourier trnsforms. For ineulities involving the p-norms where p, the choice of x =(+ )/ proves tht the ineulities re shrp. We summrise the results in this section. COROLLARY 3. Let f :, C e n solutely continuous function on,, α = β + iγ C nd < p <. Let> e rel numer such tht p + =. If β 0, then the following ineulities hold: f ( ) + ( ) e αt dt e β + β + e β β f α f, +, () e α(+) + e β + + β e β β f α f +,, β e β + e β e + β f α f,,.

10 956 P. CERONE, S.S.DRAGOMIR AND E. KIKIANTY For the cse of the p-norms ( < p < ), we hve: f ( ) + ( ) e α(+) e αt dt (3) ( Ψ +,α, + ) ( ) + f α f, +,p + Ψ,α, f α f +,,p ( Ψ +,α, + ) ( ) + + Ψ,α, f α f,,p. For the cse of the -norm, we hve: f ( ) + ( ) e α(+) e αt dt (4) β e (β +) f + α f, +, + e β f α f +,,, β > 0& + β x, + e β f α f, +, + β e (β +) f α f +,,, β < 0& + β x, + e β f α f, +, + f α f +,,, otherwise, β e (β +) + + e β f α f,,, β > 0& + β x, + e β + β e (β +) f α f,,, β < 0& + β x, ( )e + β f α f,,, otherwise, If β = 0, then the following ineulities hold: f ( ) + ( ) dt e i + γ eitγ 8 ( ) f iγ f, +, + f iγ f +,, 4 ( ) f iγ f,,. (5) For the cse of the p-norms ( < p < ), we hve: f ( ) + + ( ) e ixγ ( ) dt eitγ f iγ f ( + ), +,p + f iγ f +,,p + ( ) ( + ) For the cse of the -norm, we hve: f ( ) + ( ) dt e i + γ eitγ f iγ f,,p. (6) f iγ f, +, + f iγ f +,, ( ) f iγ f,,. (7)

11 INEQUALITIES WITH COMPLEX EXPONENTIAL WEIGHT 957 PROPOSITION. Ineulities (5) nd (6) re shrp. Proof. We show tht the constnts 8 nd 4 in (5) re shrp. We ssume tht the ineulities hold for A,B > 0 insted of 8 nd 4, respectively: f ( ) + ( ) dt e i + γ eitγ A( ) f iγ f, +, + f iγ f +,, Let γ = 0 nd choose f (x)= x + B( ) f iγ f,,. on,. Thus we hve ( ) 4 A( ) B( ) ; which yields A 8 nd B 4. It implies tht the constnts nd 4 in () reshrp. We show tht the ineulities in (6) re shrp. We ssume the ineulities hold for C,D > 0 insted of nd, respectively: f ( ) + e ixγ ( ) e itγ dt + ( ) C f iγ f ( + ), +,p + f iγ f +,,p + ( ) D ( + ) f iγ f,,p. Let γ = 0ndtke f (x)= x + on,, nd we now hve ( ) 4 + ( ) C ( + ) + ( ) D. ( + ) Tke, then we hve ( ) 4 C ( ) 4 D ( ) 4, which yields C nd D. It implies tht the ineulities in (8)reshrp. Let f :, K (K = C,R) e Leesgue integrle mpping definedonthefinite intervl,. LetL ( f ) nd F ( f ) e their finite Lplce nd Fourier trnsforms, respectively, defined y L ( f )(α) := F ( f )(t) := f (s)e αs ds, α C, f (s)e πits ds, t R. REMARK. (Lplce trnsform pproximtions) By rewriting f (s)e αt dt = L ( f )(α) in (), (3), nd (4), we otin error ounds in terms of the p-norms ( p ), for the pproximtion of L ( f )(α) y f ( + α(+) )( )e.

12 958 P. CERONE, S.S.DRAGOMIR AND E. KIKIANTY REMARK 3. (Fourier trnsform pproximtions) Let u R. Choose γ = πu in (5), (6), nd (7). By rewriting e πiut dt = F ( f )(u), we otin error ounds in terms of the p-norms ( p ), for the pproximtion of F ( f )(t) y ( ) + f ( )e iπ(+)t. COROLLARY 4. Let f :, C e n solutely continuous function on,, α = β + iγ C nd < p <. Let> e rel numer such tht p + =. If β 0, then the following ineulities hold: f () e α + f () e α e αt dt e + β + + e β + β β e β β β + e β β f α f, +, f α f +,, (8) e + β e β e β + β ( e β e β) f α f,,. For the cse of the p-norms ( < p < ), we hve f () e α + f () e α e αt dt (9) ( Ψ,α, + ) ( ) + f α f, +,p + Ψ+,α, f α f +,,p ( Ψ,α, + ) ( ) + + Ψ +,α, f α f,,p. For the cse of the -norm, we hve f () e α + f () e α e αt dt (30) e β f α f, +,+ + β e ( β +) f α f +,,, β > 0&x+ β, + β e ( β +) f α f, +,+ e β f α f +,,, β < 0&x+ β, e β f α f, +, + e β f α f +,,, otherwise, e β + β + β e ( β +) + + e ( β +) f α f,,, β > 0&x + β,, e β f α f,,, β < 0&x + β e β + e β f α f,,, otherwise,

13 INEQUALITIES WITH COMPLEX EXPONENTIAL WEIGHT 959 If β = 0, then the following ineulities hold: f () e iγ + f () e iγ e αt dt 8 ( ) f iγ f, +, + f iγ f +,, 4 ( ) f iγ f,,. For the cse of the p-norms ( < p < ), we hve f () e iγ + f () e iγ e αt dt + ( ) f iγ f ( + ), +,p + f iγ f +,,p + ( ) f iγ f ( + ),,p. For the cse of the -norm, we hve f () e iγ + f () e iγ e αt dt f iγ f, +, + f iγ f +,, ( ) f iγ f,,. (3) (3) (33) PROPOSITION. The ineulities (3) nd (3) re shrp. The proof follows similrly to tht of Proposition ; nd we omit the proof. It follows from Proposition tht ineulities (9), nd (7)reshrp. REMARK 4. (Lplce trnsform pproximtions) By rewriting f (s)e αt dt = L ( f )(α) in (8), (9), nd (30), we otin error ounds in terms of the p-norms ( p ), for the pproximtion of L ( f )(α) y f () e α + f (). e α REMARK 5. (Fourier trnsform pproximtions) Let u R. Choose γ = πu in (3), (3) nd(33). By rewriting e πiut dt = F ( f )(u), we otin error ounds in terms of the p-norms ( p ), for the pproximtion of F ( f )(t) y f () f () + eπti e πti.

14 960 P. CERONE, S.S.DRAGOMIR AND E. KIKIANTY 4. Some new nd refined Ostrowski nd trpezoid type ineulities Using the results in Section, we otin ineulities of Ostrowski nd trpezoid type; nd we present the result in the following susections. 4.. Refinements of Ostrowski s ineulities If α = 0 in Corollry, Theorems 6 nd 7, respectively, then we hve refinements for the Ostrowski ineulity, s follows: f (x)( ) dt (x ) f,x, +( x) f x,,, (x ) + + f ( x) (+),x,p + f (+) x,,p, p >, p + =, (x ) f,x, +( x) f x,,, 4 ( ) + ( x + (x ) + +( x) + (+) ( ) f,,, ) f,,, f,,p, p >, p + =, for ny x,. The constnts in cses nd in (34) nd(35) re shrp (cf. Proposition ). (34) (35) 4.. New Ostrowski type ineulities Let h α (t)=e αt for t,. If =g(t)h α (t)=g(t)e αt in Theorem 6, Corollry, ndtheorem7, then we hve the Ostrowski ineulities: If β 0, we hve g(x)( ) g(t)dt Ψ+,α (,x) g h α,x,p + Ψ,α (x,) g h α x,,p where p > nd p + =, Ψ +,α(,x)+ψ,α(x,) g h α,,p, (36) g(x)( ) g(t)dt (37) e β (x )β + e xβ β g h α,x, + e β +( x)β e xβ β g h α x,, (( ) + β e β + e β + x β )e xβ g h α,,,

15 INEQUALITIES WITH COMPLEX EXPONENTIAL WEIGHT 96 nd g(x)( ) g(t)dt (38) β e (β +) g h α,x, +( x)e xβ g h α x,,, β > 0& + β x, (x )e xβ g h α,x, + β e (β +) g h α x,,, β < 0& + β x, (x )e xβ g h α,x, +( x)e xβ g h α x,,, otherwise, β e (β +) +( x)e xβ g h α,,, β > 0& + β x, (x )e xβ + β e (β +) g h α,,, β < 0& + β x, ( )e xβ g h α,,, otherwise, for x,. Ifβ = 0, then we hve g(x)( ) g(t)dt (x ) g h α,x, +( x) g h α x,,, (x ) + + g ( x) h (+) α,x,p + g h (+) α x,,p, p >, p + =, (x ) g h α,x, +( x) g h α x,,, 4 ( ) + ( x + ) g h α,,, (x ) + +( x) + g h (+) α,,p, p >, p + =, ( ) g h α,,, (39) (40) for ny x, Refinements of the trpezoid ineulities If α = 0 in Corollry, Theorems 8 nd 9, respectively, then we hve refinements for the trpezoid ineulity s follows: f ()(x )+ f ()( x) dt (x ) f,x, +( x) f x,,, (x ) + + f ( x) (+),x,p + f (+) x,,p, p >, p + =, (4) (x ) f,x, +( x) f x,,, 4 ( ) + ( x + (x ) + +( x) + (+) ( ) f,,, ) f,,, f,,p, p >, p + =, (4)

16 96 P. CERONE, S.S.DRAGOMIR AND E. KIKIANTY for ny x,. The constnts in cses nd in (4) nd(4) re shrp (cf. Proposition ) New trpezoid type ineulities Let h α (t)=e αt for t,. If =g(t)h α (t)=g(t)e αt in Theorem 8, Corollry, ndtheorem9, then we hve the trpezoid ineulities: If β 0, then we hve g()(x )+g()( x) g(t)dt Ψ,α (,x) g h α,x,p + Ψ +,α (x,) g h α x,,p (43) Ψ,α (,x)+ψ+,α (x,) g h α,,p, where p > nd p + =, g()(x )+g()( x) g(t)dt (44) e xβ +(x )β e β β g h α,x, + e xβ ( x)β + e β β g h α x,, β e xβ +(x )β e β ( x)β + e β g h α,,, nd g()(x )+g()( x) g(t)dt (45) (x )e β g h α,x, + β e (xβ +) g h α x,,, β > 0&x + β, β e (xβ +) g h α,x, +( x)e β g h α x,,, β < 0&x + β, (x )e β g h α,x, +( x)e β g h α x,,, otherwise, (x )e β + β e (xβ +) g h α,,, β > 0&x + β, β e (xβ +) +( x)e β g h α,,, β < 0&x + β, (x )e β +( x)e β g h α,,, otherwise, for x,. Ifβ = 0, then g()(x )+g()( x) g(t)dt (x ) g h α,x, +( x) g h α x,,, (x ) + + g ( x) h (+) α,x,p + g h (+) α x,,p, p >, p + =, (x ) g h α,x, +( x) g h α x,,, (46)

17 for x,. INEQUALITIES WITH COMPLEX EXPONENTIAL WEIGHT ( ) + ( x + ) g h α,,, (x ) + +( x) + g h (+) α,,p, p >, p + =, ( ) g h α,,, Acknowledgement. The uthors would like to thnk the nonymous referees for vlule suggestions tht hve een incorported in the finl version of the mnuscript. (47) REFERENCES A.M.ACU AND F. D. SOFONEA, On n ineulity of Ostrowski type, J. Sci. Arts, no. 3 (6) (0), A. M. ACU, A. BABOŞ AND F. D. SOFONEA, The men vlue theorems nd ineulities of Ostrowski type, Sci. Stud. Res. Ser. Mth. Inform. () (0), E. O. BRIGHAM, The Fst Fourier Trnsform nd its Applictions, Englewood Cliffs, NJ: Prentice- Hll Inc., N. S. BARNETT AND S. S. DRAGOMIR, An pproximtion for the Fourier trnsform of solutely continuous mppings, Proc. 4th Int. Conf. on Modelling nd Simultion. Victori University (00), RGMIA Res. Rep. Coll. 5 (Supplement) (00), Article N. S. BARNETT, S. S. DRAGOMIR AND G. HANNA, Error estimtes for pproximting the Fourier trnsform of functions of ounded vrition, RGMIA Res. Rep. Coll. 7 () (004), Article. 6 P. CERONE, On reltionships etween Ostrowski, trpezoidl nd Cheychev identities nd ineulities, Soochow J. Mth. 8 (3) (00), P. CERONE AND S. S. DRAGOMIR, Trpezoidl-type rules from n ineulities point of view, Hndook of nlytic-computtionl methods in pplied mthemtics, Chpmn & Hll/CRC, Boc Rton, FL, 000, P. CERONE, S. S. DRAGOMIR AND E. KIKIANTY, Ostrowski nd Trpezoid Type Ineulities Relted to Pompeiu s Men Vlue Theorem, J. Mth. Ineul. 9 (3) (05), S. S. DRAGOMIR, An ineulity of Ostrowski type vi Pompeiu s men vlue theorem, J. Ineul. Pure Appl. Mth. 6 (3) (005), Article S. S. DRAGOMIR, Another Ostrowski type ineulity vi Pompeiu s men vlue theorem, Generl Mthemtics (3) (03), 3 5 S. S. DRAGOMIR, Ineulities of Pompeiu s type for solutely continuous functions with pplictions to Ostrowski s ineulity, Act Mth. Acd. Pedgog. Nyházi. (N.S.) 30 () (04), S. S. DRAGOMIR, Exponentil Pompeiu s type ineulities with pplictions to Ostrowski s ineulity, Act Mth. Univ. Comenin. (N.S.) 84 () (05), S. S. DRAGOMIR, Y. J. CHO AND S. S. KIM, An pproximtion for the Fourier trnsform of Leesgue integrle mppings, Fixed point Theory nd Applictions 4. Y. J. Cho, J. K. Kim, nd S. M. Kong (Eds.), Nov Science Pulishers Inc., 003, S. S. DRAGOMIR, G. HANNA AND J. ROUMELIOTIS, A reverse of the Cuchy-Bunykovsky-Schwrz integrl ineulity for complex-vlued functions nd pplictions for Fourier trnsform, Bull. Koren Mth. Soc., 4 (4) (005), S. S. DRAGOMIR AND T. M. RASSIAS, Generlistions of the Ostrowski ineulity nd pplictions, Ostrowski type ineulities nd pplictions in numericl integrtion. Kluwer Acd. Pul., Dordrecht, 00, D. S. MITRINOVIĆ, J. E. PEČARIĆ, A. M. FINK, Ineulities involving functions nd their integrls nd derivtives, Mthemtics nd its Applictions (Est Europen Series) 53. Kluwer Acdemic Pulishers Group, Dordrecht, A. OSTROWSKI, Üer die Asolutweichung einer differentienren Funktionen von ihren Integrlmittelwert, Comment. Mth. Helv. 0 (938), 6 7.

18 964 P. CERONE, S.S.DRAGOMIR AND E. KIKIANTY 8 J. PEČARIĆ AND Š. UNGAR, On n ineulity of Ostrowski type, J. Ine. Pure Appl. Mth.7 (4) (006), Article 5. 9 D. POMPEIU, Sur une proposition nlogue u théorème des ccroissements finis, Mthemtic, Timişor (946), E. C. POPA, An ineulity of Ostrowski type vi men vlue theorem, Gen. Mth. 5 () (007), P. K. SAHOO AND T. RIEDEL, Men Vlue Theorems nd Functionl Eutions, World Scientific, Singpore, New Jersey, London, Hong Kong, 000. (Received Jnury 7, 05) Pietro Cerone Deprtment of Mthemtics nd Sttistics L Troe University Bundoor 3086, Austrli e-mil: p.cerone@ltroe.edu.u Sever Drgomir School of Engineering nd Science Victori University PO Box 448, Melourne 800, Victori, Austrli, nd DST-NRF Centre of Excellence in the Mthemticl nd Sttisticl Sciences School of Computer Science nd Applied Mthemtics University of the Witwtersrnd Privte Bg X3, 050 Wits, South Afric e-mil: sever.drgomir@vu.edu.u Eder Kikinty Deprtment of Mthemtics nd Applied Mthemtics University of Pretori Privte g X0, Htfield 008, South Afric e-mil: eder.kikinty@up.c.z; eder.kikinty@gmil.com Journl of Mthemticl Ineulities jmi@ele-mth.com