On some refinements of companions of Fejér s inequality via superquadratic functions

Transcription

1 Proyecciones Journl o Mthemtics Vol. 3, N o, pp , December. Universidd Ctólic del Norte Antogst - Chile On some reinements o compnions o Fejér s inequlity vi superqudrtic unctions Muhmmd Amer Lti University o Hil, Sudi Arbi Received : December. Accepted : September Abstrct In this pper some compnions o Fejér s inequlity or superqudrtic unctions re given, we lso get reinements o some known results proved in [8]. Subjclss : [] 6D5. Keywords : Hermite-Hdmrd Inequlity, convex unction, Fejér inequlity, superqudrtic unction.

2 3 Muhmmd Amer Lti. Introduction Let 6= I R,, b I with <b,let : I R be convex unction nd p :[, b] R be non-negtive integrble nd symmetric bout x = +b. The ollowing two inequlities re o gret signiicnce in literture: the irst known s Hermite-Hdmrd inequlity: + b (.) (x)dx ()+(b) b with the reversed inequlity or the concve unction, nd the second, known s Fejér s inequlity: (.) Z + b b p(x)dx (x)p(x)dx ()+(b) p(x)dx. These inequlities ttrcted the ttention o mny mthemticins over the decdes nd they generlize, improve nd extend these inequlities in number o wys, see [6, 7, 8, 9,, 9]. Let us now deine some mppings nd quote the results estblished by K.L. Tseng, S. R. Hwng nd S.S. Drgomir in [8]: G(t) = L p (t) = t +( t) + b H(t) = b I(t) = + Zb L p (t) = (b ) t x + b t x + + tx +( t) + b tb +( t) + b dx, +( t) + b +( t) + b p(x)dx,, [ (t +( t) x)+ (tb +( t) x)] p(x)dx, Zb [ (t +( t) x)+ (tb +( t) x)] dx

3 Compnions o Fejér s Inequlity For Superqudrtic Functions 3 nd S p (t) = + tb +( t) x + t +( t) x tb +( t) x + b t +( t) x + b p(x)dx, where :[, b] R is convex unction nd p :[, b] R is non-negtive integrble nd symmetric bout x = +b, t [, ]. Now we quote some results rom [8]: Theorem. [8] Let, p, I be deined s bove. Then:. The ollowing inequlity holds: (.3) Z + b b p (x) dx " Z +b 3+b Z # +3b (x)p(x b)dx + (x)p(x b)dx +b Z I (t) dt " Z + b b p (x) dx + x + + x + b # p (x) dx.. I is dierentible on [, b] nd p is bounded on [, b], thenorll t [, ] we hve the inequlity x + x + b + p (x) dx I(t) (.) " ()+(b) ( t) (b ) where kpk =sup p (x). x [,b] # (x)dx kpk,

4 3 Muhmmd Amer Lti 3. I is dierentible on [, b], then or ll t [, ] we hve the inequlity ()+(b) (.5) p (x) dx I(t) ( (b) ()) (b ) Theorem. [8] Let, p, G, I be deined s bove. Then:. The ollowing inequlity holds or ll t [, ] : p (x) dx. (.6) I(t) G(t) p(x)dx.. I is dierentible on [, b] nd p is bounded on [, b], thenorll t [, ] we hve the inequlity (.7) Z + b b I(t) p (x) dx (b )[G(t) H(t)] kpk, where kpk =sup p (x). x [,b] Theorem 3. [8] Let, p, G, I, S p be deined s bove. Then we hve the ollowing results:. S p is convex on [, ].. The ollowing inequlities hold or ll t [, ] : (.8) G(t) p (x) dx S p (t) ( t) x + + x + b p (x) dx

5 Compnions o Fejér s Inequlity For Superqudrtic Functions 33 +t ()+(b) p (x) dx ()+(b) p (x) dx, (.9) I( t) S p (t) nd I(t)+I( t) (.) 3. The ollowing equlity holds: S p (t). (.) sup S p (t) =S p () = ()+(b) x [,] p (x) dx. They used the ollowing Lemm to prove the bove results: Lemm. [7, p. 3] :[, b] R be convex unction nd let A C D B b with A + B = C + D. Then (A)+(B) (C)+(D). Let us now recll the deinition, some o the properties nd results relted to superqudrtic unctions to be used in the sequel. Deinition 5. [3, Deintion.] Let I =[,] or [, ) be n intervl in R. A unction : I R is superqudrtic i or ech x in I there exists rel number C(x) such tht (.) (y) (x) C(x)(y x)+ ( y x ) or ll y I. I is superqudrtic then is clled subqudrtic. For exmples o superqudrtic unctions see [, p. 9]. Theorem 6. [3, Theorem.3] The inequlity Z Z Z (.3) gd (g(s)) g(s) gd d(s) holds or ll probbility mesure nd ll non-negtive -integrble unction g, i nd only i is superqudrtic.

6 3 Muhmmd Amer Lti The ollowing discrete version o the bove theorem will be helpul in the sequel o the pper: Lemm 7. [, Lemm A, p.9] Suppose tht is superqudrtic. Let x r, r n, ndlet x = P n r= λ r x r where λ r nd P n r= λ r =. Then nx nx (.) λ r (x r ) ( x)+ λ r ( x r x ). r= The ollowing Lemm shows tht positive superqudrtic unctions re lso convex: Lemm 8. [3, Lemm.] Let be superqudrtic unction with C(x) s in Deinition. Then. ().. I () = () = then C(x) = (x) whenever is dierentible t x>. r= 3. I, then convex nd () = () =. In [] converse o Jensen s inequlity or superqudrtic unctions ws proved: Theorem 9. [, Theorem ] Let (Ω,A,) be mesurble spce with < (Ω) < nd let :[, ) R be superqudrtic unction. I g : Ω [m, M] [, ) is such tht g, g L (), thenwehveor ḡ = (Ω) (.5) R gd, Z (Ω) (g)d M ḡ M m (m)+ ḡ m M m (M) Z (Ω) M m ((M g) (g m)+(g m) (M g)) d. The discrete version o this theorem is: Theorem. [, Theorem ] Let :[, ) R be superqudrtic unction. Let (x,...,x n ) be n n-tuple in [m, M] n ( m M< ), nd (p,...,p n ) be non-negtive n-tuple such tht P n = P n i= p i >. Denote x = P n P ni= p i x i,then

7 Compnions o Fejér s Inequlity For Superqudrtic Functions 35 (.6) nx P n i= p i (x i ) M x x m (m)+ M m M m (M) P n (M m) nx p i [(M x i ) (x i m)+(x i m) (M x i )] i= For recent results on Fejér nd Hermite-Hdmrd type inequlities or superqudrtic unctions, we reer interested reders to [], [5] nd []. In this pper we del with mppings G(t), I(t), S p (t) ndl(t) when is superqudrtic unction. In cse when superqudrtic unction is lso non-negtive nd hence convex we get reinements o some prts o Theorem, Theorem nd o Theorem 3.. Min Results In this section we prove our min results by using the sme techniques s used in [7] nd []. Moreover, we ssume tht ll the considered integrls in this section exist. In order to prove our min results we go through some clcultions. From Lemm nd Theorem 6 or n =, we get tht (z) M z M m (m)+ z m M m (M) M z z m (z m) (M z) M m M m (.) nd (M+m z) z m M m (m)+ M z M m (M) z m M m (M z) M z M m (z m) (.) hold or superqudrtic unction, m z M, m<m. Thereore rom (.) nd (.), we hve (z)+(m +m z) (m)+(m) z m M m (.3) Now or t ollowing inequlities: (M z) M z M m (z m). +b nd x, we obtin rom (.3) the

8 36 Muhmmd Amer Lti By setting z = +b 3(+b),M = x,m= x + +b in (.3), we hve tht + b (.) x + + b 3( + b) + x + b x holds. Also, by replcing z = x + +b in (.3), we get tht,m = tx+( t) +b +b,m= t +( t) x x + + b t + b +( t) x + tx +( t) + b + b (.5) t x holds. Further, or z = 3(+b) x +b,m = t +( t)( + b x), m = t ( + b x)+( t) +b in (.3), we observe tht 3( + b) x t ( + b x)+( t) + b (.6) + t + b + b +( t)( + b x) t x holds. Agin, or z = t +b +( t) x, M = +b,m = x in (.3), we observe tht t + b +( t) x + tx +( t) + b (.7) + b + b + b (x)+ t ( t) x ( t) t x holds. Finlly, by setting z = t ( + b x)+( t) +b +b,m = +b x, m = in (.3), we get tht

9 Compnions o Fejér s Inequlity For Superqudrtic Functions 37 (.8) t ( + b x)+( t) + b + b + t + b +( t)( + b x) + b + ( + b x) t ( t) ( t) + b t x x holds. Now we re redy to stte nd prove our min results bsed on the clcultions done bove. Theorem. Let be superqudrtic integrble unction on [,b] nd p(x) be non-negtive integrble nd symmetric bout x = +b, <b. Let I be deined s bove, then we hve the ollowing inequlities: (.9) Z + b b p (x) dx " Z +b 3+b (x)p(x b)dx Z +3b # + +b (x)p(x b)dx (b x) p(x)dx, (.) nd (.) " Z +b Z 3+b Z # +3b (x)p(x b)dx + (x)p(x b)dx +b Z I(t)dt Z I(t)dt " Z + b b t b x p (x) dx + p (x) dtdx x + x + b + p (x) dx Z ( t) t b x p (x) dtdx.

10 38 Muhmmd Amer Lti Proo. Using simple techniques o integrtion nd by the ssumptions on p, wehve Z + b b Z +b Z + b p (x) dx = p(x )dtdx. Thereore rom (.), we get tht Z + b b (.) p (x) dx Z +b Z x + + b 3( + b) + x p(x )dtdx Z +b Z + b x p(x )dtdx. But (.3) Z +b Z x + + b 3( + b) + x p(x )dtdx = " Z +b 3+b Z +b = [(x)+( + b x)] p(x b)dx 3+b Z +3b # (x)p(x b)dx + +b (x)p(x b)dx. From (.), (.3) nd by the chnge o vrible x x+, we get (.9). From (.5), (.6) nd (.3), we hve (.) " Z +b 3+b Z # +3b (x) p (x b) dx + (x) p (x b) dx +b

11 Compnions o Fejér s Inequlity For Superqudrtic Functions 39 ³ R +b But Z (.5) Z +b Z t + b +( t) x + t ( + b x)+( t) +b ³ t +b +( t)( + b x) R I(t)dt = i p (x ) dtdx ³³ t ³ +b x p (x ) dtdx. Z +b Z t + b +( t) x + tx +( t) + b tx +( t) + b + t ( + b x)+( t) + b ³ i + t +b +( t)( + b x) p (x ) dtdx. From (.) nd (.5), we get tht (.6) " Z +3b 3+b Z # +3b (x) p (x b) dx + (x) p (x b) dx 3+b Z Z +b I(t)dt Z t + b x p (x ) dtdx. By the chnge o vribles t t nd x x+ in (.6), we get (.). From (.7), (.8) nd (.5), we hve Z (.7) I(t)dt Z Z +b (x)+ + b + b t ( t) x

12 3 Muhmmd Amer Lti ( t) But " Z + b b (.8) + b t x + ( + b x) p (x ) dtdx. p (x) dx + x + + x + b # p (x) dx = Z +b Z (x)+ + b + ( + b x) p (x ) dtdx. From (.7), (.8) nd by the chnge o vribles x x+ nd t t, we get (.). This completes the proo o the theorem s well. Remrk. I the superqudrtic unction is non-negtive nd hence convex, then rom (.9) we get reinement o the irst inequlity o (.3) in Theorem ; rom (.) we get reinement o the middle inequlity o (.3) in Theorem nd rom (.) we get reinement o the lst inequlity o (.3) in Theorem. Corollry 3. Let be superqudrtic integrble unction on [,b]. p(x) = b, x [, b] nd <b,thenwehve + b (.9) b (.) nd (.) where Z +3b 3+b b (x)dx Z Z Z +3b 3+b (x)dx b H(t)dt b Z H(t)dt " + b + b b H(t) = b Z ( t) t b x tx +( t) + b (b x) dx, t b x # (x)dx dtdx, dx, t [, ]. I dtdx

13 Compnions o Fejér s Inequlity For Superqudrtic Functions 3 Proo. I p(x) = b, x [, b], then I(t) = H(t), t [, ], nd thereore the proo o the corollry ollows directly rom the bove theorem. Remrk. I the superqudrtic unction is non-negtive nd thereore convex, then the inequlities in Corollry reine the inequlities in (.3) o Theorem B rom [8, p. ]. To proceed to our next result, we go gin through the similr clcultions s given beore Theorem 7. For x +b, t [, ],wehvetht t +( t) + b tx +( t) + b t ( + b x)+( t) + b tb +( t) + b b. Thereore, by replcing z = tx +( t) +b t +( t) +b in (.3), we get tht (.) t( + b x)+( t) + b tb +( t) + b (x ) b + +,M = tb +( t) +b,m= tx +( t) + b t +( t) + b (b x) (t(b x)) (t(x )) b holds. Now we re redy to stte nd prove our next result bsed on the bove clcultions. Theorem 5. Let be superqudrtic integrble unction on [,b] nd p(x) be non-negtive integrble nd symmetric bout x = +b, <b. Let I nd G be deined s bove, then the ollowing inequlity holds or ll t [, ] : x b x (.3) I(t) G(t) p (x) dx b t + b x x t p(x)dx. b

14 3 Muhmmd Amer Lti Proo. Using simple techniques o integrtion nd by the ssumptions on p, we hve tht the ollowing identity holds or ll t [, ]: (.) G(t) + p(x)dx = Z +b tb +( t) + b t +( t) + b p(x )dx. Arguing similrly s in obtining (.5), by using (.) nd (.), we get tht (.5) I(t) G(t) p(x)dx Z +b (x ) b (t(b x)) + (b x) b (t(x )) p(x )dx, or ll [, ]. By the chnge o vrible x x+ in (.5), we get (.3). This completes the proo o the theorem. Remrk 6. I the superqudrtic unction is non-negtive nd hence convex, then the inequlity (.3) represents reinement o the inequlity (.6) in Theorem. Corollry 7. Let be superqudrtic integrble unction on [,b], let p(x) = b, < b nd G, H be deined s bove. Then or ll t [, ], we hve the ollowing inequlity (.6) H(t) G(t) (b ) + b x b t x b t x b x dx. Proo. This is direct consequence o the bove theorem, since or p(x) = b, x [, b], I(t) =H(t), or ll t [, ]. Remrk 8. I the superqudrtic unction is non-negtive nd hence convex, then the inequlity (.6) represents reinement o the inequlity (.6) in [8, Theorem C, p. ].

15 Compnions o Fejér s Inequlity For Superqudrtic Functions 33 Now gin we give some clcultions or our next result. For t +b nd x, we obtin rom (.3) the ollowing inequlities: By setting m = t +( t)x, M = t +( t)( + b x) ndz = t +( t) +b in (.3), we obseve tht t +( t) + b (t +( t)x)+(t +( t)( + b x)) + b (.7) ( t) x holds. Also, by replcing m = tb +( t)x, M = tb +( t)( + b x) nd z = tb +( t) +b in (.3), we get tht tb +( t) + b (tb +( t)x)+(tb +( t)( + b x)) (.8) holds. + b ( t) x Theorem 9. Let be superqudrtic integrble unction on [,b] nd p(x) be non-negtive integrble nd symmetric bout x = +b, <b. Let S p nd G be deined s bove, then the ollowing inequlity holds or ll t [, ] : (.9) G(t) p(x)dx S p (t) ( t) b x p(x)dx. Proo. By the simple techniques o integrtion nd by the ssumptions on p, we hve the ollowing identity or ll t [, ]: (.3) S p (t) = Z +b [(t +( t)x)+(t +( t)( + b x)) +(tb +( t)x)+(tb +( t)( + b x))] p(x )dx.

16 3 Muhmmd Amer Lti From (.7), (.8) nd (.3), we hve tht Z +b t +( t) + b + (.3) tb +( t) + b p(x )dx Z +b + b S p (t) ( t) x p(x )dx, holds or ll t [, ]. From (.) nd by the chnge o vrible x +x, we get rom (.3) tht G(t) p(x)dx S p (t) ( t) b x p(x)dx, or ll t [, ]. Which is (.9) nd this completes the proo o the theorem s well. Remrk. The result o the bove theorem reines the irst inequlity o Theorem 3, when superqudrtic unction is non-negtive nd hence convex. Corollry. Let be superqudrtic integrble unction on [,b] nd let p(x) = b, x [, b], <b. Let G be deined s bove, then the ollowing inequlity holds or ll t [, ] : (.3) G(t) L(t) b ( t) b x dx. Proo. Since or p(x) = b, x [, b], S p(t) =L p (t) =L(t), or ll t [, b]. Thereore the proo o the crollry ollows directly rom the bove theorem. 3. Inequlities or dierentible superqudrtic unctions In this section we give results when is dierentible superqudrtic unction. Those results give reinements o (.) nd (.5) in Theorem ndreine (.7) o Theorem when superqudrtic unction is nonnegtive nd hence convex. Here we quote very importnt result which will be helpul in the sequel o the pper.

17 Compnions o Fejér s Inequlity For Superqudrtic Functions 35 Theorem. [, Theorem, p. 5] Let be superqudrtic integrble unction on [,b] nd p(x) be non-negtive integrble nd symmetric bout x = +b, <b.letibe deined s bove nd let p be integrble on [, b], thenor s t, t>, we hve the ollowing inequlity: (3.) t + s I(s) I(t) t s t (b x) p (x) dx t s t + s t (b x) p (x) dx. Now we stte nd prove the irst result o this section. Theorem. Let be superqudrtic unction on [,b] nd p(x) be nonnegtive integrble nd symmetric bout x = +b, <b. Let be dierentible on [, b] such tht () = () = nd p is bounded on [, b], then the ollowing inequlities hold or ll t [, ]: x + x + b (3.) + p(x)dx I(t) " Z # ()+(b) b ( t) (b ) (x)dx kpk ( t) b x where kpk =sup p (x) nd x [,b] Z ()+(b) b (3.3) p (x) dx I(t) Proo. Z +b (3.) b p(x)dx p(x)dx, ( (b) ()) (b ) t By integrtion by prts, we hve tht + b x ( + b x) (x) dx = b x p(x)dx. x + b (x)dx

18 36 Muhmmd Amer Lti = ()+(b) (b ) (x)dx. Using the substitution rules or integrtion, under the ssumptions on p, wehve (3.5) x + + x + b p(x)dx = +b x + p(x)dx x + nd (3.6) = Z +b I(t) = + + [(x)+( + b x)] p(x )dx t +b x = Z +b t x + +( t) + b +( t) + b p(x)dx tx +( t) + b t( + b x)+( t) + b or ll t [, ]. Now by the ssumptions on, wehvetht (3.7) (x) tx +( t) + b t( + b x)+( t) + b + b ( t) + b ( t) p(x )dx, p(x )+[( + b x) p(x ) x ( + b x) (x) p(x ) x p(x )

19 Compnions o Fejér s Inequlity For Superqudrtic Functions 37 + b ( t) x [ ( + b x) (x)] kpk h or ll t [, ] nd x + b ( t), +b i. x p (x ), From (3.), (3.5), (3.6) nd (3.7) nd by the chnge o vrible x +x, we get (3.). By the ssumptions on nd rom Lemm 3, we get tht ³ () +b b () b nd ³ (b) +b b (b) b. Adding these inequlities we get tht ³ ()+(b) + b (b) () (b ) b. Thus Z ()+(b) b Z + b b (3.8) p (x) dx p (x) dx ³ () (b) (b ) b p (x) dx. From (3.), or s =,wehve Z + b b (3.9) p (x) dx I(t) or ll t [, ]. From (3.8) nd (3.9), we get (3.3). theorem. t b x p(x)dx, This completes the proo o the Remrk 3. The inequlities (3.) nd (3.3) reine the inequlities (.) nd (.5) in Theorem, when the superqudrtic unction is non-negtive nd thereore convex.

20 38 Muhmmd Amer Lti Corollry. Let be superqudrtic unction on [,b] nd dierentible on [, b] such tht () = () =. I p(x) = b,thenwehvethe ollowing inequlities: (3.) (b ) x + + x + b dx H(t) nd t " ()+(b) (b ) b b ( t) b x # (x)dx dx (3.) ()+(b) H(t) ( (b) ()) (b ) b dx or ll t [, ]. b t b x dx, Now we give our lst result nd summrize the results relted to it in the remrk ollowed by Theorem. Theorem 5. Let be superqudrtic unction on [,b] nd p(x) be nonnegtive integrble nd symmetric bout x = +b, <b. Let be dierentible on [, b] such tht () = () = nd p is bounded on [, b], then or ll t [, ] we hve the inequlity: Z + b b (3.) I(t) p (x) dx (b )[G(t) H(t)] kpk t (b x) p(x)dx, where kpk =sup p (x). x [,b]

21 Compnions o Fejér s Inequlity For Superqudrtic Functions 39 Proo. (3.3) By integrtion by prts, we observe tht Z +b t x + b + b + x = t x + b tx +( t) + b t ( + b x)+( t) + b tx +( t) + b dx dx =(b ) G(t) H(t), hold or sll t [, ]. Under the ssumptions on, wehvetht (3.) tx +( t) + b + b p(x ) + t ( + b x)+( t) + b t x + b + b +t x tx +( t) + b + b t ( + b x)+( t) + b t + b x p(x ) p(x ) p(x ) p(x ) + b = t x t ( + b x)+( t) + b tx +( t) + b p(x ) t + b x p(x ) + b t x t ( + b x)+( t) + b

22 33 Muhmmd Amer Lti tx +( t) + b kpk t + b x p(x ) hold or ll t [, ] nd x h h, +b i i. Integrting (3.) over x on, +b, using (3.3), by the chnge o vrible x x+, under the ssumptions on p, we get (3.). This completes the prooothetheorem. Remrk 6. The result o Theorem reines (.7) o Theorem, when superqudrtic unction is non-negtive nd thereore convex. Corollry 7. Let be superqudrtic unction on [,b]. Let be dierentible on [, b] such tht () = () =. I p(x) = b, x [, b], then (3.5) + b H(t) [G(t) H(t)] b t b x dx, or ll t [, ]. Reerences [] S. Abrmovich, S. Bnić, M. Mtić, J. Pečrić, Jensen Steensen s nd relted inequlities or superqudrtic unctions, Mth. Ineq. Appl.,, pp. 3, (8). [] S. Abrmovich, J. Brić, J. Pečrić, Fejér nd Hermite-Hdmrd type inequlities or superqudrtic unctions, Mth. J. Anl. Appl., 3, pp. 8-56, (8). [3] S. Abrmovich, G. Jmeson, G. Sinnmon, Reining Jensen s inequlity, Bull. Mth. Soc. Sci. Mth. Roumnie (N.S.) 7 (95), pp. 3, ().

23 Compnions o Fejér s Inequlity For Superqudrtic Functions 33 [] S. Bnić, J. Pečrić, S. Vrošnec, Superqudrtic unctions nd reinements o some clssicl inequlities, J. Koren Mth. Soc. 5, pp , (8). [5] S. Bnić, Superqudrtic unctions, PhD thesis, Zgreb (in Crotin), (7). [6] S. S. Drgomir, Two mppings in connection to Hdmrd s inequlities, J. Mth. Anl. Appl., 67, pp. 9 56, (99). [7] S. S. Drgomir, Further properties o some mppings ssocited with Hermite-Hdmrd inequlities, Tmkng. J. Mth., 3 (), pp. 5 57, (3). [8] S. S. Drgomir, Y.J. Cho nd S.S. Kim, Inequlities o Hdmrd s type or Lipschitzin mppings nd their pplictions, J. Mth. Anl. Appl., 5, pp. 89 5, (). [9] S. S. Drgomir, D.S. Milošević nd J. Sándor, On some reinements o Hdmrd s inequlities nd pplictions, Univ. Belgrd. Publ. Elek. Fk. Sci. Mth.,, pp. 3, (993). [] L. Fejér, Über die Fourierreihen, II, Mth. Nturwiss. Anz Ungr. Akd. Wiss.,, pp , (96). (In Hungrin). [] Ming-In Ho, Fejer inequlities or Wright-convex unctions, JIPAM. J. Inequl. Pure Appl. Mth. 8 (), (7), rticle 9. [] J. Hdmrd, Étude sur les propriétés des onctions entières en prticulier d une unction considérée pr Riemnn J. Mth. Pures nd Appl., 58, pp. 7-5, (983). [3] M. A. Lti, On some reinements o Fejér type inequlities vi superqudrtic unctions.(to pper) [] M. A. Lti, On some new Fejér-type inequlities or superqudrtic unctions. (to pper) [5] K. L. Tseng, S. R. Hwng nd S.S. Drgomir, On some new inequlities o Hermite-Hdmrd- Fejér type involving convex unctions, Demonstrtio Mth., XL (), pp. 5 6, (7). [6] K. L. Tseng, S. R. Hwng nd S.S. Drgomir, Fejér-type Inequlities (I), (Submitted) Preprint RGMIA Res. Rep. Coll. (9), No., Article 5. [Online

24 33 Muhmmd Amer Lti [7] K. L. Tseng, S.R. Hwng nd S.S. Drgomir, Fejér-type Inequlities (II), (Submitted) Preprint RGMIA Res. Rep. Coll. (9), Supplement, Article 6, pp.-. [Online [8] K. L. Tseng, S. R. Hwng nd S.S. Drgomir, Some compnions o Fejér s inequlity or convex unctions, (Submitted) Preprint RGMIA Res. Rep. Coll. (9), Supplement, Article 9, pp.-. [Online [9] G.S. Yng nd K.L. Tseng, Inequlities o Hermite-Hdmrd-Fejér type or convex unctions nd Lipschitzin unctions, Tiwnese J. Mth., 7(3), pp. 33, (3). M. A. Lti Deprtment o Mthemtics, University o Hil, Hil, Sudi Arbi e-mil : m mer lti@hotmil.com