Bulletin of the. Iranian Mathematical Society

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1 ISSN: X Print ISSN: Online Bulletin of the Irnin Mthemticl Society Vol 3 07, No, pp 09 5 Title: Some extended Simpson-type ineulities nd pplictions Authors: K-C Hsu, S-R Hwng nd K-L Tseng Published by Irnin Mthemticl Society

2 Bull Irnin Mth Soc Vol 3 07, No, pp 09 5 Online ISSN: SOME EXTENDED SIMPSON-TYPE INEQUALITIES AND APPLICATIONS K-C HSU, S-R HWANG AND K-L TSENG Communicted by Abbs Slemi Abstrct In this pper, we shll estblish some extended Simpson-type ineulities for differentible convex functions nd differentible concve functions which re connected with Hermite-Hdmrd ineulity Some error estimtes for the midpoint, trpezoidl nd Simpson formul re lso given Keywords: Hermite-Hdmrd ineulity, Simpson ineulity, midpoint ineulity, trpezoid ineulity, convex function, concve functions, specil mens, udrture rules MSC00: Primry; 6D5; Secondry: 6A5 Introduction Throughout this pper, let < b in R The ineulity + b f b f + f b which holds for ll convex concve functions f : [, b] R, is known in the literture s Hermite-Hdmrd ineulity [8] For some results which generlize, improve, nd extend the ineulity, see [ 7] nd [9 6] In [], Tseng et l estblished the following Hermite-Hdmrd-type ineulity which refines the ineulity Article electroniclly published on 30 April, 07 Received: 5 November 0, Accepted: 9 November 05 Corresponding uthor 09 c 07 Irnin Mthemticl Society

3 Some extended Simpson-type ineulities 0 Theorem Suppose tht f : [, b] R is convex function on [, b] Then we hve the ineulity + b f [ ] 3 + b + 3b f + f f x dx b [ ] + b f + f b f + f + f b The third ineulity in is known in the literture s Bullen s ineulity Using the similr proof of Theorem, we lso note tht the ineulities in re reversed when f is concve on [, b] In [], Drgomir nd Agrwl estblished the following results connected with the second ineulity in the ineulity Theorem Let f : [, b] R be differentible function on, b If f is convex on [, b], then we hve 3 f b + f b b f + f b 8 which is the trpezoid ineulity provided f is convex on [, b] Theorem 3 Let f : [, b] R be differentible function on, b nd let p > If f p/p is convex on [, b], then we hve f b + f b [ b f p p + f p ] p p b p p + p which is the trpezoid ineulity provided f p/p is convex on [, b] In [], Perce nd Pečrić estblished the following results tht give n improved nd simplified constnt in Theorem 3 to obtin Theorem s follows: Theorem Let f : [, b] R be differentible function on, b nd If the mpping f is convex on [, b], then we hve 5 f + f b b b [ f + f b ], which is the trpezoid ineulity provided f is convex on [, b] Theorem 5 Under the ssumptions of Theorem, we hve 6 + b f b b [ f + f b ],

4 Hsu, Hwng nd Tseng which is the midpoint ineulity provided f is convex on [, b] The comprble results to Theorem nd Theorem 5 with concvity property insted of convexity Theorem 6 Under the ssumptions of Theorem nd f is concve on [, b], we hve 7 f + f b b b f + b nd 8 + b f b b + b f, which re the trpezoid ineulity nd the midpoint ineulity provided f is concve on [, b], respectively From the bove results, it is nturl to consider the extended Simpson-type formul in the following lemm Remrk 7 Let 0 α, x [, +b nd y +b, b] Then we hve the extended Simpson-type formuls s follows: The trpezoid-type formul f x + f y + b α + α f f γ + γb + f γ + γ b b b s α, x γ + γb nd y γ + γ b The trpezoid formul f x + f y + b α + α f b f + f b b s α, x nd y b 3 The midpoint formul f x + f y + b α + α f b + b f b s α 0

5 Some extended Simpson-type ineulities The Simpson-type formul f x + f y + b α + α f b f γ + γb + f γ + γ b α + b + α f b s 0 γ, x γ + γb nd y γ + γ b 5 The Simpson formul [ f x + f y α [ + b f + f 6 ] + α f s α 3, x nd y b 6 The Bullen formul f x + f y α + α f [ f + f + b + b ] + f b b + b ] + f b b b b s α, x nd y b In this pper, we estblish some extended Simpson-type ineulities which reduce the Simpson-type, trpezoid-type, midpoint-type, Bullen-type ineulities, nd generlize Theorems nd -6 Some pplictions to specil mens of rel numbers re given Finlly, the pproximtions for udrture formul re lso given Extended Simpson-type ineulity Throughout this section, let 0 α, x [, +b J, J, h t, h t t [, b] be defined s follows [ α J 3 b 3 x 3b + b x + + b y 3 + y + b ] 5b y + α 8 ], y [ +b, b], nd let 5b x x

6 3 Hsu, Hwng nd Tseng [ α J 3 b 3 x 3 + b + x x 5 b + b y y 3 b h t α t +b α t +b + y + b y 5 b t, t < x + α t, x t < +b + α t b, +b t < y t b, y t b t, t < x α +b t + α t, x t < +b h t α t +b + α b t, +b t < y b t, y t b ] + α 8 In order to prove our min results, we need the following lemm nd remrk whose proof cn be obtined by simple computtion Lemm Let, b, x, y, α, J, J, h t, h t t [, b] be defined s bove Then we hve J b 3 h t b t dt, J b 3 h t t dt, J + J b h t dt [ α b x + b + [ x +b α x b y b + x + b y + y + b y +b b ], ] + α J + bj b b h t tdt [ α 6 b x + b x b y y + b + y + b x x + + b y + + b ] + α + b, 0 < J J + J nd 0 < J J + J

7 Some extended Simpson-type ineulities Remrk Let 0 γ, ρ, x γ + γb nd y ρ + ρ b in the identities nd Then we hve J 8 αγ γ, J 8 αρ ρ nd J + J [γ α ] + ρ ρ γ Further, if γ ρ, then J J 8 αγ γ nd J + J αγ γ Now, we re redy to stte nd prove the min results Theorem 3 Let, b, x, y, α, J, J, h t, h t t [, b] be defined s bove nd let, f be defined s in Theorem Then we hve the extended Simpsontype ineulity 3 α [ f x + f y ] + α f + b b J f + J f b J + J b J + J Proof Using the integrtion by prts nd simple computtion, we hve the following identity: h t f t dt b f x + f y + b α + α f b Now, using Hölder s ineulity, the convexity of f nd Lemm, we hve the ineulity b 5 h t f t dt b b b b b h t f t dt h t f t dt h t dt h t f t dt h t dt h t b t f b + t b b dt

8 5 Hsu, Hwng nd Tseng b b b [ h t dt h t b t b h t dt b 3 f + h t t f b ] dt b h t b t dt f + b 3 h t t dt f b b J + J J f + J f b b J f + J f b J + J b J + J The ineulity 3 follows from the identity nd the ineulity 5 This completes the proof Under the conditions of Theorem 3 nd Remrk, we hve the following corollries nd remrks Corollry Using Theorem 3 nd Remrk 7, we hve f γ + γb + f γ + γ b α + b + α f b f + f b αγ γ b which is the Simpson-type ineulity provided f is convex on [, b] Corollry 5 In Corollry, let γ 0 Then, we hve f + f b + b 6 α + α f b b f + f b Remrk 6 In Corollry 5, let α 3 Then, we hve [ ] + b f + f + f b 6 b b f + f b, which is the Simpson ineulity [5] provided f is convex on [, b]

9 Some extended Simpson-type ineulities 6 Remrk 7 In Corollry 5, let α Then, we hve [ ] + b f + f + f b b b f + f b, which is the Bullen s ineulity provided f is convex on [, b] Remrk 8 If we choose α in Corollry 5, then the ineulity 6 reduces the trpezoid ineulity 5 Remrk 9 If we choose α 0 in Corollry 5, then the ineulity 6 reduces the midpoint ineulity 6 Corollry 0 In Corollry, let α Then, we hve f γ + γb + f γ + γ b b f + f b γ γ b, which is the trpezoid-type ineulity provided f is convex on [, b] Remrk In Corollry 0, let γ Then, we hve f 3+b b 8 + f +3b f + f b b, which is the second ineulity in provided f is convex on [, b] Theorem Let, b, x, y, α, J, J, h t, h t t [, b] be defined s bove nd let, f be defined s in Theorem 6 Then we hve the extended Simpsontype ineulity 7 f x + f y + b α + α f J + J b f J + J b J + J b

10 7 Hsu, Hwng nd Tseng Proof Since > nd f is concve on [, b], f is lso concve on [, b] Using the Jensen s integrl ineulity nd Lemm, we hve b 8 h t f t dt b b b b h t f t dt h t f t dt h t dt f h t tdt h t dt b b b h t dt f b J + J f J + J b J + J b b h t tdt b b h t dt The ineulity 7 follows from the identity nd the ineulity 8 This completes the proof Under the conditions of Theorem nd Remrk 7, we hve the following corollries nd remrks Corollry 3 In Theorem, let x γ +γb nd y γ+ γ b where 0 γ Then, using Remrk, we hve J J 8 αγ γ, J + J αγ γ nd f γ + γb + f γ + γ b α + b + α f b αγ γ b + b f, which is the Simpson-type ineulity provided f is concve on [, b]

11 Some extended Simpson-type ineulities 8 Corollry In Corollry 3, let γ 0 Then, we hve f + f b + b 9 α + α f b b + b f Remrk 5 In Corollry, let α 3 Then, we hve [ ] + b f + f + f b 6 b b + b f, which is the Simpson ineulity [5] provided f is concve on [, b] Remrk 6 In Corollry, let α Then, we hve + b b [ f + f + b f, ] + f b b which is the Bullen s ineulity provided f is concve on [, b] Remrk 7 If we choose α in Corollry, then the ineulity 9 reduces the trpezoid ineulity 7 Remrk 8 If we choose α 0 in Corollry, then the ineulity 9 reduces the midpoint ineulity 8 Corollry 9 In Corollry 3, let α Then, we hve f γ + γb + f γ + γ b b γ γ b f + b, which is the trpezoid-type ineulity provided f is concve on [, b] Remrk 0 In Corollry 9, let γ Then, we hve f 3+b + f +3b b b + b 8 f, which is the second ineulity in provided f is concve on [, b]

12 9 Hsu, Hwng nd Tseng 3 Applictions for specil mens In the literture, let us recll the following specil mens of the two rel numbers u nd v: The weighted rithmetic men A α u, v : αu + α v, u, v R The unweighted rithmetic men A u, v : u + v, u, v R 3 The hrmonic men H u, v : u +, u, v > 0 v The identric men { v v v u I u, v : e u if u v u, u, v > 0 u if u v 5 The logrithmic men L u, v : 6 The p-logrithmic men [ L p u, v : u v p+ u p+ p+v u { v u ln v ln u ] p if u v, u, v > 0 u if u v if u v if u v, u, v > 0, p R\ {, 0} 7 The p-power men u p + v p p M p u, v :, u, v > 0, p R\ {0} Using the bove results, we hve the following results bout the bove specil mens: Proposition 3 In Corollry, let s, ] [ +, \ {, 0},, > 0, b > 0 nd let f t t s on [, b] Then we hve Aα A A s γ b,, A s γ, b, A s, b L s s, b αγ γ s b M s, b s Corollry 3 In Proposition 3, let γ 0 Then, we hve 3 A α A s, b s, A s, b L s s, b s b M s, b s

13 Some extended Simpson-type ineulities 0 Corollry 33 In Proposition 3, let α Then, we hve 3 A A s γ b,, A s γ, b L s s, b γ γ s b M s, b s Proposition 3 In Corollry 3, let s, +,, 0, b 0 nd let f t t s on [, b] Then we hve A α A A s γ b,, A s γ, b, A s, b L s s, b αγ γ s b A s, b Corollry 35 In Proposition 3, let γ 0 Then, we hve 33 A α A s, b s, A s, b L s s, b s b A s, b Corollry 36 In Proposition 3, let α Then, we hve 3 A A s γ b,, A s γ, b L s s, b γ γ s b A s, b Proposition 37 In Corollry, let, > 0, b > 0 nd let f t t on [, b] Then we hve A α H A γ b,, A γ, b, A, b L, b αγ γ b M, b Corollry 38 In Proposition 37, let γ 0 Then, we hve 35 A α H, b, A, b L, b b M, b Corollry 39 In Proposition 37, let α Then, we hve 36 H A γ b,, A γ, b L, b γ γ b M, b Proposition 30 In Corollry 3, let > 0, b > 0 nd let f t ln t on [, b] Then we hve A α A ln A γ b,, ln A γ, b, ln A, b ln I, b αγ γ b A, b

14 Hsu, Hwng nd Tseng Corollry 3 In Proposition 30, let γ 0 Then, we hve 37 A α A ln, ln b, ln A, b ln I, b b A, b Corollry 3 In Proposition 30, let α Then, we hve 38 A ln A γ b,, ln A γ, b ln I, b γ γ b A, b Applictions for the extended Simpson udrture formul Throughout this section, let 0 α, I n : x 0 < x < [ < x n < ] x n b be prtition of the intervl [, b], l i x i+ x i, ξ i x i, x i+x i+ nd ζ i xi +x i+, x i+ i 0,,, n Define the extended Simpson udrture formul where n S α f, I n, ξ, ζ : α ftdt S α f, I n, ξ, ζ + R α f, I n, ξ, ζ, f ξ i + f ζ i n l i + α xi + x i+ f l i, nd ξ i [x i, x i + x i+ /], ζ i [x i + x i+ /, x i+ ], nd the reminder term R α f, I n, ξ i, ζ i denotes the ssocited pproximtion error of ftdt by S α f, I n, ξ, ζ Let α { 0, 3, } in the identity Then we hve the following specil formule The midpoint formul n xi + x i+ S 0 f, I n, ξ, ζ f l i The trpezoid formul S f, I n, ξ, ζ n f x i + f x i+ l i, where ξ i x i nd ζ i x i+ i 0,,, n 3 The Simpson formul n S f, I [ ] xi + x i+ n, ξ, ζ f x 3 i + f + f x i+ l i, 6

15 Some extended Simpson-type ineulities where ξ i x i nd ζ i x i+ i 0,,, n Theorem Let f be defined s in Theorem 3 nd let ftdt, S αf, I n, ξ, ζ nd R α f, I n, ξ, ζ be defined s in the identity Then, the reminder term R α f, I n, ξ, ζ stisfies the estimte R α f, I n, ξ, ζ n T i + T i li T i f x i + T i f x i+ T i + T i n mx { f, f b } T i + T i li, where nd T i α 3l 3 i T i α 3l 3 i [ ξ i x i 3xi+ x i ξ i xi + x i+ 5xi+ x i + ξ i + ζ i x i + x i+ 5xi+ x i [ xi + x i+ ξ i + x i+ ζ i 3 ζ i ] + α 8 ξ i ξ i 5x i x i+ + ξ i x i 3 + x i+ ζ i ζ i 3x i x i+ + ζ i x i + x i+ ζ i 5x i x i+ ] + α 8 for ll i 0,,, n Proof Apply Theorem 3 on the intervls [x i, x i+ ] i 0,,, n to get [ α f ξ ] i + f ζ i xi + x xi+ i+ 3 + α f l i ftdt x i T i + T i li T i f x i + T i f x i+ T i + T i for ll i 0,,, n

16 3 Hsu, Hwng nd Tseng Using the convexity of f, we hve T i f x i + T i f x i+ T i + T i [ T i b xi T i + T i b f + x i T i b xi+ + T i + T i b f + x i+ b b f b mx { f, f b } mx { f, f b } ] f b for ll i 0,,, n The ineulity follows from the ineulities 3, nd the generlized tringle ineulity This completes the proof Corollry In Theorem, let α 3 nd ξ i x i, ζ i x i+ i 0,,, n Then T i T i 8 i 0,,, n nd the Simpsontype error stisfies n R f, I l [ i f x i + f x i+ ] n, ξ, ζ 3 n mx { f, f b } Corollry 3 In Theorem, let α nd ξ i x i, ζ i x i+ i 0,,, n Then T i T i 8 i 0,,, n nd the trpzoidtype error stisfies n l [ i f x i + f x i+ ] R f, I n, ξ, ζ which is Proposition 3 in [] n mx { f, f b } Corollry In Theorem, let α 0 nd ξ i x i, ζ i x i+ i 0,,, n Then T i T i 8 i 0,,, n nd the midpointtype error stisfies n l [ i f x i + f x i+ ] R 0 f, I n, ξ, ζ n mx { f, f b } l i l i l i

17 Some extended Simpson-type ineulities Similrly, using Theorem we cn prove the following theorem Theorem 5 Let f be defined s in Theorem, T i, T i ti 0,,, n be defined s in Theorem nd let ftdt, S α f, I n, ξ, ζ nd R α f, I n, ξ, ζ be defined s in the identity Then, the reminder term R α f, I n, ξ, ζ stisfies the estimte n R α f, I n, ξ, ζ T i + T i li f T i x i + T i x i+ T i + T i for ll i 0,,, n Corollry 6 In Theorem 5, let α 3 nd ξ i x i, ζ i x i+ i 0,,, n Then T i T i 8 i 0,,, n nd the Simpsontype error stisfies n R f, I l i n, ξ, ζ 3 f xi + x i+ Corollry 7 In Theorem 5, let α nd ξ i x i, ζ i x i+ i 0,,, n Then T i T i 8 i 0,,, n nd the trpzoidtype error stisfies n l i R f, I n, ξ, ζ f xi + x i+ Corollry 8 In Theorem 5, let α 0 nd ξ i x i, ζ i x i+ i 0,,, n Then T i T i 8 i 0,,, n nd the midpointtype error stisfies n l i R 0 f, I n, ξ, ζ f xi + x i+ which is Proposition in [] References [] M Alomri nd M Drus, On the Hdmrd s ineulity for log-convex functions on the coordintes, J Ineul Appl , Article ID 837, 7 pges [] SS Drgomir, Two mppings in connection to Hdmrd s ineulities, J Mth Anl Appl 67 99, no, 9 56 [3] SS Drgomir, On the Hdmrd s ineulity for convex functions on the co-ordintes in rectngle from the plne, Tiwnese J Mth 5 00, no, [] SS Drgomir nd RP Agrwl, Two ineulities for differentible mppings nd pplictions to specil mens of rel numbers nd to trpezoidl formul, Appl Mth Lett 998, no 5, 9 95 [5] SS Drgomir, RP Agrwl nd P Cerone, On Simpson s ineulity nd pplictions, J Ineul Appl 5 000, no 6,

18 5 Hsu, Hwng nd Tseng [6] SS Drgomir, YJ Cho nd SS Kim, Ineulities of Hdmrd s type for Lipschitzin mppings nd their pplictions, J Mth Anl Appl 5 000, no, [7] L Fejér, Über die Fourierreihen, II, Mth Nturwiss Anz Ungr Akd Wiss [8] J Hdmrd, Étude sur les propriétés des fonctions entières en prticulier d une fonction considérée pr Riemnn, J Mth Pures Appl [9] US Kirmci, Ineulities for differentible mppings nd pplictions to specil mens of rel numbers nd to midpoint formul, Appl Mth Comput 7 00, no, 37 6 [0] US Kirmci nd ME Özdemir, On some ineulities for differentible mppings nd pplictions to specil mens of rel numbers nd to midpoint formul, Appl Mth Comput 53 00, no, [] CEM Perce nd J Pečrić, Ineulities for differentible mppings with ppliction to specil mens nd udrture formul, Appl Mth Lett 3 000, no, 5 55 [] KL Tseng, SR Hwng, nd SS Drgomir, Fejér-Type Ineulities I, J Ineul Appl 00 00, Article ID 53976, 7 pges [3] KL Tseng, GS Yng nd KC Hsu, On some ineulities of Hdmrd s type nd pplictions, Tiwnese J Mth 3 009, no 6, [] GS Yng nd KL Tseng, On Certin Integrl Ineulities Relted to Hermite- Hdmrd Ineulities, J Mth Anl Appl , no, [5] GS Yng nd KL Tseng, Ineulities of Hdmrd s type for Lipschitzin mppings, J Mth Anl Appl 60 00, no, [6] GS Yng nd KL Tseng, Ineulities of Hermite-Hdmrd-Fejér type for convex functions nd Lipschitzin functions, Tiwnese J Mth 7 003, no 3, 33 0 Ki-Chen Hsu Deprtment of Business Administrtion, Alethei University, Tmsui, New Tipei City 503, Tiwn E-mil ddress: victor8@miluedutw Shiow-Ru Hwng Chin University of Science nd Technology, Nnkng, Tipei 5, Tiwn E-mil ddress: hsru@cccustedutw Kuei-Lin Tseng Deprtment of Applied Mthemtics, Alethei University, Tmsui, New Tipei City 503, Tiwn E-mil ddress: kltseng@miluedutw nd kltseng@gmilcom