A Note on Saigo s Fractional Integral Inequalities
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1 Turkish Joural of Aalysis ad Number Theory, 214, Vol 2, No 3, Available olie a hp://pubssciepubcom/ja/2/3/2 Sciece ad Educaio Publishig DOI:112691/ja A Noe o Saigo s Fracioal Iegral Iequaliies Guoao Wag 1,*, Harshvardha Harsh 2, SD Purohi 3, Trilok Gupa 4 1 School of Mahemaics ad Compuer Sciece, Shaxi Normal Uiversiy, Life, Shaxi, People s Republic of Chia 2 Deparme of Mahemaics, Amiy Uiversiy, Jaipur, Idia 3 Deparme of Basic Scieces (Mahemaics, College of Techology ad Egieerig, MP Uiversiy of Agriculure ad Techology, Udaipur, Idia 4 Deparme of Civil Egieerig, College of Techology ad Egieerig, MP Uiversiy of Agriculure ad Techology, Udaipur, Idia *Correspodig auhor: wg2512@163com Received May 1, 214; Revised Jue 3, 214; Acceped Jue 12, 214 Absrac I his paper, some ew iegral iequaliies relaed o he bouded fucios, ivolvig Saigo s fracioal iegral operaors, are eshablished Special cases of he mai resuls are also poied ou Keywords: Iegral iequaliies, Gauss hypergeomeric fucio, Saigo s fracioal iegral operaors Cie This Aricle: Guoao Wag, Harshvardha Harsh, SD Purohi, ad Trilok Gupa, A Noe o Saigo s Fracioal Iegral Iequaliies Turkish Joural of Aalysis ad Number Theory, vol 2, o 3 (214: doi: /ja Iroducio ad Prelimiaries Uder various assumpios (Chebyshev iequaliy, Grüss iequaliy, Mikowski iequaliy, Hermie- Hadamard iequaliy, Osrowski iequaliy ec, iequaliies are playig a very sigifica role i all fields of mahemaics, paricularly i he heory of approximaios (see [2,6,7,13,14,17,23] Therefore, i he lieraure we foud several exesios ad geeralizaios of hese iegral iequaliies for he fucios of bouded variaio, sychroous, Lipschizia, moooic, absoluely coiuous ad -imes differeiable mappigs ec ([1,11,12,15,16,19,2,21,22,26,27,28] I he pas rece years, oe more dimesio have bee added o his sudy, by iroducig umber of iegral iequaliies ivolvig various fracioal calculus ad q- calculus operaors For deailed accou, oe may refer [1,3,4,5,8,9,18,24,25,29-35] ad he refereces cied herei Recely, Tariboo e al [33] ivesigaed cerai ew iegral iequaliies for he iegrable fucios, whose bouds are also iegrable fucios, ivolvig he Riema-Liouville fracioal iegral operaors Our aim i his paper, is o obai a geeral exesios of he resuls due o Tariboo e al [33] Mai resuls ivesigaed here provide cerai ew iegral iequaliies associaed wih he iegrable fucios, whose bouds are also iegrable fucios, ivolvig he Saigo s fracioal iegral operaors We also give some coseque resuls ad special cases of he mai resuls Firsly, we meio below he basic defiiios ad oaios of some well-kow operaors of fracioal calculus, which shall be used i he sequel Le α>, βη,, he he Saigo fracioal iegral,, I αβη, of order α for a real-valued coiuous fucio f ( is defied by ([36], see also [[37], p 19]: α ( τ = τ τ, Γ α (11 2F1α+ β, ηα ; ;1 f ( τ I f d where, he fucio F 2 1 appearig as a kerel for he operaor (11 is he Gaussia hypergeomeric fucio defied by F 2 1 ( abc = ( c, ; ; a b =!, (12 ad ( a is he Pochhammer symbol ( a a( a ( a ( a The operaor = , = 1,, I αβη, icludes boh he Riema- Liouville ad he Erdélyi-Kober fracioal iegral operaors give by he followig relaioships:,, = α α α I f I f 1 (13 = ( τ f ( τ dτ ( α > Γ( α ad αη, α,, η I f ( = I, f ( α η (14 = ( τ τ f ( τ dτ ( α >, η Γ ( α Followig [36], for f ( = µ i (11, we ge µ Γ ( µ + 1 Γ ( µ + 1 β + η ( µ 1 β ( µ 1 α η I = Γ + Γ ( α >, mi µµ, β+ η > 1, > µ (15
2 Turkish Joural of Aalysis ad Number Theory 66 2 Mai Resuls I his secio, we obai cerai iegral iequaliies, relaed o he iegrable fucios, whose bouds are also iegrable fucios, ivolvig Saigo s fracioal hypergeomeric operaors The resuls are give i he form of he followig heorems: Theorem 1 Le f, ϕ 1, ad ϕ 2 are iegrable fucios defied o [,, such ha ( f ( ϕ ( for all [ ϕ1 2,, (21 The, for >, we have + ϕ ϕ γδ,, γδ,, I ϕ1 I f I 2 I, f γδ,, γδ,, I ϕ2 I 1 I f I f, (22 where α > max{, β}, β < 1, β 1 < η <, γ > max{, δ}, δ < 1 ad δ 1 < < Proof By he hypohesis of iequaliy (21, for ay τ, ρ >, we have which follows ha Cosider ( ϕ ( τ f ( τ ( f ( ρ ϕ ( ρ 2 1, f + f f f ϕ2 τ ρ ϕ1 ρ τ ϕ ρ ϕ τ + τ ρ 1 2 α ( τ 2F1 ( α ( τ ; > 1 α 1 ( τ α ( α + β( η ( τ Γ( α α+ β Γ ( α + 1 α+ β+ 1 ( α + β( α + β + 1( η( η+ 1 ( τ 2Γ ( α + 2 α β τ F, τ = α+ β, ηα ; ;1 Γ = + α , which remais posiive, for all ( ( (23 (24 τ, >, uder he codiios saed wih Theorem 1 Muliplyig boh sides, F, τ is give by (24 ad of (23 by F( τ (where iegraig he resulig ideiy wih respec o τ from o, ad usig (11, we ge ( ρ ϕ2 + ϕ1( ρ I ( f I f ( f I I f ϕ ρ ϕ + ρ 1 2 Nex, o muliplyig boh sides of (25 by γ ( ρ Γ( γ ( γ 1 ( ρ, ; >, (25 H(, ρ = 2F1( γ + δ, ; γ;1 ρ (26 which also remais posiive, for all ρ ( ( >, Upo iegraig he resulig iequaliy so obaied wih respec o ρ from o, ad usig he operaor (11, we easily arrive a he desired resul (21 I may be oed ha, for γ = α, δ = β, = η, he Theorem 1 immediaely reduces o he followig resul: Corollary 1 Le ϕ 1 ad ϕ 2 are iegrable fucios defied o [, ad saisfyig iequaliy (21 The, for >, we have + ϕ ϕ ϕ1 2 I ϕ2 I 1 I f, I I f I I f where { } (27 α > max, β, β < 1 ad β 1 < η < Theorem 2 Le f ad g be wo iegrable fucios, ad ϕ1, ϕ2, ψ1 ad ψ 2 are four iegrable o [ fucios o [,, such ha ( f ( ( ( g( ( for all [, ϕ ϕ, ψ ψ, The, for { } { } ad (28 >, α > max, β, β < 1, β 1 < η <, γ > max, δ, δ < 1 δ 1 < < he followig iequaliies holds rue: + ϕ ψ γδ,, γδ,, I ψ1 I f I 2 I, g γδ,, γδ,, I ϕ2 I 1 I f I g, + ψ ϕ γδ,, γδ,, I ϕ1 I g I 2 I, f γδ,, γδ,, I ψ2 I 1 I g I f, ψ + ψ γδ,, γδ,, I ϕ2 I 2 I f I, g γδ,, γδ,, I ϕ2 I g I f I 2, ψ + ψ γδ,, γδ,, I ϕ1 I 1 I f I, g γδ,, γδ,, I ϕ1 I f I f I 1 (29 (21 (211 (212 Proof Le f ad g are wo iegrable fucios ad saisfyig iequaliy (28, he o prove (29, we ca wrie which follows ha ( ϕ ( τ f ( τ ( g( ρ ψ ( ρ 2 1, g + f f g ϕ2 τ ρ ψ1 ρ τ ψ ρ ϕ τ + τ ρ 1 2 (213 O muliplyig boh sides of (213 by F(, τ (where F(, τ is give by (24 ad iegraig wih respec o τ from o, he by makig use of (11, we ge ( ρ ϕ2 + ψ1( ρ I ( g I f ( g I I f ψ ρ ϕ + ρ 1 2 (214 Nex, muliplyig boh sides of (213 by H(, ρ ( ρ, >, where H(, ρ is give by (26, ad iegraig wih respec o ρ from o, we easily arrive a he desired resul (29
3 67 Turkish Joural of Aalysis ad Number Theory Followig he similar procedure, oe ca easily esablish he remaiig iequaliies (21 o (212 by usig he followig iequaliies, respecively ad ( ψ2( τ g( τ ( f ( ρ ϕ1( ρ ( ϕ ( τ f ( τ ( g( ρ ψ ( ρ, 2 2 ( ϕ ( τ f ( τ ( g( ρ ψ ( ρ 1 1 Therefore, we omi he furher deails of he proof of hese resuls 3 Coseque Resuls ad Special Cases The Saigo s fracioal iegral operaor defied by (11, possess he advaage ha he Erdélyi-Kober ad he Riema-Liouville ype fracioal iegral operaors happe o be he paricular cases of his operaor Therefore, by suiably specializig he parameers, we ow briefly cosider some special cases of he resul derived i he precedig secio To his ed, le us se β = ad δ =, ad make use of he relaio (14, he Theorems 1 & 2 yields he followig iequaliies ivolvig he Erdélyi-Kober ype fracioal iegral operaors: Corollary 2 Le f, ϕ ad ϕ 2 are iegrable fucios defied o [, ad saisfyig iequaliy (21, he for >, we have + ϕ ϕ γ, αη, αη, γ, ϕ1 2 αη, γ, αη, γ, ϕ2 1, I I f I I f I I + I f I f (31 where α >, 1 < η <, γ > ad 1 < < Corollary 3 Le f ad g be wo iegrable fucios o [, ad ϕ1, ϕ2, ψ 1 ad ψ 2 are four iegrable fucios o [,, ad saisfyig iequaliy (28 The, for >, α >, 1 < η <, γ > ad 1 < < he followig iequaliies holds rue: + ϕ ψ γ, αη, αη, γ, ψ1 2 αη, γ, αη, γ, ϕ2 1, I I f I I g I I + I f I g + ψ ϕ γ, αη, αη, γ, ϕ1 2 αη, γ, αη, γ, ψ2 1, I I g I I f I I + I g I f ψ + ψ γ, αη, αη, γ, ϕ2 2 αη, γ, αη, γ, ϕ2 2, I I I f I g I I g + I f I ψ + ψ γ, αη, αη, γ, ϕ1 1 αη, γ, αη, γ, ϕ1 1 I I I f I g I I f + I f I (32 (33 (34 (35 Nex, if we replace β by αδ, by γ ad make use of he relaio (13, he Theorems 1 & 2 correspods o he kow iegral iequaliies ivolvig Riema- Liouville ype fracioal iegral operaors, due o Tariboo e al [33] Furher, if we pu ϕ1, ϕ2, ψ1 ψ ( = P where mm,, pp,, [, 2, = m = M = p ad ad make use of formula (15, he he Theorems 1 & 2 leads o he followig paricluar resuls: Corollary 4 Le f be a iegrable fucio defied o [,, such ha [ m f M, m, M for all, (36 The, for >, we have Γ( 1 δ + ( 1 δ ( 1 γ Γ( 1 β + η γδ,, ( 1 β ( 1 α η Γ( 1 β + η Γ( 1 δ + Γ( 1 Γ ( 1+ α + η Γ( 1 Γ ( 1+ γ + γδ, m I f + M I f mm,, + I f I f (37 where α > max{, β}, β < 1, β 1 < η <, γ > max{, δ}, δ < 1 ad δ 1 < < Corollary 5 Le f ad g be wo iegrable fucios o [,, such ha m f M, p g P, m, p, M, P for all, The, for { } > max{, δ}, 1 δ [ (38 >, α < max, β, β < 1, β 1 < µ <, γ δ < ad 1 < < he followig iequaliies holds rue: Γ( 1 δ + ( 1 δ ( 1 γ Γ( 1 β + η γδ,, ( 1 β ( 1 α η Γ( 1 β + η Γ( 1 δ + Γ( 1 Γ ( 1+ α + η Γ( 1 Γ ( 1+ γ + γδ, p I f + M I g pm,, + I f I g Γ( 1 δ + ( 1 δ ( 1 γ Γ( 1 β + η γδ,, ( 1 β ( 1 α η Γ( 1 β + η Γ( 1 δ + Γ( 1 Γ ( 1+ α + η Γ( 1 Γ ( 1+ γ + γδ, m I g + P I f mp,, + I g I f Γ( 1 β + η Γ( 1 δ + ( 1 β ( 1 α η ( 1 δ ( 1 γ γδ MP,, + I f I g (39 (31
4 Turkish Joural of Aalysis ad Number Theory 68 Γ( 1 β + η ( 1 β ( 1 α η Γ( 1 δ + ( 1 δ ( 1 γ γδ,, M I g (311 + P I f, Γ( 1 β + η Γ( 1 δ + ( 1 β ( 1 α η ( 1 δ ( 1 γ mp γδ,, + I f I g Γ( 1 β + η ( 1 β ( 1 α η Γ( 1 δ + ( 1 δ ( 1 γ γδ,, m I g + p I f Agai, if we se ϕ = ad ϕ ( ( = + 1 ad make use of formula (15, he he Theorem 1 ad Corollary 1, furher leads o he followig iegral iequaliies: Corollary 6 Le f be a iegrable fucio defied o [,, such ha 1, [, f + for all The, for { } max{, }, 1 δ >, α > max, β, β < 1, β 1 < η <, γ > δ δ < ad 1< < we have 1 Γ( 2 δ + ( 2 δ ( 2 γ 1 Γ( 2 β + η ( 2 β ( 2 α η Γ( 1 β + η ( 1 β ( 1 α η + + γδ,, 1 Γ( 2 β + η 1 Γ( 2 Γ ( 2+ α + η Γ( 2 δ + ( 1 Γ β + η Γ( 2 Γ ( 2+ γ + + Γ ( 1 β Γ ( 1 + α + η γδ,, + I f ( I f ( I f I f (313 Corollary 7 Le f be a iegrable fucio defied o [,, such ha 1, [, f + for all The, for { } we have >, α > max, β, β < 1 ad β 1 < η <, 1 2Γ( 2 β + η Γ( 2 Γ ( 2+ α + η Γ( 1 β + η + Γ ( 1 β Γ ( 1 + α + η 2 ( I, f ( I f 1 Γ( 2 β + η 1 Γ( 2 Γ ( 2+ α + η Γ( 2 β + η Γ( 1 β + η ( 2 β ( 2 α η + Γ ( 1 β Γ ( 1 + α + η + (314 I his paper, we have iroduced cerai geeral iegral iequaliies, relaed o he iegrable ad bouded fucios f ad g, ivolvig Saigo s fracioal iegral operaors Therefore, we coclude wih he remark ha, by ϕ, ϕ, suiably specializig he arbirary fucio 1 2 ψ ad ( 1 ψ oe ca furher easily obai addiioal 2, iegral iequaliies ivolvig he Riema-Liouville, Erd elyi-kober ad Saigo ype fracioal iegral operaors from our mai resuls Coflic of Ieress The auhors declare ha here is o coflic of ieress regardig he publicaio of his aricle Refereces [1] Aasassiou, GA: Advaces o Fracioal Iequaliies, Spriger Briefs i Mahemaics, Spriger, New York, 211 [2] Ahmadmir, N ad Ullah, R: Some iequaliies of Osrowski ad Gr uss ype for riple iegrals o ime scales, Tamkag J Mah, 42(4 (211, [3] Baleau, D ad Purohi, SD: Chebyshev ype iegral iequaliies ivolvig he fracioal hypergeomeric operaors, Absrac Appl Aal, 214, Aricle ID 6916, 1 (pp [4] Baleau, D, Purohi, SD ad Agarwal, P: O fracioal iegral iequaliies ivolvig hypergeomeric operaors, Chiese Joural of Mahemaics, 214, Aricle ID 69476, 5(pp [5] Belarbi, S ad Dahmai, Z: O some ew fracioal iegral iequaliies, J Iequal Pure Appl Mah, 1(3(29, Ar 86, 5 (pp [6] Ceroe, P ad Dragomir, SS: New upper ad lower bouds for he Chebyshev fucioal, J Iequal Pure App Mah, 3 (22, Aricle 77 [7] Chebyshev, PL: Sur les expressios approximaives des iegrales defiies par les aures prises ere les mêmes limies, Proc Mah Soc Charkov, 2(1882, [8] Dahmai, Z ad Bezidae, A: New weighed Gruss ype iequaliies via (α, β fracioal qiegral iequaliies, Ieraioal Joural of Iovaio ad Applied Sudies, 1(1(212, [9] Dahmai, Z, Tabhari, L ad Taf, S: New geeralisaios of Gr uss iequaliy usig Riema- Liouville fracioal iegrals, Bull Mah Aal Appl, 2 (3(21, [1] Dragomir, SS: A geeralizaio of Grüss s iequaliy i ier produc spaces ad applicaios, J Mah Aal Appl, 237 (1999, [11] Dragomir, SS: A Grüss ype iequaliy for sequeces of vecors i ier produc spaces ad applicaios, J Iequal Pure Appl Mah, 1(2 (2, 1-11 [12] Dragomir, SS: Some iegral iequaliies of Grüss ype, Idia J Pure Appl Mah, 31(4(2, [13] Dragomir, SS: Operaor Iequaliies of he Jese,Čebyšev ad Grüss Type, Spriger Briefs i Mahemaics, Spriger, New York, 212 [14] Dragomir, SS ad Wag, S: A iequaliy of Osrowski-Grüss ype ad is applicaios o he esimaio of error bouds for some special meas ad for some umerical quadraure rules, Compu Mah Appl, 13(11 (1997, 15-2 [15] Gauchma, H: Iegral iequaliies i q-calculus, Compu Mah Appl, 47 (24, [16] Gavrea, B: Improveme of some iequaliies of Chebysev-Grüss ype, Compu Mah Appl, 64 (212, [17] Grüss, D: Uber das maximum des absolue Berages vo 1 b 1 b b f ( x g ( x dx, a 2 f x dx g x dx a Mah a b a ( b a Z, 39(1935,
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4 4 Vol 4 No 4 26 7 Journal of Jiangxi Normal Universiy Naural Science Jul 26-5862 26 4-349-5 3 2 6 2 67 3 3 O 77 9 A DOI 6357 /j cnki issn-5862 26 4 4 C q x' x /q G s = { α 2 - s -9 2 β 2 2 s α 2 - s