Hermite-Hadamard and Simpson Type Inequalities for Differentiable Quasi-Geometrically Convex Functions

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1 Trkish Jornl o Anlysis nd Nmer Theory, 4, Vol, No, 4-46 Aville online h://ssciecom/jn/// Science nd Edcion Plishing DOI:69/jn--- Hermie-Hdmrd nd Simson Tye Ineliies or Dierenile Qsi-Geomericlly Convex Fncions İmd İşcn *, Kerim Bekr, Selim Nmn Dermen o Mhemics, Fcly o Ars nd Sciences, Giresn Universiy, Giresn, Trkey *Corresonding hor: imdiscn@giresnedr Received Mrch, 4; Revised Aril 6, 4; Acceed Aril, 4 Asrc In his er, he hors deine new ideniy or dierenile ncions By sing o his ideniy, hors oin new esimes on generlizion o Hdmrd nd Simson ye ineliies or si-geomericlly convex ncions Keywords: si-geomericlly convex ncions, hermie hdmrd ye ineliies, simson ye ineliy Cie This Aricle: İmd İşcn, Kerim Bekr, nd Selim Nmn, Hermie-Hdmrd nd Simson Tye Ineliies or Dierenile Qsi-Geomericlly Convex Fncions Trkish Jornl o Anlysis nd Nmer Theory, vol, no (4): 4-46 doi: 69/jn--- Inrodcion Le rel ncion e deined on some nonemy inervl I o rel line R The ncion is sid o e convex on I i ineliy ( x + ( ) y) ( x) + ( ) ( y) holds or ll xy, Ind [, ] Following ineliies re well known in he lierre s Hermie-Hdmrd ineliy nd Simson ineliy resecively: Theorem Le : I e convex ncion deined on he inervl I o rel nmers nd, I wih < The ollowing dole ineliy holds + ( ) + ( ) ( x) dx Theorem Le : [, ] e or imes coninosly dierenile ming on (, ) nd (4) (4) = s ( x) < Then he ollowing x (, ) ineliy holds: ( ) + ( ) + ( x) dx + 88 (4) ( ) 4 In recen yers, mny hors hve sdied errors esimions or Hermie-Hdmrd, Osrowski nd Simson ineliies; or reinemens, conerrs, generlizion see [,9,] The ollowing deiniions re well known in he lierre : I, is Deiniion ([7,8]) A ncion sid o e GA-convex (geomeric-rihmiclly convex) i ( x y ) ( x) + ( y) or ll xy, Ind [,] Deiniion ([7,8]) A ncion : I (, ) (, ) is sid o e GG-convex (clled in [] geomericlly convex ncion) i ( ( x y ) ( x) ( y) ) or ll xy, Ind [,] In [], İşcn gve deiniion o si-geomericlly convexiy s ollows: : I, is sid o Deiniion A ncion e si-geomericlly convex on I i { xy s ( x), ( y), or ny xy, Ind [, ] Clerly, ny GA-convex nd geomericlly convex ncions re si-geomericlly convex ncions Frhermore, here exis si-geomericlly convex ncions which re neiher GA-convex nor GG-convex [] For some recen resls concerning Hermie-Hdmrd ye ineliies or GA-convex, GG-convex, sigeomericlly convex ncions we reer inereses reder o [,,4,,6,,,4] The gol o his ricle is o eslish some new generl inegrl ineliies o Hermie-Hdmrd nd Simson ye or si-geomericlly convex ncions y sing new inegrl ideniy

2 Trkish Jornl o Anlysis nd Nmer Theory 4 Min Resls Le : I (, ) e dierenile ncion on I, he inerior o I, hrogho his secion we will ke I λµ,,, = λ µ + µ ( ) ( ) + ( λ ) ( ) d ln( / ), I wih < nd λµ, In order o rove or min resls we need he ollowing ideniy : I, e dierenile Lemm Le ncion on I sch h L [, ],, I wih < Then or ll λµ, we hve: / I ( λµ,,, ) = ln( / ) ( µ ) ( ) d () + ( λ ) ( ) d / Proo By inegrion y rs nd chnging he vrile, we cn se / / ( µ ) ln( / ) d = ( µ ) d / / =( µ ) d = µ + µ ( ) d ln( / ) nd similrly we ge / / ( λ ) ln( / ) d = ( λ) d =( λ) d / ( λ) λ / = ( ) d ln( / ) Adding he resling ideniies we oin he desired resl : I, e dierenile Theorem Le ncion on I sch h L [, ],, I wih < I is si-geomericlly convex on [, ] or some ixed he ollowing ineliy holds nd µ /λ, hen { I ( λµ,,, ) ln( / ) s ( ), ( ) / / / / { C ( µ ) C ( µ,,, ) + C ( λ) C4 ( λ,,, ) λ C( λ)= λ +, 8 C( µ,,, ) 4 () µ C ( µ )= µ +, () 8 + ln( / ) / / / = L(, ) µ, < µ /, / / / / L(, ), µ = ln( / ) C ( λ,,, )= / ( µ ) µ µ ( µ )( ) 4 µ L(, ) ( ), ln( / ) / ( λ) λ ( ) L( ),,, ( λ) λ λ / / / + 4λ L( ) L( ) nd L(, ) is logrihmic men deined y L(, ) = ( ) / ( ln ln ) Proo Since or ll [,] [, ], is si-geomericlly convex on { s, Hence, sing Lemm nd ower men ineliy we ge I λµ,,, ln( / ) / / µ ( ) µ d s { ( ), ( ) d λ ( ) + λ d / /s { ( ), ( ) d { ln( / ) s, / / µ d µ ( ) d λ d λ ( ) + d / /

3 44 Trkish Jornl o Anlysis nd Nmer Theory / / / / µ µ d = C ( µ )= µ +, 8 λ λ d = C( λ)= λ +, 8 µ d = C ( µ,,, ), λ d = C ( λ,,, ), which comlees he roo Corollry Under he ssmions o Theorem wih λ = µ = /, he ineliy () redced o he ollowing ineliy ( ) + ( ) ( ) d ln( / ) / ( ), / ln( / ) s { C (/,,, ) 8 ( ) / ( ), ln( / ) s 8 ( ) { 4 / / / 4 4 C (,,, ) + C (,,, ) + C (/,,, ) Corollry Under he ssmions o Theorem wih µ = nd λ =, he ineliy () redced o he ollowing ineliy ( ) d ln( / ) 8 / ( { ) / / { C C4 ln( / ) s, (,,, ) + (,,, ) Corollry Under he ssmions o Theorem wih µ =/6 nd λ = / 6, he ineliy () redced o he ollowing ineliy ( ) + ( ) ( ) + d ln( / ) / ( ), ln( / ) s 7 ( ) / C (/6,,, ) / + C4 (/6,,, ) Theorem 4 Le : I (, ) e dierenile ncion on I sch h L [, ],, I wih < I is si-geomericlly convex on [, ] or some ixed > nd µ /λ, hen he ollowing ineliy holds { I ( λµ,,, ) ln( / ) s ( ), ( ) / / / / { C (, µ ) C7 (,, ) C6 (, λ) C8 (,, ) C (, µ )= µ + ( µ ), C6 (, λ) = ( λ ) + ( λ), + C7 (,, )= L,, C(,, )= L, C (,, ) nd + = / / / 8 7 Proo Since (4) is si-geomericlly convex on [, ] nd sing Lemm nd Hölder ineliy, we ge ( λµ ) I,,, ln( / ) / /, µ d ( ) s d ( ), + λ d ( ) s d / / ( ) { ln( / ) s, / / µ d ( ) d λ d ( ) + d / / here i is seen y simle comion h / / / / + + µ d = µ + ( µ ), λ d = ( λ ) + ( λ), + / / ( ) d = L(, ) / / / / nd d = L, L,

4 Trkish Jornl o Anlysis nd Nmer Theory 4 Hence, he roo is comleed Corollry 4 Under he ssmions o Theorem 4 wih λ = µ = /, he ineliy (4) redced o he ollowing ineliy ( ) + ( ) ( ) d ln( / ) { / ln( / ) s, / / { C7 C8 + ( + ) Corollry Under he ssmions o Theorem 4 wih µ = nd λ =, he ineliy (4) redced o he ollowing ineliy ( ) d ln( / ) { / ln( / ) s, / / { C7 C8 + ( + ) Corollry 6 Under he ssmions o Theorem 4 wih µ =/6 nd λ = / 6, he ineliy (4) redced o he ollowing ineliy ( ) + ( ) ( ) + d ln( / ) { ( ) ( ) / ln( / ) s, / / { C7 C ( + ) Theorem Le : I (, ) e dierenile ncion on I sch h L [, ],, I wih < I is si-geomericlly convex on [, ] or some ixed > nd µ /λ, hen he ollowing ineliy holds I ( λµ,,, ) ( ), ln( / ) s ( ) / / / / { C7 ( C,, ) (, µ ) C8 ( C,, ) 6 (, λ) + () C, C6, C7, C 8 re deined s in Theorem 4 nd + = Proo Since is si-geomericlly convex on [, ] nd sing Lemm nd Hölder ineliy, we ge ( λµ ) I,,, ln( / ) / /, ( ) d µ s d ( ), + ( ) d λ s d / / ( ) { ln( / ) s, / / ( ) d µ d ( ) + d λ d / / { ln( / ) s, / / / / { C7 C µ + C8 C6 λ (,, ) (, ) (,, ) (, ) Hence, he roo is comleed Corollry 7 Under he ssmions o Theorem wih λ = µ = /, he ineliy () redced o he ollowing ineliy ( ) + ( ) ( ) d ln( / ) { ln( / ) s, / / / { C7 C8 + ( + ) Corollry 8 Under he ssmions o Theorem wih µ = nd λ =, he ineliy () redced o he ollowing ineliy ( ) d ln( / ) { / ln( / ) s, / / { C7 C8 + ( + )

5 46 Trkish Jornl o Anlysis nd Nmer Theory Corollry 9 Under he ssmions o Theorem wih µ =/6 nd λ = / 6, he ineliy () redced o he ollowing ineliy ( ) + ( ) ( ) + d ln( / ) { ( ) ( ) / ln( / ) s, / / { C7 C ( + ) Reerences [] H, J, Xi, B-Y nd Qi, F, "Hermie-Hdmrd ye ineliies or Geomeric-rihmeiclly S-convex ncions", Commn Koren Mh Soc, 9 () -6 4 [] İşcn, İ,"Generlizion o dieren ye inegrl ineliies or s -convex ncions vi rcionl inegrls", Alicle Anlysis, [] İşcn, İ, "New generl inegrl ineliies or sigeomericlly convex ncions vi rcionl inegrls", Jornl o Ineliies nd Alicions, 49 ges [4] İşcn, İ, "Some New Hermie-Hdmrd Tye Ineliies or Geomericlly Convex Fncions", Mhemics nd Sisics, () 86-9 [] İşcn, İ, "Some Generlized Hermie-Hdmrd Tye Ineliies or Qsi-Geomericlly Convex Fncions", Americn Jornl o Mhemicl Anlysis, () 48- [6] İşcn, İ, "Hermie-Hdmrd ye ineliies or GA-S-convex ncions", Le Memiche, 4 Acceed or licion [7] Niclesc, C P, "Convexiy ccording o he geomeric men", Mh Inel Al, () -67 [8] Niclesc, C P, "Convexiy ccording o mens", Mh Inel Al, 6 (4) 7-79 [9] Srıky, M Z nd Akn, N, "On he generlizion o some inegrl ineliies nd heir licions", Mh Com Modelling, [] Srıky, M Z, Se, E nd Ozdemir, M E, "On new ineliies o Simson s ye or S-convex ncions", Com Mh Al [] Shng, Y, Yin, H-P nd Qi, F, "Hermie-Hdmrd ye inegrl ineliies or geomeric-rihmeiclly S-convex ncions", Anlysis (Mnich), () 97-8 [] Zhng, X-M, Ch, Y-M nd Zhng, X-H "The Hermie- Hdmrd Tye Ineliy o GA-Convex Fncions nd Is Alicion", Jornl o Ineliies nd Alicions Aricle ID 76 ges [] Zhng, T-Y, Ji, A-P nd Qi, F, "On Inegrl Ineliies o Hermie-Hdmrd Tye or S-Geomericlly Convex Fncions", Asr Al Anl, Aricle ID ges [4] Zhng, T-Y, Ji, A-P nd Qi, F, "Some ineliies o Hermie- Hdmrd ye or GA-convex ncions wih licions o mens", Le Memiche, LXVIII- Fsc I 9-9