MUHAMMAD MUDDASSAR AND AHSAN ALI

Transcription

1 NEW INTEGRAL INEQUALITIES THROUGH GENERALIZED CONVEX FUNCTIONS WITH APPLICATION rxiv: v1 [th.ca] 19 Aug 213 MUHAMMAD MUDDASSAR AND AHSAN ALI Abstrct. In this pper, we estblish vrious inequlities for soe ppings tht re lined with the illustrious Herite-Hdrd integrl inequlity for ppings whose bsolute vlues belong to the clss K α,s,1 nd Kα,s,2. 1. Introduction The role of theticl inequlities within the theticl brnches s well s their enorous pplictions cn not be underestited. The ppernce of the new theticl inequlity often puts on fir foundtion for the heuristic lgoriths nd procedures used in pplied sciences. Aong others, one of the in inequlity which gives us n explicit error bounds in the trpezoidl nd idpoint rules of sooth function, clled Herit-Hdrd s inequlity, is defined s [11, p. 53]: ) +b f 1 fx)dx f)+fb), 1) 2 b 2 where f : [,b] R is convex function. Both inequlities hold in the reversed direction for f to be concve. We note tht Herit-Hdrd s inequlity 1) y be regrded s refineent of the concept of convexity nd it follows esily fro Jensens inequlity. Inequlity 1) hs received renewed ttention in recent yers nd vriety of refineents nd generliztions cn be found in ny Articles, Boos, Volues nd Journls. The notion of qusi-convex functions generlizes the notion of convex functions. More precisely, function f : [,b] R is sid to be qusi-convex on [,b] if ftx+1 t)y) x{fx),fy)} holds for ny x,y [,b] nd t [,1]. Clerly, ny convex function is qusiconvex function. Furtherore, there exist qusi-convex functions which re not convex see [5]). In [1], Özdeir et l. estblished severl integrl inequlities respecting soe inds of convexity. Especilly, they discussed the following result connecting with qusi-convex functions: Theore 1. Let f : [,b] R be continuous on [,b] such tht f L[,b]), with < b <. If f is qusi-convex on [,b], then for soe fixed p,q >, we hve x ) p b x) q fx)dx = b ) p+q+1 βp+1,q +1)x{f),fb)} Dte: My 11, Mthetics Subject Clssifiction. 26A51, 26D15, 26D1. Key words nd phrses. Convex functions, Generlized Convexity, Herite-Hdrd inequlity, Jensens inequlity, Hölder inequlity, Power-en inequlity, Specil ens. 1

2 2 M. MUDDASSAR AND A. ALI where βx,y) is the Euler Bet function. Recently, Liu in [7] estblished soe new integrl inequlities for qusiconvex functions s follows: Theore 2. Let f : [,b] R be continuous on [,b] such tht f L[,b]), with < b < nd let > 1. If f 1 is qusi-convex on [,b], for soe fixed p,q >, then x ) p b x) q fx)dx = b ) p+q+1 βp+1,q +1)) 1 1 x{ f), fb) ) 1 1 } Theore 3. Let f : [,b] R be continuous on [,b] such tht f L[,b]), with < b < nd let l 1. If f l is qusi-convex on [,b], for soe fixed p,q >, then x ) p b x) q fx)dx = b ) p+q+1 βp+1,q +1) x{ f) l, fb) l )1 l } Tht is, this study is further continution nd generliztion of [8] nd [1] vi generlized convexity. 2. Min Results In this section, we generlize the bove theores nd produced soe ore results using the following le described in [12]. Le1. Let f : [,b] [, ) R be continuouson [,b] suchtht f L[,b]), with < b. Then the equlity x ) p b x) q fx)dx = b ) p+q+1 1 t) p t q ft+1 t)b)dt 2) holds for soe fixed p,q >. To prove our in results, we follow the following definitions first defined in [1] by M. Muddssr et. l., ned s s α, )-convex functions s reproduced below; Definition 1. A function f : [, ) [, ) is sid to be s α,)-convex function in the first sense or f belongs to the clss K α,s,1, if for ll x,y [, ) nd µ [,1], the following inequlity holds: y fµx+1 µ)y) µ αs )fx)+1 µ αs )f ) where α,) [,1] 2 nd for soe fixed s,1]. Definition 2. A function f : [, ) [, ) is sid to be s α,)-convex function in the second sense or f belongs to the clss K α,s,2, if for ll x,y [, ) nd µ,ν [,1], the following inequlity holds: fµx+1 µ)y) µ α ) s fx)+1 µ α ) s y f ) where α,) [,1] 2 nd for soe fixed s,1]. Note tht for s = 1, we get K α I) clss of convex functions nd for α = 1 nd = 1, we get K 1 si) nd K 2 si) clss of convex functions.

3 NEW INTEGRAL INEQUALITIES 3 Theore 4. Let f : [,b] R be continuous on [,b] such tht f L[,b]), with < b <. If f belongs to the clss K α,s,1 on [,b], then for soe fixed p,q >, we hve ) ) x ) p b x) q fx)dx b ) {βq p+q+1 +αs+1,p+1) f) f ) } +βq +1,p+1) f 3) Proof. Ting bsolute vlue of Le 1, we hve x ) p b x) q fx)dx b ) p+q+1 1 t) p t q ft+1 t)b) dt 4) Since f belongs to the clss K α,s,1 inequlity 4) cn be written s 1 t) p t q ft+1 t)b) dt Now Here, on [,b], then the integrl on the right side of 1 t) p t q t αs f) +1 t αs ) fb) )dt 5) 1 t) p t q+αs dt = βq +αs+1,p+1) 6) nd 1 t) p t q 1 t αs )dt = βq +1,p+1) βq +αs+1,p+1) 7) Using 5), 6) nd 7) in 4) nd doing soe lgebric opertions, we get 3). Theore 5. Let f : [,b] R be continuous on [,b] such tht f L[,b]), with < b < nd let > 1. If f 1 belongs to the clss K α,s,1 on [,b], then for soe fixed p,q >, we hve x ) p b x) q fx)dx b ) p+q+1 βαs+1,1)) 1 [ f) 1 + βq +1,p+1)) 1 b f ) ] 1 1 8) Proof. Applying the Hölder s Integrl Inequlity on the integrl on the rightside of 4), we get [ ] 1 1 t) p t q ft+1 t)b) dt 1 t) p t q ) dt Here, [ ft+1 t)b) 1 dt ] 1 1 9) 1 t) p t q dt = βq +1,p+1) 1) Since f 1 belongs to the clss K α,s,1 on [,b] for > 1, we hve 1 1 ft+1 t)b) dt t αs f) 1 +1 t αs ) fb) 1 ) dt 11)

4 4 M. MUDDASSAR AND A. ALI furtherore, t αs dt = 1 t) αs dt = βαs+1,1) 12) Inequlities 4), 9), 16) nd Equtions 1),12) together iplies 8). Theore 6. Let f : [,b] R be continuous on [,b] such tht f L[,b]), with < b < nd let l 1. If f l belongs to the clss K α,s,1 on [,b], then for soe fixed p,q >, we hve x ) p b x) q fx)dx b ) p+q+1 βq +1,p+1)) l 1 l [βq +αs+1,p+1) { f) l ) l} ) f +βq+1,p+1) l ]1 l f 13) Proof. Now pplying the Hölder s Integrl Inequlity on the integrl on the rightside of 4) in notherwy, we get [ ]1 1 l 1 t) p t q ft+1 t)b) dt 1 t) p t q dt Here, Since f l belongs to the clss K α,s,1 1 t) p t q ft+1 t)b) l dt [ 1 t) p t q ft+1 t)b) l dt ] 1 l 14) 1 t) p t q dt = βq +1,p+1) 15) on [,b] for l 1, we hve 1 t) p t q t αs f) l +1 t αs ) fb) l) dt 16) which copletes the proof. Soe ore integrl inequlities cn be found using K α,s,2 clss of convex functions. References [1] S. S. Drgoir, On soe new inequlities of Herite-Hdrd type for -convex functions, Tng J. Mth ), no. 1, [2] S. S. Drgoir nd C.E.M. Perce, Selected Topics on Herite-Hdrd Inequlities nd Applictions, RGMIA Monogrphs, Victori University, 2. [3] S. S. Drgoir nd C. E. M. Perce, Qusi-convex functions nd Hdrds inequlity, Bull. Austrl. Mth. Soc ), no. 3, [4] S. S. Drgoir, J. Pečrić nd L. E. Persson, Soe inequlities of Hdrd type, Soochow J. Mth ), no. 3, [5] D. A. Ion, Soe estites on the Herite-Hdrd inequlity through qusiconvex functions, An. Univ. Criov Ser. Mt. Infor ), [6] U. S. Kirci nd M.E. Ozedeir, on soe inequlities for differentible ppings nd pplictioons to specil ens of rel nubers nd to idpoint forul, App.Mth.Cop, 15324), [7] W. J. Liu, New integrl inequlities vi α, )-convexity nd qusi-convexity, rxiv: v1 [th.fa] [8] W. Liu, Soe New Integrl Inequlities vi P-convexity, rxiv: v1 [th.fa] 1 FEB 212,

5 NEW INTEGRAL INEQUALITIES 5 [9] M.Muddssr, M.I. Bhtti nd M.Iqbl. Soe new s-herite Hdrd Type Inequlities for differentible functions nd their Applictions, Proceedings of Pistn Acdey of Science 491)212,9-17 [1] M.Muddssr, M.I. Bhtti nd Wjeeh Irshd, Generliztion of integrl inequlities of the type of Herit- Hdrd through convexity, Bull. Austrl. Mth. Soc., Avilble on CJO 212 doi: 1.117/SOOO [11] C.Niculescu nd L. E. Persson, Convex functions nd their pplictions, Springer, Berlin Heidelberg NewYor, 24). On-line: jborwein/preprints/boos/cup/cupold/np-convex.pdf. [12] M. E. Özdeir, E. Set nd M. Alori, Integrl inequlities vi severl inds of convexity, Cret. Mth. Infor ), no. 1, [13] C. E. M. Perce nd J. Pečrić, Inequlities for differentible ppings nd pplictions to specil ens of rel nubers nd to idpoint forul App. Mth. Lett, 132)2), Deprtent of Mthetics, University of Engineering nd Technology, Txil, Pistn E-il ddress: li.uddssr@gil.co Deprtent of Electronic Engineering, University of Engineering nd Technology, Txil, Pistn E-il ddress: hsn.li@uettxil.edu.p