New Integral Inequalities through Generalized Convex Functions

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1 Punjb University Journ of Mthetics ISSN ) Vo. 462)214) pp New Integr Inequities through Generized Convex Functions Muhd Muddssr, Deprtent of Mthetics, University of Engineering nd Technoogy, Txi. Pistn. Ei: Ahsn Ai, Deprtent of Eectronic Engineering, University of Engineering nd Technoogy, Txi. Pistn. Ei: Abstrct. In this rtice, we founded sever inequities for soe singere-vued function, reted to the fous Herite-Hdrd s H H) inequity for ppings who hs positive vues ies in the csses K α,s,1 nd K α,s,2. AMS MOS)[21] Subject Cssifiction Codes: 26A51, 26D15, 26D1. Key Words: Generised Convexity, H H inequity, Jensens inequity, Höder inequity. 1. INTRODUCTION With the outgrowth of ccuus during the 19th century, the concern of inequities hs rpidy incresed. Inequities hve gined significnt iportnce not ony in Mthetics itsef but so in Engineering nd nery res of Sciences. Such s, in nueric nysis, the estition of definite integr of re vued function over n interv [, b] is very interesting probe. An eeent inequity tht contributes error bounds for qudrture forue of continuous convex singe-vued ppings, ned Herit- Hdrd s H H) inequity, is set s [11, p. 53]: ) + b f 2 1 b fx) dx f) + fb), 1.1) 2 whence f : [, b] R is convex singe-vued function. Both inequities turned bc for f to be concve. The ipression of qusi-convex singe-vued function infer genery the picture of convex singe-vued function. To greter extent, excty singe-vued p f : [, b] R is qusi-convex on [, b] if fλu + 1 λ)v) x{fu), fv)}, hods for ny u, v [, b] nd λ [, 1]. Inteigiby, singe-vued convex function y be considered s qusi-convex function. Moreover, qusi-convex singe-vued functions ight be convex excty see [5]). In [12], Özdeir et. estbished sever integr 47

2 48 Muhd Muddssr nd Ahsn Ai inequities respecting soe inds of convexity. Especiy, they discussed the foowing resut connecting with qusi-convex functions: Theore 1. Let continuous p f : [, b] φ [, ) R so tht f L 1 [, b]). If f is qusi-convex on [, b] for p, q >, induces x ) p b x) q fx)dx = b ) p+q+1 βp + 1, q + 1) x{f), fb)}, where βx, y) is the Euer Bet function. Recenty, Liu [7] gve cose to new integr inequities for qusi-convex functions s coes: Theore 2. Let continuous p f : [, b] φ [, ) R so tht f L 1 [, b]). If for ny > 1, f 1 is qusi-convex on [, b] for p, q >, induces 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 continuous p f : [, b] φ [, ) R so tht f L 1 [, b]) nd et 1. If f is qusi-convex on [, b] for soe fixed p, q >, induces ) x ) p b x) q fx)dx = b ) p+q+1 βp + 1, q + 1) x{ f), fb) 1 }. Tht is, this study is further continution of [8], where we generise the resuts discussed in [8] by ween the condition of convexity discussed in [1].. 2. PRINCIPLE OUTCOMES In this segent, we generize the bove theores nd produce soe ore resuts using the foowing e described in [12]. Le 4. Let f : I = [, b] φ [, ) R is continuous p on [, b] so tht f L 1 [, b]), induces equity x ) p b x) q fx)dx = b ) p+q+1 1 t) p t q ft + 1 t)b)dt 2.1) hods for soe fixed p, q >. Here we rec the foowing definitions fro [1] by Muddssr et ned s s α, )-convex functions s reproduced beow; Definition 5. A function f : [, ) [, ) is supposed to s α, )-convex function in the first sense or f K α,s,1, if u, v [, ) β [, 1] the coing inequity grees: fβu + 1 β)v) β αs) fu) + 1 β αs) v f, ) where α, ) [, 1] 2 for s, 1].

3 New Integr Inequities through Generized Convex Functions 49 Definition 6. A function f : [, ) [, ) is supposed to s α, )-convex function in the second sense or f K α,s,2, if u, v [, ) β [, 1] the coing inequity grees: fβu + 1 β)v) β α ) s fu) + 1 β α ) s v f, ) where α, ) [, 1] 2 for s, 1]. Note tht for s = 1, we get K α I) css of convex functions nd for α = 1 nd = 1, we get K 1 s I) nd K 2 s I) css of convex functions. Theore 7. Let f : I = [, b] φ [, ) R is continuous p on [, b] so tht f L 1 [, b]). If f K α,s,1 on [, b] for p, q >, ) ) x ) p b x) q fx)dx b ) {βq p+q+1 + αs + 1, p + 1) f) f ) } +βq + 1, p + 1) f 2.2) Proof. Ting bsoute vue of Le 4, x ) p b x) q fx)dx b ) p+q+1 1 t) p t q ft + 1 t)b) dt. 2.3) Since f K α,s,1 As, nd on [, b], then the inequity 2.3) cn be written s 1 t) p t q ft + 1 t)b) dt 1 t) p t q t αs f) + 1 t αs ) fb) ) dt, 2.4) 1 t) p t q+αs dt = βq + αs + 1, p + 1) 2.5) 1 t) p t q 1 t αs )dt = βq + 1, p + 1) βq + αs + 1, p + 1). 2.6) Using 2.4), 2.5) nd 2.6) in 2.3), we get 2.2). Theore 8. Let f : I = [, b] φ [, ) R is continuous p on [, b] so tht f L 1 [, b]) nd et > 1. If f 1 K α,s,1 on [, b] for p, q >, x ) p b x) q fx)dx b ) p+q+1 βαs + 1, 1)) 1 [ f) Proof. Appying the Höder s Inequity on 2.3), ipies 1 βq + 1, p + 1)) 1 + b f ) ] ) [ 1 t) p t q ft + 1 t)b) dt 1 t) p t q ) dt ] 1 [ ft + 1 t)b) 1 dt ] )

4 5 Muhd Muddssr nd Ahsn Ai here, Since f 1 K α,s,1 on [, b] for > 1, therefore 1 1 ft + 1 t)b) dt t αs f) furtherore, 1 t) p t q dt = βq + 1, p + 1). 2.9) t αs dt = t αs ) fb) 1 ) dt, 2.1) 1 t) αs dt = βαs + 1, 1). 2.11) Inequities 2.3), 2.8), 2.1) nd equtions 2.9),2.11) together ipies 2.7). Theore 9. Let f : I = [, b] φ [, ) R is continuous p on [, b] so tht f L 1 [, b]) nd et 1. If f K α,s,1 on [, b] for p, q >, x ) p b x) q fx)dx b ) p+q+1 βq + 1, p + 1)) 1 [βq + αs + 1, p + 1) { f) ) } ) f ] + βq+1, p+1) 1 f 2.12). Proof. Now ppying the Höder s Inequity on 2.3), we get [ 1 t) p t q ft + 1 t)b) dt 1 t) p t q dt here, Since f K α,s on [, b] for 1, therefore,1 1 t) p t q ft+1 t)b) dt which copetes the proof. [ ] 1 1 ] 1 1 t) p t q ft + 1 t)b) dt 2.13) 1 t) p t q dt = βq + 1, p + 1). 2.14) 1 t) p t q t αs f) +1 t αs ) fb) ) dt 2.15) Soe ore integr inequities cn be found using K α,s,2 css of convex functions in siir wy. 3. CONCLUSION It is ong-fiir tht the convexity hs been bringing ey roe in thetic progring, engineering, nd optiistion theory. The generistion of convexity is one of the ost significnt pnor in thetic progring nd optiistion theory. There hve been ny efforts to ween the convexity presuption in the iterture. A substnti generistion of convex functions is tht of s α, ) functions brought in by Muddssr et in [1]. In [12], Özdeir et ted bout soe integr inequities for different inds of convexity. In this pper we deveoped soe ore resuts on herite-hdrd s type inequities by ween the condition of convexity discussed in [1].

5 New Integr Inequities through Generized Convex Functions ACKNOWLEDGEMENTS The uthors woud ie to offer their hertiest thns to the nonyous reviewers for pprecibe notices nd rers unified in the terin edition of this rtice. REFERENCES [1] S.S. Drgoir, On soe new inequities of Herite-Hdrd type for -convex functions, Tng J. Mth. 33, No. 1 22) [2] S.S. Drgoir nd C.E.M. Perce: Seected Topics on Herite-Hdrd Inequities nd Appictions, RGMIA Monogrphs, Victori University, 2. [3] S.S. Drgoir nd C.E.M. Perce, Qusi-convex functions nd Hdrds inequity, Bu. Austr. Mth. Soc. 57, No ) [4] S.S. Drgoir, J. Pečrić nd L.E. Persson, Soe inequities of Hdrd type, Soochow J. Mth. 21, No ) [5] D.A. Ion, Soe estites on the Herite-Hdrd inequity through qusiconvex functions, An. Univ. Criov Ser. Mt. Infor. 34, 27) [6] U.S. Kirci nd M.E. Ozedeir, on soe inequities for differentibe ppings nd ppictioons to speci ens of re nubers nd to idpoint foru, App.Mth.Cop, 153, 24) [7] W.J. Liu, New integr inequities vi α, )-convexity nd qusi-convexity, rxiv: v1 [th.fa] [8] W. Liu, Soe New Integr Inequities vi P -convexity, rxiv: v1 [th.fa] 1 FEB 212, [9] M. Muddssr, M.I. Bhtti nd M.Iqb. Soe new s-herite Hdrd Type Inequities for differentibe functions nd their Appictions, Proceedings of Pistn Acdey of Science 49, No ) [1] M. Muddssr, M.I. Bhtti nd Wjeeh Irshd, Generiztion of integr inequities of the type of Herit- Hdrd through convexity, Bu. Austr. Mth. Soc. 88, 213) [11] C. Nicuescu nd L. E. Persson, Convex functions nd their ppictions, Springer, Berin Heideberg NewYor, 24. [12] M.E. Özdeir, E. Set nd M. Aori, Integr inequities vi sever inds of convexity, Cret. Mth. Infor. 2, No ) [13] C.E.M. Perce nd J. Pečrić, Inequities for differentibe ppings nd ppictions to speci ens of re nubers nd to idpoint foru App. Mth. Lett, 13, No. 2 2)