NTMSCI 6, No 1, 1-7 18 1 New Trends in Mthemticl Sciences http://dxdoiorg/185/ntmsci1739 New generl integrl ineulities for usiconvex functions Cetin Yildiz Atturk University, K K Eduction Fculty, Deprtment of Mthemtics, Erzurum, Turkey Received: 6 July 16, Accepted: 11 November 16 Published online: 6 December 17 Abstrct: In this pper, by using n integrl identity nd the Hölder integrl ineulity we estblish severl new ineulities for n times differentible mppings tht re connected with the usiconvex functions Keywords: Hermite-Hdmrd ineulity, Hölder ineulity, QusiConvex functions 1 Introduction Let f : I R R be convex function defined on n intervl I of rel numbers,,b I nd <b The following double ineulity is well known in the literture s Hdmrd s ineulity: +b f 1 fxdx b Both ineulities hold in the reversed direction if f is concve f+ fb 1 The ineulities in 1 hve become n importnt cornerstone in mthemticl nlysis nd optimiztion Mny uses of these ineulities hve been discovered in vriety of settings Moreover, mny ineulities of specil mens cn be obtined for prticulr choice of the function f Due to the rich geometricl significnce of Hermite-Hdmrd ineulity, there is growing literture providing its new proofs, extensions, refinements nd generliztions, see for exmple [5,[9-[13 nd the references therein Definition 1 A function f : [,b R R is sid to be convex if whenever x,y [,b nd t [,1, the following ineulity holds: ftx+1 ty tfx+1 t fy We sy tht f is concve if f is convex This definition hs its origins in Jensen s results from [8 nd hs opened up the most extended, useful nd multi-disciplinry domin of mthemtics, nmely, convex nlysis Convex curves nd convex bodies hve ppered in mthemticl literture since ntiuity nd there re mny importnt results relted to them We recll tht the notion of usiconvex functions generlizes the notion of convex functions Definition A function f :[, b R R is sid to be usiconvex on[, b if ftx+1 ty mx{ fx, fy}, Corresponding uthor e-mil: cetin@tuniedutr c 18 BISKA Bilisim Technology
C Yildiz: New generl integrl ineulities for usi convex functions for ll x, y [,b nd t [,1 Clerly, ny convex function is usiconvex Furthermore, there exist usiconvex functions which re not convex see [7,[14 For exmple, consider the following Let f :R + R, fx=lnx, x R + This function is usiconvex However f is not convex function For other recent results concerning the n times differentible functions see [- [4,[6,[9,[11,[15-[17 where further references re given In [1, Alomri et l proved the following theorems for usiconvex functions Theorem 1 Let f : I [, R be differentible mpping on I such tht f L[,b, where,b I with <b If f is n usi-convex on[,b, then the following ineulity holds f+ fb 1 b fxdx b 8 [ mx { +b f, f } + mx { +b f, f b } Theorem Let f : I [, R be differentible mpping on I such tht f L[,b, where,b I with <b If f p/p 1 is n usi-convex on[,b, for p>1, then the following ineulity holds: f+ fb 1 b fxdx b mx{ 4p+1 1/p + mx{ +b p/p 1 f +b p/p 1, f f } p 1/p p/p 1, f b } p 1/p p/p 1 3 Theorem 3 Let f : I R R be differentible mpping on I such tht f L[,b, where,b I with <b If f is n usi-convex on[,b, for p 1, then the following ineulity holds f+ fb 1 b [ fxdx b mx 8 { f +b, f } 1 + mx { f +b, f b } 1 4 The min purpose of the present pper is to prove severl new ineulities for usiconvex functions tht re connected with the celebrted Hermite-Hdmrd integrl ineulity Min results Lemm 1 Let f :[,b R be n-times differentible functions If f n L[,b, then ft = + k= f k + 1 k f k b k+1! 1 t n f n t +b where n nturl number, n 1 b k+1 { + 1 nb n+1 n+1 t 1 n f n t +b n! } +1 tb +1 t 5 c 18 BISKA Bilisim Technology
NTMSCI 6, No 1, 1-7 18 / wwwntmscicom 3 Proof The proof is by mthemticl induction For n = 1, we hve to prove the eulity f+ fb 1 ft = b { t 1 f b 4 Integrting by prts, we hve bove eulity t +b +1 t + 1 t f Assume tht 5 holds for n nd let us prove it for n+1 Tht is, we hve to prove the eulity n ft = k= + Then, we cn write f k + 1 k f k b k+ 1! b 1 t n+1 f n+1 t +b +1 tb { I = b n+ 1 n+ t 1 n+1 f n+1 n+1! nd integrting by prts gives k+1 n+1 b n+ + 1 n+ n+1! t +b { t +b } +1 tb t 1 n+1 f n+1 t +b +1 t } 6 +1 t + 1 t n+1 f n+1 I = b n+ fn t +b +1 t 1 n+ t 1n+1 n+1! b n+1 t 1 n f n t +b b + b n+ fn t +b +1 tb 1 n+ 1 tn+1 n+1! b + n+1 1 t n f n t +b b = 1 n+ b n+1 n+1 n+1! fn b n+1 n+1 t 1 n f n t +b +1 t n! + b n+1 n+1 n+1! fn b b n+1 n+1 1 t n f n t +b +1 tb n! Now, using the mthemticl induction hypothesis, we get 1 b 1 n ft = 1 1 n k= f k + 1 k f k b k+ 1! Multiplying the both sides of 7 by 1 n, we obtin b t +b } +1 tb +1 t +1 tb k+1 n+ b n+1 + 1 n+1 n+1! fn + b n+1 n+1 n+1! fn b I 7 f ft = k + 1 k f k b b k+1 + b n+1 k= k+1! n+1 n+1! fn + 1 n b n+1 n+1 n+1! fn b 1 {ww n t 1 n+1 f n+1 t +b +1 t + ww 1 t n+1 f n+1 t +b +1 tb n f = k + 1 k f k b b k+1 { n+1 b n+ 1 + 1 k= k+1! n+ t 1 n+1 f n+1 t +b n+1! + 1 t n+1 f n+1 t +b +1 tb } +1 t where ww= b n+ Thus, the identity 6 nd the lemm is proved n+ n+1! c 18 BISKA Bilisim Technology
4 C Yildiz: New generl integrl ineulities for usi convex functions Theorem 4 For n 1, let f :[,b R R be n times differentible function nd <b If f n L[,b nd f n is usiconvex on[, b, then the following ineulity holds: ft k= f k + 1 k f k b k+1! Proof From Lemm 1, it follows tht ft k= f k + 1 k f k b k+1! Since f n is usi-convex on[,b, we obtin ft k= This completes the proof f k + 1 k f k b k+1! b k+1 [ } mx{ b n+1 f n n+1, +b n+1! fn { +b } +mx f, n f n b b k+1 b n+1 { n+1 1 t n f n t +b +1 t n! + 1 t n f n t +b } +1 tb b k+1 [ b n+1 } mx{ f n n+1, +b n+1! fn { +b } +mx f, n f n b 8 Corollry 1 Let f s in Theorem 4, if in ddition 1 f n is incresing, then we hve ft k= f n is decresing, then we hve ft k= f k + 1 k f k b k+1! f k + 1 k f k b k+1! b k+1 b k+1 b n+1 n+1 n+1! b n+1 n+1 n+1! [ +b f n + [ f n Remrk Under conditions of Theorem 4, if we choose n = 1; then we obtin ineulity Theorem 5 Let f : [,b R R be n times differentible nd <b If f n L[,b nd on[, b, then the following ineulity holds ft k= where >1 f k + 1 k f k b k+ 1! b k+1 b n+1 n+1 n! + mx 1 [ { 1 p mx f n, { f n +b, f n b } 1 f n b + +b fn f n is usiconvex +b } 1 fn 9 c 18 BISKA Bilisim Technology
NTMSCI 6, No 1, 1-7 18 / wwwntmscicom 5 Proof From Lemm 1 nd the Hölder integrl ineulity, we obtin ft k= f k + 1 k f k b k+ 1! b k+1 Since f n is usiconvex on[,b, for >1, then ft k= which completes the proof f k + 1 k f k b k+ 1! b k+1 Corollry Let f s in Theorem 5, if in ddition 1 f n is incresing, then we hve ft k= f k + 1 k f k b k+1! f n is decresing, then we hve ft k= f k + 1 k f k b k+1! { 1 1 t np p 1 fn t +b fn t +b +1 tb b n+1 n+1 n! 1 + 1 t np p 1 b n+1 n+1 n! + mx b k+1 b k+1 1 [ { 1 p mx f n, { f n +b b n+1 n+1 n! b n+1 n+1 n!, f n b } 1 1 +1 t 1} +b } 1 fn 1 [ 1 p +b f n + f n b 1 1 p [ f n + +b fn Remrk Under conditions of Theorem 5, if we choose n = 1; then we obtin ineulity 3 Theorem 6 For n 1, let f : [,b R R be n times differentible nd < b If f n L[,b nd f n is usiconvex on[, b, for 1, then the following ineulity holds ft k= f k + 1 k f k b k+1! b k+1 b n+1 n+1 n+1! + mx [ { mx f n { +b f n Proof From Lemm 1 nd using the well known power-men integrl ineulity, we hve ft k= f k + 1 k f k b k+1! + 1 t n f n wt+1 tb 1 1 + 1 t n 1 t n f n 1, +b } fn, f n b } 1 b k+1 { b n+1 1 n+1 1 t n f n t +b +1 t n! { } b n+1 1 1 1 n+1 1 t n 1 t n f n wt+1 t n! t +b 1} +1 tb 1 c 18 BISKA Bilisim Technology
6 C Yildiz: New generl integrl ineulities for usi convex functions where t +b Since f n is usiconvex on[,b, for 1, then we obtin ft k= which completes the proof f k + 1 k f k b k+1! b k+1 b n+1 n+1 n+1! + mx [ { mx f n { +b f n Remrk Under conditions of Theorem 6, if we choose n = 1; then we obtin ineulity 4, +b } 1 fn, f n b } 1 3 Conclusions In this study, we presented some generlized Hermite type ineulities for the mppings whose derivtives re usiconvex functions re estblished A further study could be ssess weighted versions of these ineulities Competing interests The uthors declre tht they hve no competing interests Authors contributions All uthors hve contributed to ll prts of the rticle All uthors red nd pproved the finl mnuscript References [1 M Alomri, M Drus nd US Kırmci, Refinements of Hdmrd-type ineulities for usi-convex functions with pplictions to trpezoidl formul nd to specil mens, Comp Mth Appl, 59 1, 5-3 [ S-P Bi, S-H Wng nd F Qi, Some Hermite-Hdmrd type ineulities for n-time differentibleα, m-convex functions, Jour of Ine nd Appl, 1, 1:67 [3 P Cerone, SS Drgomir nd J Roumeliotis, Some Ostrowski type ineulities for n-time differentible mppings nd pplictions, Demonstrtio Mth, 3 4 1999, 697-71 [4 P Cerone, SS Drgomir nd J Roumeliotis nd J Šunde, A new generliztion of the trpezoid formul for n-time differentible mppings nd pplictions, Demonstrtio Mth, 33 4, 719-736 [5 SS Drgomir nd CEM Perce, Selected Topics on Hermite-Hdmrd Ineulities nd Applictions, RGMIA Monogrphs, Victori University, Online:[http://wwwstxovueduu/RGMIA/monogrphs/hermite hdmrdhtml [6 D Y Hwng, Some Ineulities for n-time Differentible Mppings nd Applictions, Kyung Mth Jour, 43 3, 335-343 [7 DA Ion, Some estimtes on the Hermite-Hdmrd ineulities through usi-convex functions, Annls of University of Criov, Mth Comp Sci Ser, 34 7, 8-87 [8 J L W V Jensen, On konvexe funktioner og uligheder mellem middlverdier, Nyt Tidsskr Mth B, 16, 49-69, 195 [9 W-D Jing, D-W Niu, Y Hu nd F Qi, Generliztions of Hermite-Hdmrd ineulity to n-time differentible function which re s-convex in the second sense, Anlysis Munich, 3 1, 9- [1 US Kırmcı, MK Bkul, ME Özdemir nd J Pećrić, Hdmrd-type ineulities for s-convex functions, Appl Mth nd Comp, 1937, 6-35 [11 M E Özdemir, Ç Yıldız, New Ineulities for n-time differentible functions, Arxiv:144959v1 c 18 BISKA Bilisim Technology
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