Avilble online t wwwisr-ublictionscom/jns J Nonliner Sci Al, 10 017, 6141 6148 Reserch Article Journl Homege: wwwtjnscom - wwwisr-ublictionscom/jns Some new integrl ineulities for n-times differentible convex nd concve functions Selhttin Mden, Huriye Kdkl b, Mhir Kdkl c,, İmdt İşcn c Dertment of Mthemtics, Fculty of Sciences nd Arts, Ordu University-Ordu-TÜRKİYE b Institute of Science, Ordu University-Ordu-TÜRKİYE c Dertment of Mthemtics, Fculty of Sciences nd Arts, Giresun University-Giresun-TÜRKİYE Communicted by M Bohner Abstrct In this work, by using n integrl identity together with both the Hölder nd the Power-men integrl ineulities we estblish severl new ineulities for n-times differentible convex nd concve mings c 017 All rights reserved Keywords: Convex function, concve function, Hölder integrl ineulity, ower-men integrl ineulity 010 MSC: 6A51, 6D10, 6D15 1 Introduction Let f : I R R be concve function on the intervl I of rel numbers nd, b I with < b The ineulity f + f b 1 b + b f x dx f, b is well-known in the literture s Hermite-Hdmrd s ineulity for concve functions [16] Both ineulities hold in the reserved direction if f is convex The clssicl Hermite-Hdmrd ineulity rovides estimtes of the men vlue of continuous convex or concve function Hdmrd s ineulity for convex or concve functions hs received renewed ttention in recent yers nd remrkble vriety of refinements nd generliztions hve been found; for exmle see [1 16] A function f : I R R is sid to be convex if the ineulity f tx + 1 ty tf x + 1 tf y, is vlid for ll x, y I nd t [0, 1] If this ineulity reverses, then f is sid to be concve on intervl Corresonding uthor Emil ddresses: mden55@mynetcom Selhttin Mden, huriyekdkl@hotmilcom Huriye Kdkl, mhirkdkl@gmilcom Mhir Kdkl, imdtiscn@giresunedutr İmdt İşcn doi:10436/jns010101 Received 017-01-13
S Mden, H Kdkl, M Kdkl, İ İşcn, J Nonliner Sci Al, 10 017, 6141 6148 614 I This definition is well-known in the literture Convexity theory hs ered s owerful techniue to study wide clss of unrelted roblems in ure nd lied sciences For some ineulities, generliztions nd lictions concerning convexity see [3, 5 11, 15 18] Recently, in the literture there re so mny ers bout n-times differentible functions on severl kinds of convexities In references [ 4, 6, 13, 17, 19], reders cn find some results bout this issue Mny ers hve been written by number of mthemticins concerning ineulities for different clsses of convex functions see for instnce the recent ers [1, 5, 7 1, 14] nd the references within these ers Let 0 < < b, throughout this er we will use A, b = + b, G, b = b, b +1 +1 L, b = + 1b 1, b, R, 1, 0, for the rithmetic, geometric, generlized logrithmic mens, resectively Min results We will use the following for obtin our min results Lemm 1 Let f : I R R be n-times differentible ming on I for n N nd f n L [, b], where, b I with < b We hve the identity 1 k f k bb k+1 f k k+1 fxdx = 1n+1 x n f n xdx, 1 k + 1! where n emty sum is understood to be nil Proof To rove, we shll use the induction method For n = 1, by integrtion by rts we hve fbb f fxdx = xf xdx This coincides with the eulity 1 for n = 1 Similrly for n = nd using integrtion by rts s bove, we hve 1 1 k f k bb k+1 f k k+1 fxdx = 1+1 x f xdx, k + 1!! fbb f 1! f bb f! x f xdx = f bb f [ fxdx = 1! xfx b fxdx = f bb f [bfb f] + Eution coincides with the eulity 1 for n = Suose 1 holds for n = t Tht is t1 1 k f k bb k+1 f k k+1 k + 1! fxdx = 1t+1 t! x f xdx, ] fxdx x t f t xdx 3
S Mden, H Kdkl, M Kdkl, İ İşcn, J Nonliner Sci Al, 10 017, 6141 6148 6143 Using the integrtion by rts we hve 1 t+ { b } x t+1 f t+1 xdx = 1t+ x t+1 f t x b b t + 1! t + 1! t + 1 x t f t xdx = 1t [f t bb t+1 f t t+1] t + 1 1t+ x t f t xdx t + 1! t + 1! = 1t+ [f t bb t+1 f t t+1] + 1t+1 x t f t xdx, t + 1! t! 1 t+1 t! x t f t xdx = 1t+ t + 1! Substituting 4 in 3 we obtin t1 1 k f k bb k+1 f k k+1 k + 1! t1 1 k f k bb k+1 f k k+1 k + 1! x t+1 f t+1 xdx 1t [ f t bb t+1 f t t+1] 4 t + 1! fxdx = 1t+1 t! fxdx = 1t+ t + 1! 1t t + 1! x t f t xdx, x t+1 f t+1 xdx [ f t bb t+1 f t t+1], tht is, t1 1 k f k bb k+1 f k k+1 k + 1! = 1t+ t + 1! x t+1 f t+1 xdx fxdx + 1t [f t bb t+1 f t t+1] t + 1! This comletes the roof of Lemm Theorem For n N, let f : I 0, R be n-times differentible function on I nd, b I with < b If f n L [, b] nd f n for > 1 is convex on [, b], then the following ineulity holds: 1 k f k bb k+1 f k k+1 fxdx k + 1! 1 b Ln n, ba 1 f n, f n b Proof If f n for > 1 is convex on [, b], using Lemm 1, the Hölder integrl ineulity nd f n x = x fn b b + b x b x b f n b + b x f n, b we hve 1 k f k bb k+1 f k k+1 k + 1! fxdx
S Mden, H Kdkl, M Kdkl, İ İşcn, J Nonliner Sci Al, 10 017, 6141 6148 6144 This comletes the roof of theorem 1 x n f n x dx 1 x n dx f n x dx 1 [ x x n dx f n b + b x f n ] dx b b = 1 [ b 1 b 1 b n+1 n+1 ] [ 1 f n + f n b n + 1b = 1 b Ln n, ba 1 f n, f n b Corollry 3 Under the conditions of Theorem for n = 1, we hve the following ineulity: fbb f b 1 b 1 [ f fxdx L + f b ] 1, b Proosition 4 Let, b 0, with < b, > 1 nd m, 0] [1, \ {, }, we hve L m +1 m +1, b L, ba 1 m, b m Proof Under the ssumtion of the roosition, let fx = m+ x m +1, x 0, Then f x = x m, is convex on 0, nd the result follows directly from Corollry 3 Theorem 5 For n N, let f : I 0, R be n-times differentible function on I nd, b I with < b If f n L [, b] nd f n for 1 is convex on [, b], then the following ineulity holds: 1 k f k bb k+1 f k k+1 k + 1! 1 n fxdx 1 b 1 1 Ln, b { f n b [ L n+1 n+1, b Ln n, b ] + f n [ 1 bl n n, b L n+1 n+1, b]} Proof From Lemm 1 nd ower-men integrl ineulity, we obtin 1 k f k bb k+1 f k k+1 fxdx k + 1! 1 x n f n x dx 1 1 1 b x n dx x n f n x dx 1 ] 1
S Mden, H Kdkl, M Kdkl, İ İşcn, J Nonliner Sci Al, 10 017, 6141 6148 6145 1 1 x n dx x n [ x b f n b = 1 [ b 1 1 b n+1 n+1 ] 1 1 n + 1b { f n [ b b n+ n+ n + b bn+1 n+1 ] n + 1b + f n [ b bn+1 n+1 n + 1b bn+ n+ n + b 1 n + b x f n ] dx b = 1 b 1 1 Ln, b { f n b [ L n+1 n+1, b Ln n, b ] + f n [ 1 bl n n, b L n+1 n+1, b]} This comletes the roof of theorem Corollry 6 Under the conditions of Theorem 5 for n = 1, we hve the following ineulity: fbb f 1 fxdx b b 1 6 + b 1 1 [ b + f b + b + f ] 1 Proosition 7 Let, b 0, with < b, > 1 nd m, 0] [1, \ {, }, we hve ]} 1 L m +1 m +1, b 3 1 A 1 1, b [ A m+1, b m+1 + G, ba m1, b m1] 1 Proof The result follows directly from Corollry 6 for the function fx = This comletes the roof of roosition m + x m +1, x 0, Corollry 8 Using Proosition 7 for m = 1, we hve the following ineulity: L 1 +1 1 +1, b 1 3 A 1 1, b [ A, b + G, b ] 1 Corollry 9 Using Proosition 7 for = 1, we hve the following ineulity: Lm+1, b 1 [ A m+1, b m+1 + G, ba m1, b m1] 3 Corollry 10 Using Corollry 9 for m = 1, we hve the following ineulity: L, b 1 3 [ A, b + G, b ] Corollry 11 Under the conditions of Theorem 5 for = 1, we hve the following ineulity: 1 k f k bb k+1 f k k+1 fxdx k + 1! 1 { f n b [ L n+1 n+1, b Ln n, b ] + f n [ bl n n, b L n+1 n+1, b]} 1
S Mden, H Kdkl, M Kdkl, İ İşcn, J Nonliner Sci Al, 10 017, 6141 6148 6146 Theorem 1 For n N, let f : I 0, R be n-times differentible function on I nd, b I with < b If f n L [, b] nd f n for > 1 is convex on [, b], then the following ineulity holds: 1 k f k bb k+1 f k k+1 fxdx k + 1! 1 { b 1 f n b [ L n+1 n+1 ], b Ln, b + f n [ bl n ]} 1 n, b L n+1 n+1, b Proof Since f n for > 1 is convex on [, b], using Lemm 1 nd the Hölder integrl ineulity, we hve the following ineulity: 1 k f k bb k+1 f k k+1 fxdx k + 1! 1 1x n f n x dx 1 1 b 1 dx x n f n x dx 1 = 1 b x 1dx x n f n b b + b x 1 b dx 1 [ x f 1dx b xn n b + b x f b xn n ] 1 dx = 1 { f b 1 n [ b b n+ n+ n + b bn+1 n+1 ] n + 1b + f n [ b bn+1 n+1 n + 1b bn+ n+ ]} 1 n + b = 1 { b 1 f n b [ ] L n+1 n+1, b Ln, b + f n [ ]} 1 bl n n, b L n+1 n+1, b This comletes the roof of theorem Corollry 13 Under the conditions of Theorem 1 for n = 1, we hve the following ineulity: fbb f 1 fxdx b b { f b [ ] L +1 b +1, b L, b + f [ ] } 1 bl b, b L +1 +1, b Proosition 14 Let, b 0, with < b, > 1 nd m, 0] [1, \ {, }, then we hve L m +1 { m +1, b b 1 b m m L +1 +1, b G, b b m1 m1 1 L, b} Proof The result follows directly from Corollry 13 for the function fx = This comletes the roof of roosition m + x m +1, x 0,
S Mden, H Kdkl, M Kdkl, İ İşcn, J Nonliner Sci Al, 10 017, 6141 6148 6147 Corollry 15 For m = 1 from Proosition 14, we obtin the following ineulity: L 1 +1 [ ] 1 +1, b L +1 1 +1 +1, b = L+1, b Theorem 16 For n N, let f : I 0, R be n-times differentible function on I nd, b I with < b If f n L [, b] nd f n for > 1 is concve on [, b], then the following ineulity holds: 1 k f k bb k+1 f k k+1 k + 1! fxdx b L n n, b + b fn Proof Since f n for > 1 is concve on [, b], with resect to Hermite-Hdmrd ineulity we get b f n x dx b + b fn Using Lemm 1 nd the Hölder integrl ineulity we hve This comletes the roof of theorem 1 k f k bb k+1 f k k+1 k + 1! 1 x n f n x dx 1 n x n dx f n x dx 1 fxdx 1 1 x n dx b + b fn = b [ b n+1 n+1 ] 1 + b f n n + 1b = b L n n, b + b fn Corollry 17 Under the conditions Theorem 16 for n = 1, we hve the following ineulity: fbb f 1 fxdx b b L, b + b f Proosition 18 Let, b 0, with < b, > 1 nd m [0, 1], then we hve L m +1 m +1, b L, ba m, b Proof Under the ssumtion of the roosition, let fx = m+ x m +1, x 0, Then f x = x m, is concve on 0, nd the result follows directly from Corollry 17
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