Interntionl Journl of Mtheticl Anlysis Vol, 6, no 6, 795-85 HIKARI Ltd, www-hikrico http://dxdoiorg/988/ij6575 Properties of Jensen -Convex Functions Teodoro Lr Deprtento de Físic y Mteátics Universidd de los Andes N U Rfel Rngel, Trujillo, Venezuel Nelson Merentes Universidd Centrl de Venezuel Escuel de teátics, Crcs, Venezuel Roy Quintero nd Edgr Rosles Deprtento de Físic y Mteátics Universidd de los Andes N U Rfel Rngel, Trujillo, Venezuel Copyright c 5 Teodoro Lr, Nelson Merentes, Roy Quintero nd Edgr Rosles This rticle is distributed under the Cretive Coons Attribution License, which perits unrestricted use, distribution, nd reproduction in ny ediu, provided the originl work is properly cited Abstrct In this reserch we present ineulities for Jensen -convex functions, these re siple ineulities, Jensen type ineulities nd those involving integrls A result on iniu of this clss of functions is exhibit s well Mthetics Subject Clssifiction: 6A5, 39B6 Keywords: -convex, Jensen convex, Jensen -convex, Jensen type ineulities, Fejér type ineulities This reserch hs been prtilly supported by Centrl Bnk of Venezuel
796 Teodoro Lr et l Introduction According to Hrdy, Littlewood, nd Póly 3 the foundtions of the theory of convex functions re due to Jensen They lso express tht the originl definition of rel vlued convex function estblished by Jensen hiself ws wht nowdys is better known s idconvex function or ore deutely convex function in Jensen s sense or just Jensen convex for short Well, following the, function f : I R which stisfies the ineulity ( x + y f f(x + f(y for ll x, y I ( is clled Jensen convex in the intervl I Now we will introduce new type of rel vlued function tht hs the property of stisfying ineulity ( for = With this ide in ind, for ech (, (the cse = hs been excluded becuse the next rguenttion does not ke sense for this uniue sitution let us try to find prticulr vlue of t(= t in the open intervl (, for which given ny pir (x, y, b, b, t x + ( t y cn be represented s: t x + ( t y = x + y ( c( where c( is constnt depending only on Rewriting eution ( s ( t ( x + ( t y = c( c( nd solving for c( nd t, we obtin c( = + / nd t = /( + (3 Now we recll definition which is useful in the clss of functions we re bout to introduce (7,,,, 3 Definition A function f :, b R (b > is sid to be -convex in the intervl, b,, if for ny x, y, b nd t, we hve f(tx + ( ty tf(x + ( tf(y (4 Rerk In the foregoing definition it y hppen tht f :, + R nd everything runs in the se fshion Unless nother thing is stted, in this reserch, functions re defined fro, + to the set of rel nubers R, (, We begin with couple of definitions which will be used in our reserch, soe of the re known nd soe others re reltively new Bsed on these siple fcts nd denoting c( by we set the following (,,,
Properties of Jensen -convex functions 797 Definition 3 Let (, A function f :, + R which stisfies the ineulity ( x + y f(x + f(y f for ll x, y, + (5 will be clled Jensen -convex in the intervl, + The clss of ll Jensen -convex functions in the intervl, + is denoted s J + In the se fshion, J b will denote the clss of ll Jensen - convex functions defined in the intervl, b Definition 4 A function A :, + R is sid to be subliner if A(tx ta(x nd A(x + y A(x + A(y for ny t nd x, y, + The forecoing result shows wy to generte Jensen -convex functions fro known ones, tht is, -convex nd subliner functions Le 5 Let F, A :, + R be functions, F is -convex nd A is subliner nd consider g :, + R given s g(x = F (x + A(x, x, then g is Jensen -convex (g J + Proof For ny x, y, + ( x + y g F (x + F (y F (x + F (y = g(x + g(y ( x ( y + A + A + A(x + A(y Let f :, b R, f J b nd I = (, α, α b rbitrry We re going to build nontrivil Jensen -convex function strting fro f (6 Here is the wy to do it For ny x I let us choose r > sll enough such tht set I r x = (x r, x + r I, now set ϕ x (r = inf t I r x {f(t} Then ϕ x is decresing function, hence the following liit exits f (x = li ϕ x(r = li inf f (6 r + r + Ix r Function f : I, + is clled the lower hull of f nd its vlue t x I is clled the infiu of f t x Siilrly the upper hull of f cn be defined Theore 6 Let f J b, I s before, f Then f : I R is in J α for ny < α b Proof Let z I nd ɛ >, then there exists δ > such tht f (z ϕ z (r + ɛ for r < δ (7
798 Teodoro Lr et l Hence, for t I r z, r < δ f(t inf {f(w} f (z ɛ (8 w Iz r Tke now x, y I, z = x+y nd ɛ > fix; choose δ > s in (7 Then (8 holds for t Iz r Moreover, we y find points u I cr x, v I cr y (of course we lwys cn pick r in such wy tht I cr x nd f(u f(v inf cr w I x inf cr w I y, I cr y I such tht {f(w} + ɛ ϕ x (r + ɛ f (x + ɛ (9 {f(w} + ɛ ϕ y (r + ɛ f (y + ɛ ( since ϕ x nd ϕ y re decresing functions On the other hnd, u + v z u x + v y < r + r = r, but then u+v Iz r nd by (8 ( x + y ( u + v f f + ɛ f(u + f(v Conclusion follows fro (9, ( nd letting ɛ + Jensen Type Ineulities The results we show here re ostly inspired in nd 6 Proposition 7 Let f :, + R, f J + nd strshped, then f(tx+( ty tf( x+( tf( y, for ny x, y, +, t, Proof Given x, y, + nd t,, we hve ( c tx + ( ty f(tx + ( ty = f f( tx + f( ( ty tf( x + ( tf( y Proposition 8 Let f :, + R in J + nd strshped function, (, Then for ny < b < +, f is bounded bove on, b
Properties of Jensen -convex functions 799 Proof We proceed s in foregoing proposition, let z, b rbitrry, then z = t + ( tb for t, b nd M = x{f(, f( b}, then f(z tf( + ( tf( b M Theore 9 Let f :, + R, f J + nd strshped function Then for ny integer n nd x, x,, x n, +, the following ineulity tkes plce, ( n f x k n f(c n f ( c n k x k+ n n c n x + c n k Proof First of ll we notice tht becuse f is strshped, for ny n, ( n f x k ( n n n f x k, therefore is enough to show tht ( n f x k c n n f(c n x + f ( c n k x k+ c n k for ny integer n This foregoing ineulity is proven by induction on n Actully for n = or 3 it is obvious Let us ssue tht the result is true for n nd show for n + Indeed, ( n+ ( n f x k = f c x k + x n+ ( n f = c (n+ x k + f f(c n c n x + ( x n+, nd by hypothesis n x k+ f(c n k ( (n+ f (n+ x + c n k f(c (n+ k x k+ c (n+ k 3 Ineulities involving Integrls + f ( x n+ The present section is devoted to show soe ineulities for integrls, we include here those differentible functions whose derivtives re integrble nd follows ides fro, 4, 5,8,9 nd 3
8 Teodoro Lr et l Theore Let f,, nd b s before, if f L b, then b c c f(udu f( + f(b c Proof For ny t,, f ( t+( tb c tf(+( tf(b, now we integrte the foregoing ineulity over t on,, or ( t + ( tb f b c c dt tf( + ( tf(bdt f(udu f( + f(b Theore Let f,,, b s in theore nd dditionlly f L, b, then ( + b f ( f(udu f(c + f(c c b (3 Proof First we set x = t + ( tb, y = tb + ( t, then + b = x + y nd conseuently ( + b f(x + f(y f = f ( ( t + ( tb + f tb + ( t, by integrting this ineulity over t in, produces ( + b f f ( t + ( tb dt + f ( tb + ( t dt, hence ( + b f ( f(udu The second prt of the ineulity is obtined by proceeding s in proposition 7 by noticing tht f(t + ( tb tf( + ( tf( b, gin we perfor integrtion s before nd get f(udu f(c + f( b
Properties of Jensen -convex functions 8 Theore Let f,, nd b s in theore If g :, b, + is integrble nd syetric bout x = +b, then the following ineulity holds ( + b b f g(xdx f(xg(xdx f( +f( b g(xdx Additionlly, if g is density function on, b; tht is ( + b f (3 g(tdt =, then f(xg(xdx f( + f( b (33 Proof As in the proof of theore, ( + b f f(t + ( tb + f(tb + ( t (34 Multiplying ech eber of (34 by g(t + ( tb, tking into ccount tht g(t + ( tb = g( + b (tb + ( t = g(tb + ( t, nd integrting the result on,, we obtin, by considering chnge of vrible, f ( + b g(xdx f(xg(xdx And the ineulity on the left hnd side is proven For the proof of the right hnd side of (3 we proceed s in proof of theore, f(t + ( tb tf( + ( tf( b (35 Agin s before, ultiplying ech eber of (35 by g(t+( tb, integrting the result on,, nd tking into ccount tht +b b xg(xdx = g(xdx, we hve, fter grouping ppropritely, f(xg(xdx f( + f( b g(xdx If g is syetric density function on, b, (33 is redily obtined Rerk 3 Note tht for g(x =, (3 coincides with (3 b The following result is siilr to one given in Le 4 Let f J +, strshped nd differentible in (, +, < < b < + If f L, b then f( + f(b f( + f( b Proof First of ll we get, integrting by prts, ( tf (t+( tbdt = f( + f(b ( tf (t + ( tbdt f(t+( tbdt, (36
8 Teodoro Lr et l nd by using hypotheses, conclusion is strightforwrd f(t + ( tb tf(c + ( tf( b The following theores follow ides fro nd 5 Theore 5 Let f J +, differentible in (, + nd < < b < + If f J b nd strshped, then f( + f(b f(xdx f (c + f (c b Proof Eulity (36 in previous le cn be rewritten s ( f( + f(b Hence, f( + f(b f(xdx = f(xdx = ( tf (t + ( tbdt (37 ( tf (t + ( tbdt t f (t + ( tbdt t f ( + f ( b dt = f ( + f ( b In the following results we shll use the following ineulity, n n ( k + b k k + n b k, if >, k, b k, k =, n (38 Theore 6 Let us ssue f s in previous theore, p > nd f p p Jensen -convex in (, b nd strshped, then f( + f(b with p + = f(xdx ( ( p f ( + f ( b, p +
Properties of Jensen -convex functions 83 Proof We use (37, Hölder ineulity with = p, of course the hypotheses p nd proceeding s in previous theore f( + f(b f(xdx = ( tf (t + ( tbdt t f (t + ( tbdt ( t p p dt ( f (t + ( tbdt ( ( conclusion follows by using t p p dt = The following result (4 will be used, ( f ( + f ( b ( t p p dt, p+ p nd (38 Le 7 Let f : (α, β R, (, b (α, β If f L, b then for ny x (, b, (b xf(b + (x f( (x (t f (tx+( tdt+ f(udu = (b x ( tf (tx+( tbdt Theore 8 Let f J +, differentible in (, + nd < < b < + If f J b nd strshped, then (b xf(b + (x f( f(udu (x f (c x + f (c (b x f (c x + f (c b + Proof We use le(7 nd get (b xf(b + (x f( (x (t f (tx+( tdt + (x f(udu (b x t f (b x (tx+( tdt+ ( tf (tx+( tbdt t f (tx+( tbdt
84 Teodoro Lr et l (x f (c x + f (c (b x f (c x + f (c b + Corollry 9 If x = +b, the following ineulity shows up, f(b + f( b f(udu f ( ( + b + f (c + f (c b 4 Theore If f is s in previous theore, p > nd f p p convex in (, +, then (b xf(b + (x f( f(udu ( p p + { (x f ( x+ f ( + Jensen (b x f ( x+ f ( b} Proof We use le7 nd Hölder ineulity with = p, proof goes in p siilr wy to previous theore (b xf(b + (x f( f(udu (x t f (b x (tx+( tdt+ t f (tx+( tbdt (x ( t p ( p dt f (b ( x (tx+( tdt + t p p dt ( f (tx + ( tbdt ( p p + { (x f (c x + f ( f ( x + f ( b + }, (b x now use (38 Corollry If x = +b in theore then we get f(b + f( ( ( ( p f(udu p + ( f ( ( + b + f ( + f ( b
Properties of Jensen -convex functions 85 Proof We substitute x = +b into ineulity described by theore ( nd use (38 References S S Drgoir nd R P Agrwl, Two ineulities for differentible ppings nd pplictions to specil ens of rel nubers nd to trpezoidl forul, App Mth Lett, (998, no 5, 9-95 http://dxdoiorg/6/s893-9659(9886-x S S Drgoir nd G Toder, Soe ineulities for -convex functions, Studi Univ Bbes-Bolyi Mthetic, 38 (993, -8 3 G H Hrdy, J E Litlewood nd G Póly, Ineulities, Cbridge University Press, London, 934 4 H Kvurci, M Avci nd M E Özdeir, New ineulities of Herite-Hdrd type for convex functions with pplictions, J Ine nd Applic, (, no, - http://dxdoiorg/86/9-4x--86 5 M Klričić, M E Özdeir nd J Pečrić, Hdrd type ineulities for -convex nd (α, convex functions, J Ine in Pure nd Appl Mth, 8 (8, no 4, - 6 M Kucz, Introduction to the Theory of Functionl Eutions nd Ineulities, Second Edition, Edited by Attil Gillányi, Birkhäuser, Bsel, Boston, Berlin, 7 T Lr nd E Rosles nd J L Sánchez, New properties of -convex functions, Interntionl Journl of Mtheticl Anlysis, 9 (5, no 5, 735-74 http://dxdoiorg/988/ij54389 8 N Minculete nd F C Mitroi, Fejér-type ineulities, (, -9, rxiv:55778v 9 M E Özdfeir nd E Set nd M Z Sriky, Soe new Hdrd type ineulities for coordinted -convex nd (α, -convex functions, Hcettepe J of Mth nd Sttistics, 4 (, no, 9-9 G Toder, The hierrchy of convexity nd soe clssic ineulities, J Mth Ine, 3 (9, no 3, 35-33 http://dxdoiorg/753/ji-3-3 G Toder, On generliztion of the convexity, Mthetic, 3 (988, no 53, 83-87 G Toder, Soe generliztions of the convexity, Proc Collo Approx Opti Cluj- Nploc, Roni, (984, 39-338 3 C Yildiz, M Gürbüz nd A O Akdeir, The Hdrd type ineulities for -convex functions, Konurlp J of Mth, (3, no, 4-47 Received: Noveber, 5; Published: June, 6