A Simple Proof of the Jensen-Type Inequality of Fink and Jodeit

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1 Mediterr. J. Mth. 13 (2016, DOI /s X/16/ published online October 16, 2014 c Springer Bsel 2014 A Simple Proof of the Jensen-Type Inequlity of Fink nd Jodeit Mrcel V. Mihi nd Constntin P. Niculescu Dedicted to Tudor Zmfirescu on the occsion of his 70th birthdy Abstrct. We discuss the extension of Jensen s inequlity to the frmework of qusiconvex functions. Moreover, it is proved tht our results work for clss of signed mesures lrger thn the clss of probbility mesures. Mthemtics Subject Clssifiction. Primry 26A51; Secondry 26D15. Keywords. Jensen s inequlity, bsolutely continuous function, convex function, qusiconvex function, signed mesure, subdifferentil. From time to time, it is worth looking t old clssicl results. And surprises re not fr wy. The im of this pper is to discuss the cse of Jensen s inequlity, bsic result in rel nlysis, known to chrcterize the convex functions. Trying to understnd result stted without proof by Fink nd Jodeit [4], we discovered tht Jensen s inequlity ctully works in much more generl frmework relted to qusiconvexity nd the restriction to probbility mesures cn be relxed, llowing suitble signed mesures. The detils re given below. We strt by reclling tht rel-vlued function f defined on n intervl I is clled qusiconvex if f ((1 λx + λy mx {f(x,f(y} for ll x, y I nd λ [0, 1]. Qusiconvexity is equivlent to the fct tht ll level sets L λ = {x I : f(x λ} re convex, whenever λ R. Clerly, every convex function is lso qusiconvex, but the converse fils. For exmple, every monotonic function is qusiconvex. The continuous qusiconvex functions hve nice monotonic behvior, first noticed by Mrtos [7]: Lemm 1. A continuous rel-vlued function f defined on n intervl I is qusiconvex if nd only if it is either monotonic or there exists n interior The second uthor ws supported by grnt of the Romnin Ntionl Authority for Scientific Reserch, CNCS UEFISCDI, Project Number PN-II-ID-PCE

2 120 M. V. Mihi nd C. P. Niculescu MJOM Figure 1. The grph of the function xe x point c I such tht f is nonincresing on (,c] I nd nondecresing on [c, I. For detils, see the book of Cmbini nd Mrtein [2], Theorem 2.5.2, p. 37. Tht book lso contins welth of exmples nd pplictions. Figure 1 shows the grph of the qusiconvex function xe x. This function is decresing on the intervl (, 1] nd incresing on the intervl [ 1,. It is concve on (, 2] nd convex on [ 2,. An importnt feture of this function tht will be used in this pper is the fct tht the tngent line to the grph t ny point x 1 is support line. Jensen s inequlity is usully stted in the frmework of positive mesures of totl mss 1 (tht is, of probbility mesures. The min feture of positive mesure is the fct tht the integrl of nonnegtive function is nonnegtive number. Surprisingly, this property still works for some signed mesures when restricted to suitble subcones of the cone of positive integrble functions. A simple exmple in the discrete cse is offered by the following consequence of the Abel summtion formul: If ( k n nd (b k n re two fmilies of rel numbers such tht then 1 2 n 0 nd j b k 0 k b k 0. for ll j {1, 2,...,n}, Indeed, ccording to the Abel summtion formul, we hve n 1 k k b k = ( k k+1 + n 0. b j j=1 b j j=1 This remrk cn be esily extended to the frmework of Lebesgue integrbility.

3 Vol. 13 (2016 The Jensen-Type Inequlity of Fink nd Jodeit 121 Lemm 2. Suppose tht f :[, b] R is nonnegtive bsolutely continuous function nd g :[, b] R is n integrble function. Then, f(xg(xdx 0, in ech of the following two cses: (i f is decresing nd x g(tdt 0 for ll x [, b]; or, (ii f is incresing nd g(tdt 0 for ll x [, b]. x Proof. (i. Indeed, f(xg(xdx = = [ f(x = f(b ( x f(xd x g(tdt ] x=b ( x g(tdt f (x g(tdt dx x= ( x g(tdt + ( f (x g(tdt dx 0, s sum of nonnegtive numbers. The integrtion by prts for bsolutely continuous functions is motivted by Theorem 18.19, p. 287, in the monogrph of Hewitt nd Stromberg [5]. The cse (ii follows in similr mnner. By combining Lemm 1 nd Lemm 2, we rrive t the following result: Theorem 1. Suppose tht f :[, b] R is nonnegtive bsolutely continuous qusiconvex function nd g :[, b] R is n integrble function such tht Then, x g(tdt 0 nd x g(tdt 0 f(xg(xdx 0. for every x [, b]. Corollry 1. Under the hypotheses of Theorem 1 for the function g, f(xg(xdx 0 for every nonnegtive continuous convex function f :[, b] R. Proof. The bsolute continuity of continuous convex functions f :[, b] R is proved in [8], Proposition 1.6.1, p. 37. Corollry 2. Under the hypotheses of Theorem 1, x (x tg(tdt 0 nd x (t xg(tdt 0 for every x [, b].

4 122 M. V. Mihi nd C. P. Niculescu MJOM A strightforwrd computtion shows tht the hypotheses of Theorem 1 re fulfilled by the function g(x =x 2 1 6,forx [ 1, 1]. Moreover, 1 ( x 2 1 dx = > 0. This fct mkes the mesure ( x dx very specil in the clss of signed mesures. Definition 1. (Niculescu nd Persson [8], p. 179 A signed Borel mesure μ on n intervl I is clled Steffensen Popoviciu mesure if μ (I > 0nd f(xdμ(x 0 I for every nonnegtive continuous convex function f : I R. According to Corollry 1, n exmple of such mesure on the intervl [ 1, 1] is ( x dx. Using the pushing-forwrd technique of constructing imge mesures, one cn indicte exmples of Steffensen Popoviciu mesures tht hve n rbitrrily given compct intervl s support. Corollry 2 is relted to Lemm in [8], p. 179 (see lso [3], which shows tht signed Borel mesure μ is Steffensen-Popoviciu mesure on n intervl [, b] if nd only if μ ([, b] > 0nd x (x tdμ(t 0 nd x (t xg(tdt 0 for every x [, b]. (SP If g :[, b] R is n integrble function such tht g(xdx =1, we define the brycenter of the bsolutely continuous mesure g(xdx s its moment of first order, β g(xdx = xg(xdx. The brycenter belongs to [, b] when g(xdx is Steffensen Popoviciu mesure. This follows from (SP, by tking x = nd x = b. Alterntively, the brycenter cn be chrcterized s the unique solution β g(xdx of the following eqution involving the clss of ffine functions on [, b] : Aβ g(xdx + B = (Ax + Bg(xdx, for every A, B R. Theorem 1 esily yields the Jensen-type inequlity stted by Fink nd Jodeit [4]: Theorem 2. Suppose tht g :[, b] R is n integrble function tht verifies the hypotheses of Theorem 1 nd lso the condition g(xdx =1. Then,

5 Vol. 13 (2016 The Jensen-Type Inequlity of Fink nd Jodeit 123 f ( b β g(xdx f(xg(xdx, for every continuous convex function f :[, b] R. Proof. Since f is the uniform limit of sequence of convex polygonl functions, we my ssume tht f itself is of this prticulr type. This ssures tht the subdifferentil f(x, of f t ny point x [, b], is nonempty. Let λ f(β g(xdx. Then, f(x f ( ( β g(xdx + λ x βg(xdx for every x [, b]. nd tking into ccount tht g(xdx is Steffensen Popoviciu mesure, we conclude tht ( ( ( f(xg(xdx f βg(xdx + λ x βg(xdx g(xdx = f ( b ( ( β g(xdx + λ x βg(xdx g(xdx = f βg(xdx. An inspection of the rgument of Theorem 2 revels tht similr result works for open intervls. Indeed, in this cse, convexity implies continuity nd lso the nonemptiness of the subdifferentil t ny point. See [8], Theorem 1.3.3, p. 21, nd Lemm 1.5.1, p. 30. Theorem 3. Suppose tht g is rel-vlued integrble function defined on n open intervl I such tht g(xdx =1 I β g(xdx = xg(xdx I Then, I g(xdx is Steffensen Popoviciu mesure on I, f ( β g(xdx I f(xg(xdx for every convex function f : I R with the property tht fg L 1 (I. An exmple of function g tht fulfils the conditions of Theorem 3 is { λe x 2 +1 if x > 1 g(x = ( x if x 1, 6λ 5 where the constnt λ>0 is chosen such tht g(xdx =1. Using the error R function erf(x = 2 x e t2 dt, π one cn show tht the exct vlue of λ is 1 λ = e (1 erf (1 = π 0

6 124 M. V. Mihi nd C. P. Niculescu MJOM The brycenter of the mesure g(xdx is the origin. This exmple cn be esily modified to provide exmples of functions g on R for which g(xdx hs prescribed brycenter. Interestingly, the hypothesis concerning the convexity of the function f in Theorem 2 nd Theorem 3 cn be considerbly relxed using the concept of point of convexity, recently introduced by Niculescu nd Rovenţ [9]. Definition 2. Given rel-vlued continuous function f defined on n intervl I, point I is clled point of convexity of f reltive to neighborhood V of (clled neighborhood of convexity if f( λ k f(x k, (J for every fmily of points x 1,...,x n in V nd every fmily of nonnegtive weights λ 1,...,λ n with n λ k =1nd n λ kx k =. Reversing the inequlity (J, one obtins the notion of point of concvity. Clerly, continuous function f : I R is convex if nd only if every point of I is point of convexity reltive to the whole domin. A simple sufficient condition for point to be point of convexity is the nonemptiness of the subdifferentil of f t. Indeed, the condition λ f( mens the existence of n ffine function of the form L(x =f(+λ(x such tht f(x L(x for ll x in the domin of f. In this cse, ( n f( =L( =L λ k x k = λ k L(x k λ k f(x k, for every fmily of points x 1,...,x n in I nd every fmily of nonnegtive weights λ 1,...,λ n with n λ k =1nd n λ kx k =. Thus, every point x 1 is point of convexity reltive to the whole domin of the function xe x. See Fig. 1. In the cse of the rctngent function (which is convex on (, 0] nd concve on [0,, every point x<0is point of convexity reltive to the neighborhood (,x ], where x > 0is the bsciss of the point where the tngent t x intersects gin the grph. The phenomenon of the existence of convexity/concvity points is genuine for the clss of continuous qusiconvex/qusiconcve functions. The extension of Jensen s inequlity to the cse of points of convexity is s follows: Theorem 4. Suppose tht f :[, b] R is continuous function nd β is point of convexity of f reltive to the whole domin. Then, f(β f(xdμ(x for every Borel probbility mesure μ on [, b] hving the brycenter β.

7 Vol. 13 (2016 The Jensen-Type Inequlity of Fink nd Jodeit 125 Proof. The cse of discrete probbility mesures is covered by Definition 2. In the generl cse, we should notice tht every Borel probbility mesure μ on [, b] is the pointwise limit of net of discrete Borel probbility mesures μ α, ech hving the sme brycenter s μ. See [8], Lemm , p Theorem 4 mkes esy the computtion of the extremum of certin functionls. For exmple, combining it with the technique of Dirc sequences (see [6], Chpter XI, one cn prove tht for every β 1, the infimum of the functionl F (g = xe x g(xdx over the convex set of ll nonnegtive integrble functions g :[, b] R such tht g(xdx =1 nd xg(xdx = β is βe β. We end our pper with result tht extends Theorem 4 to the frmework of qusiconvex functions nd signed mesures. This is to be done by dpting the min ingredient in deriving Theorem 2 from Theorem 1: the fct tht the sum between convex function nd liner one is qusiconvex. In generl, the sum between qusiconvex function nd liner function is not necessrily qusiconvex. See the cse of the functions x 3 nd 3x. On the other hnd, there re differentible qusiconvex functions f : R R (such s xe x for which f(x f (cx is still qusiconvex whenever c is point such tht f(c. Such functions verify the following version of Jensen s inequlity. Theorem 5. Suppose tht g :[, b] R is n integrble function tht verifies the conditions of Theorem 1 nd g(xdx =1. Then, f(β g(xdx f(xg(xdx, for every qusiconvex function f :[, b] R such tht f ( β g(xdx contins numbers λ with the property tht x f(x λx is lso qusiconvex. Proof. By our hypotheses, for λ f ( β g(xdx,wehve f(x f(β g(xdx +λ(x β g(xdx for every x [, b] nd the nonnegtive function f(x f(β g(xdx λ(x β g(xdx is qusiconvex. By Theorem 1, [ 0 f(x f(βg(xdx λ(x β g(xdx ] g(xdx = nd the proof is done. f(xg(xdx f(β g(xdx

8 126 M. V. Mihi nd C. P. Niculescu MJOM The rgument of Theorem 5 lso covers the cse of robust qusiconvex functions, recently introduced by Brron, Goebel nd Jensen [1]. Some of the results proved bove (including Theorem 2 nd Theorem 3 cn be extended esily to the context of severl vribles. However, the chrcteriztion of Steffensen Popoviciu mesures in tht context is still n open problem. Only few exmples re known. See the pper of Niculescu nd Spiridon [10]. References [1] Brron, E.N., Goebel, R., Jensen, R.R.: Functions which re qusiconvex under liner perturbtions. SIAM J. Optim 22(3, (2012 [2] Cmbini, A., Mrtein, L.: Generlized Convexity nd Optimiztion. Theory nd Applictions, Lecture Notes in Economics nd Mthemticl Systems, vol Springer, Berlin (2009 [3] Fink, A.M.: A best possible Hdmrd inequlity. Mth. Inequl. Appl. 1, (1998 [4] Fink, A.M., Jodeit, M.: On Chebyshev s other inequlity. In: Inequlities in Sttistics nd Probbility, Lecture Notes IMS, vol. 5, pp Institute of Mthemticl Sttistics, Hywood (1984 [5] Hewitt, E., Stromberg, K.: Rel nd Abstrct Anlysis. Second printing corrected, Springer, Berlin, Heidelberg, New York (1969 [6] Lng, S.: Undergrdute Anlysis, 2nd edn. Springer, New York (1997 [7] Mrtos, B.: Nonliner Progrmming Theory nd Methods. North-Hollnd, Amsterdm (1975 [8] Niculescu, C.P., Persson L.-E.: Convex Functions nd their Applictions. A Contemporry Approch, CMS Books in Mthemtics, vol. 23. Springer, New York (2006 [9] Niculescu, C.P., Rovenţ, I.: Reltive convexity nd its pplictions. Aequt. Mth. (2014. doi: /s x [10] Niculescu, C.P., Spiridon, C.: New Jensen-type inequlities. J. Mth. Anl. Appl 401(1, (2013 Mrcel V. Mihi nd Constntin P. Niculescu Deprtment of Mthemtics University of Criov Criov Romni e-mil: cpniculescu@gmil.com; mmihi58@yhoo.com Received: June 30, Revised: September 10, Accepted: October 4, 2014.