ON a NEW CONVERSE of JENSEN ' S INEQUALITY Slavko Simic

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PUBLICATIONS DE L quoteright INSTITUT MATH Eacute MATIQUE \noindentNouvelle seacuteriePUBLICATIONS comma DE tome L 85 ’ INSTITUT open parenthesis MATH 99 $ closing\acute parenthesis{E} $ MATIQUE open parenthesis 2 9 closing parenthesis comma 1 7 endash 1 1 0 .... DOI : 1 0 period 2 298 slash PIM 9991 7 S \noindentON A NEWNouvelle CONVERSE s \ ’{ OFe} JENSENrie , tome85 quoteright ( S 99 INEQUALITY ) ( 29 ) , 17 −− 1 1 0 \ h f i l l DOI : 1 0 . 2 298 / PIM 9991 7 S Slavko Simicacute \ centerlineCommunicated{ON by A Stevan NEW CONVERSE Pilipovicacute OF JENSEN ’ S INEQUALITY } Abstract periodPUBLICATIONS .. We give another DE L ’ INSTITUTglobal upper MATH boundE´ MATIQUE for Jensen quoteright s discrete inequal hyphen \ centerline { Slavko Simi \ ’{ c}} ity which is betterNouvelle than série, already tome existing85 ( 99 ) ( ones 2 9 ) , period 1 7 – 11 .. 0 For instance comma DOI we : 1 determine 0 . 2 298 / PIM a 9991 7 S new converses forON generalized A NEW A endash CONVERSE G and G endash OF H JENSEN inequalities period’ S INEQUALITY \ centerline1 period .. Introduction{Communicated by Stevan PilipoviSlavko\ ’ Simić{ c}} Throughout this paper x-tilde = open braceCommunicated x sub i closing by Stevan brace Pilipović represents a finite sequence of real numbers \ centerline { Abstract . \quad We give another global upper bound for Jensen ’ s discrete inequal − } belonging to a fixed closedAbstract interval . IWe = give open another square global bracket upper a bound comma for bJensen closing ’ s discrete square inequal bracket - comma a less b and p-tilde = open braceity p which i closing is better brace than comma already sum existing p i = ones 1is . a For instance , we determine a \ centerline { ity which is better than already existing ones . \quad For instance , we determine a } sequence of positive weights associatednew converses with for x-tilde generalized sub period A – G and G – H inequalities . If f is a convex function on I comma then the well1 . hyphen Introduction known Jensen quoteright s inequality open square \ centerline {new converses for generalized A −− G and G −− H inequalities . } bracket 1 closing square bracketThroughout comma open this paper squarex ˜ bracket= {xi} 4represents closing square a finite bracket sequence of real numbers asserts that : belonging to a fixed closed interval I = [a, b], a < b andp ˜ = {pi}, P pi = 1 is a sequence \ centerline {1 . \quad Introduction } 0 leqslant sum pof i positive f open parenthesis weights associated x sub i closing withx ˜ parenthesis. minus f open parenthesis sum p i to the power of x i closing parenthesisIf f periodis a convex function on I, then the well - known Jensen ’ s inequality [ 1 ] , [ 4 ] \ hspaceThere∗{\ are manyf i l lasserts} importantThroughout that inequalities: this paperwhich are $ particular\ tilde { casesx} of= Jensen\{ quoterightx { si }\} $ represents a finite sequence of real numbers inequality such as the weighted A endash G endash H inequality comma Cauchy quoteright s inequality comma \noindent belongingtoafixed closedX interval $IX x = [ a , b ] , the Ky Fan 0 6 pif(xi) − f( pi i). a and< Hodieresislderb $ and inequalities $ \ tilde comma{p} etc= period\{ p i \} , \sum p i = 1 $ i sOne a can see that theThere lower are bound many zero important is of global inequalities nature since which it does are particular not cases of Jensen ’ s sequencedepend on of p-tilde positiveinequality and tilde-x such weights sub as comma the weighted associated but only A – on G with f –and H theinequality $ interval\ tilde ,I Cauchy{ commax} { ’ whereupon s. inequality}$ fis , the convex Ky period An upper globalFan bound and open Hölderinequalities parenthesis i period , etc .e period comma depending in f and I only closing parenthesis for\ hspace Jensen∗{\ quoterightf i l l } If s inequality $ fOne $ can is see a convex that the lower function bound on zero is $ of I global , $ nature then since the it does well not− known Jensen ’ s inequality [ 1 ] , [ 4 ] was given by Dragomirdepend on openp ˜ and squarex ˜, but bracket only on3 closingf and square the interval bracketI, whereupon comma f is convex . \noindentTheorem 1asserts period .. If that f isAn a : differentiable upper global convex bound (mapping i . e . , dependingon I comma in thenf and weI haveonly ) for Jensen ’ s Equation: openinequality parenthesis 1 closing parenthesis .. sum p i f open parenthesis x sub i closing parenthesis minus\ [ 0 f open\ l e parenthesis q s l awas n t given sum\sum by p i Dragomir top the power i [ 3 ] of f, x i closing ( x parenthesis{ i } leqslant) − 1 dividedf ( by 4\ opensum parenthesisp i bˆ{ x } minusi ) a closing . \ ] parenthesisTheorem open parenthesis1 . If f fis to a the differentiable power of prime convex open mapping parenthesis on I, b closingthen we parenthesis have minus f to the power of prime open parenthesis a closing parenthesis closing parenthesis : = D sub f open parenthesis a comma b closing parenthesis periodX X 1 There are many importantpif inequalities(x ) − f( pix whichi) (b are− a)( particularf 0(b) − f 0(a)) cases:= D (a, of b). Jensen(1) ’ s There is a number of papers where inequalityi open parenthesis6 4 1 closing parenthesisf is utilized in applications inequalityconcerning some such parts as of the Analysis weighted comma A ..−− NumericalG −− H Analysis inequality comma Information , Cauchy Theory ’ s inequality etc , the Ky Fan andopen H\ parenthesis”{o} lder cfThere inequalities period is open a number square , of etcbracket papers . 1 where closing inequality square bracket ( 1 ) is comma utilized open in applications square bracket con- 2 closing square bracket commacerning open some square parts bracket of Analysis 3 closing , square Numerical bracket Analysis closing , Information parenthesis Theory period hline etc \ hspace2000 Mathematics∗{\ f i l l }One Subject can Classification see that the : Primary lower 26 bound B 25 comma zero 26 is D of20 period global nature since it does not Key words and phrases : Jensen quoteright s discrete inequality semicolon .. global bounds semicolon .. generalized\noindent Adepend hyphen G on hyphen $ \ Htilde {p} $ and(cf. $ [1]\ tilde, [2], [3]){.x} { , }$ but only on $ f $ andinequality the interval period $ I , $ whereupon $ f $ is convex . 1 7 \ hspace ∗{\ f i l l }An upper2000 globalMathematics bound Subject ( Classification i . e . ,: dependingPrimary 26 B 25 in , 26 D $ 20 f . $ and $ I $ only ) for JensenKey ’ words s inequality and phrases : Jensen ’ s discrete inequality ; global bounds ; generalized A - G - H inequality . \noindent was given by Dragomir [ 3 ] , 1 7

\ centerline {Theorem 1 . \quad If $ f $ is a differentiable convex mapping on $ I , $ then we have }

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 ) $}\sum p i f ( x { i } ) − f ( \sum p i ˆ{ x } i ) \ l e q s l a n t \ f r a c { 1 }{ 4 } ( b − a ) ( f ˆ{\prime } ( b ) − f ˆ{\prime } ( a ) ) : = D { f } ( a , b ) . \end{ a l i g n ∗}

There is a number of papers where inequality ( 1 ) is utilized in applications concerning some parts of Analysis , \quad Numerical Analysis , Information Theory etc

\ begin { a l i g n ∗} (cf.[1],[2],[3]). \\\ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\ centerline {2000 Mathematics Subject Classification : Primary 26 B 25 , 26 D 20 . }

Key words and phrases : Jensen ’ s discrete inequality ; \quad global bounds ; \quad generalized A − G − H inequality .

\ centerline {1 7 } 1 8 .. S IMI Cacute \noindentIn open square1 8 bracket\quad 5S closing IMI square $ \acute bracket{C} we$ obtained an upper global bound without differentiability restriction \ hspaceon f period∗{\ f Namely i l l } In comma [ 5 ] we we proved obtained the following an upper global bound without differentiability restriction Theorem 2 period .. If p-tilde sub comma tilde-x are defined as above comma we have that \noindentEquation:on open $ parenthesis f . $ 2 Namelyclosing parenthesis , we proved .. sum the p i f following open parenthesis x sub i closing parenthesis minus f open parenthesis1 8 S IMIsumC´ p i to the power of x i closing parenthesis leqslant f open parenthesis a closing parenthesis\ centerline plus{Theorem f open parenthesisIn 2 [. 5 ]\ wequad b obtained closingI f parenthesis $an\ uppertilde global minus{p} bound 2{ f open, }\ without parenthesistilde differentiability{x a} plus$ b are divided restriction defined by 2 to as the above , we have that } power of closing parenthesison f. Namely : = ,S we sub proved f open the parenthesis following a comma b closing parenthesis comma \ begin { a l i g n ∗} for any f that is convex overTheorem I : = open2 . squareIf bracketp˜, x˜ are a comma defined b as closing above square , we have bracket that period \ tagIn∗{ many$ ( cases 2 the bound) $}\ Ssum sub f openp parenthesis i f a ( comma x b{ closingi } parenthesis) − isf better ( than\sum D subp f open i ˆ{ x } i ) \ leqslant f ( a ) + f ( b ) − 2 f ( \ f r a c { a parenthesis a comma b closing parenthesis period .. For instance comma for ) X X a + b +f open b }{ parenthesis2 }ˆ{ ) x} closing: parenthesis =pif(x S ) −{f =(f minus} pix(i x) tof a the(a) power + ,f(b) b of− s2f comma( ) , 0:= lessS s(a, less b), 1 semicolon(2) f open i 6 2 f parenthesis\end{ a l i g x n closing∗} parenthesis = x to the power of s comma s greater 1 semicolon I subset R to the power of plus comma we have thatfor S any sub f openthat isparenthesis convex over a commaI := [a, b bclosing]. parenthesis leqslant D sub f open parenthesis a comma\noindent b closingforany parenthesis $f$In comma many thatisconvexover cases the bound Sf (a, b) is better $I than :D =f (a, b) [. For a instance , b , for ] .$ s s + for each s in openf(x parenthesis) = −x , 0 < 0 s comma < 1; f( 1x) closing = x , s parenthesis > 1; I ⊂ R cup, we open have parenthesis that Sf (a, 1b) comma6 Df (a, 2 b closing), parenthesis\ hspace cup∗{\ openf i parenthesis l l } In many 3 comma cases plus the infinity bound closing $ S parenthesis{ f } period( a , b )$ isbetterthan $ D { f } ( a , b ) . $ \quad For instance , for In this article we establish another globalforeachs bound T∈ sub(0, 1) f open∪ (1, parenthesis2) ∪ (3, +∞) a. comma b closing parenthesis for Jensen quoteright s inequality comma In this article we establish another global bound T (a, b) for Jensen ’ s inequality , \noindentwhich is better$ f than ( both x of aforementioned ) = − boundsx ˆ{ Ds sub} f open, parenthesis0 f < s a comma< b1 closing ; parenthesis f ( which is better than both of aforementioned bounds D (a, b) and S (a, b). andx S ) sub f= open x parenthesis ˆ{ s } a, comma s b> closing1 parenthesis ; I period\subset f R ˆ{ + f} , $ we have that Finally , we determine T (a, b) in the case of a generalized A – G inequality as $ SFinally{ f comma} ( we determine a , T b sub f ) open\ parenthesisl e q sf l a n t a comma D { bf closing} ( parenthesis a , in the b case ) of a generalized , $ a combination of some well - known classical means . As a consequence , new global A endash G inequality as upper bounds for A – G and G – H inequalities are established . \ begina combination{ a l i g n ∗} of some well hyphen known classical means period .. As a consequence comma new global 2 . Results f oupper r eachbounds for s A endash\ in G( and 0G endash , H 1 inequalities ) \cup are established( 1 period , 2 ) \cup ( Our main result is contained in the following 32 period , + .. Results\ infty ). Theorem 3 . Let f, p˜ x˜ be defined as above and p, q > 0, p + q = 1. Then \endOur{ a main l i g n result∗} is contained in the following, Theorem 3 period .. Let f comma p-tilde sub comma tilde-x be defined as above and p comma q greater 0 \ hspace ∗{\ f i l l } In thisX articleX wex establish another global bound $ T { f } ( comma p plus q = 1 period ..pif Then(xi) − f( pi i) 6 max p[pf(a) + qf(b) − f(pa + qb)] := Tf (a, b). (3) aEquation: , b open )$ parenthesis for Jensen 3 closing ’ s parenthesis inequality .. sum , p i f open parenthesis x sub i closing parenthesis For example , minus f open parenthesis sum p i to the power of x i closing parenthesis leqslant maximum p bracketleftbig pf open\noindent parenthesiswhich a closing is better parenthesis than plus both qf open of parenthesis aforementioned b closing parenthesis bounds minus $ D f{ openf } parenthesis( a pa , b ) $ and $ S { f } (2 a2 , b )2 . $ 2 1 2 plus qb closing parenthesisTx2( bracketrightbiga, b) = max p(pa : =+ Tqb sub− f(pa open+ qb parenthesis) ) = max ap( commapq(b − a b) closing) = (b parenthesis− a) , period For example comma 4 \ hspaceT sub∗{\ x 2 openf i l land} parenthesisFinally we obtain a, at comma we once determine the b closing well - parenthesisknown $ T pre{ - =f Grüssinequality} maximum( a p parenleftbig , b pa )$ to the in power the ofcase 2 ofageneralizedA −− G inequality as plus qb to the power of 2 minus open parenthesis pa plus qb closing parenthesis to the power of 2 parenrightbig X X 1 =\noindent maximum pa parenleftbig combination pq open of parenthesis some wellpi bx− minus2 −known( a closingpix classicali)2 parenthesis(b − a means)2, to the . power\quad of 2As parenrightbig a consequence = , new global i 6 4 1upper divided boundsby 4 open for parenthesis A −− G b andminus G a−− closingH inequalities parenthesis to the are power established of 2 comma . with 1 / 4 as the best possible constant . and we obtain at once the well hyphen known pre hyphen Grudieresisss inequality 2 . 1 It is easy to see that \ centerlinesum p i to the{2 power . \quad of x subResults i to the} power of 2 minus openRemark parenthesis sum. p i to the power of x i closing g(p) := pf(a) + (1 − p)f(b) − f(pa + (1 − p)b) is parenthesis 2 leqslant 1 divided by 4 open parenthesis b minus a closing parenthesis to the power of 2 comma concave for 0 p 1 with g(0) = g(1) = 0. Hence , there exists the unique positive \ centerlinewith 1 slash{ 4Our as the main best result possible6 6 constant is contained period in the following } Remark 2 period 1 period It is easy to see that g open parenthesis p closing parenthesis : = pf open parenthesis \ centerline {Theorem 3 . \quad Let $ f , \ tilde {p} { , }\ tilde {x} $ be defined as above and a closing parenthesis plus open parenthesis 1 minus pmax closingg(p) parenthesis= Tf (a, b). f open parenthesis b closing parenthesis p minus$ p f open , parenthesisq > 0 pa plus , open p parenthesis + q 1 = minus 1 p closing . $ parenthesis\quad Then b closing} parenthesis is concave for 0 leqslantThe p next leqslant theorem 1 with approves g open parenthesis that inequality 0 closing ( 3 ) parenthesisis stronger than = g open ( 1 ) parenthesis or ( 2 ) . 1 closing \ begin { a l i g n ∗} ˜ ˜ parenthesis = 0 periodTheorem Hence comma4 . thereLet existsI be the domain unique positive of a convex function f and I := [a, b] ⊂ I. \ tagmaximum∗{$ ( sub 3Then p g ) open $}\ parenthesissum p p closing i f parenthesis ( x ={ Ti sub} f) open− parenthesisf ( a comma\sum b closingp i ˆ{ x } parenthesisi ) \ periodl e q s l a n t \max{ p } [ pf (a)+qf (b) − f ( paThe + next qb theorem ) approves ] : that = inequality T { f open} ( parenthesis a , 3 closing b ) parenthesis . is stronger than open (i) Tf (a, b) 6 Df (a, b); (ii) Tf (a, b) 6 Sf (a, b), parenthesis\end{ a l i g 1n closing∗} parenthesis or open parenthesis 2 closing parenthesis period ˜ Theorem 4 period .. Let I-tilde be the domain of a convex function f andforeachI I : = open⊂ I. square bracket a comma \ centerline {For example , } b closing square bracketThe subset following tilde-I well sub periodknown A - G inequality [ 4 ] , asserts that A(˜p,x˜) > G(˜p,x˜), P x Q i Then where A(˜p,x˜) := pi i, G(˜p,x˜) := p , are the generalized arithmetic and \ [T { x } 2 ( a , b ) = \max{ p xi} ( pa ˆ{ 2 } + qb ˆ{ 2 } − open parenthesisgeometric i closing means parenthesis , respectively T sub f open . parenthesis a comma b closing parenthesis leqslant D sub f( open pa parenthesis + qb a comma ) ˆ b{ closing2 } parenthesis) = \max semicolon{ p } open( parenthesis pq ( ii closing b − parenthesisa ) T ˆ{ sub2 f} open) =parenthesis\ f r a c { a comma1 }{ 4 b closing} ( parenthesis b − leqslanta ) S ˆsub{ 2 f open} parenthesis, \ ] a comma b closing parenthesis comma for each I subset I-tilde sub period The following well known A hyphen G inequality open square bracket 4 closing square bracket comma asserts that\noindent A open parenthesisand we obtain p-tilde sub at commaonce the tilde-x well closing− known parenthesis pre geqslant− Gr\”{ Gu} openss parenthesisinequality p-tilde sub comma tilde-x closing parenthesis comma \ [ where\sum A openp parenthesis i ˆ{ x } p-tildeˆ{ 2 sub} { commai } tilde-x − ( closing\sum parenthesisp : i = ˆ sum{ x p} i to thei power ) of 2 x i comma\ l e q s l a n t G\ f open r a c { parenthesis1 }{ 4 } p-tilde( sub b comma− tilde-xa ) closing ˆ{ 2 parenthesis} , \ ] : = product p x sub i to the power of i comma .. are the generalized arithmetic and geometric means comma respectively period \noindent with 1 / 4 as the best possible constant .

\ hspace ∗{\ f i l l }Remark2.1. It iseasytoseethat $g ( p ) : = pf ( a ) + ( 1 − p ) f ( b ) − f ( pa + ( 1 − p ) b ) $ i s

\noindent concave for $ 0 \ l e q s l a n t p \ leqslant 1$ with $g ( 0 ) = g ( 1 ) = 0 .$ Hence , there exists theunique positive

\ begin { a l i g n ∗} \max { p } g ( p ) = T { f } ( a , b ) . \end{ a l i g n ∗}

\ centerline {The next theorem approves that inequality ( 3 ) is stronger than ( 1 ) or ( 2 ) . }

\ hspace ∗{\ f i l l }Theorem 4 . \quad Let $ \ tilde { I } $ be the domain of a convex function $f$and$I : = [ a , b ] \subset \ tilde { I } { . }$

\noindent Then

\ begin { a l i g n ∗} ( i ) T { f } ( a , b ) \ l e q s l a n t D { f } ( a , b ) ; ( i i ) T { f } ( a , b ) \ l e q s l a n t S { f } ( a , b ), \\ f o r each I \subset \ tilde { I } { . } \end{ a l i g n ∗}

\ hspace ∗{\ f i l l }The following well known A − G inequality [ 4 ] , asserts that $A ( \ tilde {p} { , }\ tilde {x} ) \ g e q s l a n t G ( \ tilde {p} { , }\ tilde {x} ) , $

\noindent where $ A ( \ tilde {p} { , }\ tilde {x} ) : = \sum p i ˆ{ x } i , G ( \ tilde {p} { , }\ tilde {x} ) : = \prod p { x { i }}ˆ{ i } , $ \quad are the generalized arithmetic and geometric means , respectively . ON A NEW CONVERSE OF JENSEN quoteright S INEQUALITY .. 1 9 ONApplying A NEW CONVERSE theorems .. OF 1 .. JENSEN and 2 with ’ S f INEQUALITYopen parenthesis\quad x closing1 9 parenthesis = minus log x comma .. one obtainsApplying the following theorems \quad 1 \quad and2with $f ( x ) = − $ l o g $ x , $converses\quad ofone A endash obtains G inequality the following : Equation: open parenthesis 4 closing parenthesis .. 1 leqslant A open parenthesis p-tilde sub comma x-tilde closing\noindent parenthesisconverses divided by of G A open−− parenthesisG inequality tilde-p : sub comma tilde-x closing parenthesis leqslant exponent parenleftbigg open parenthesisON A NEW b CONVERSE minus a closing OF JENSEN parenthesis ’ S INEQUALITY to the power1 of 9 2 Applying divided by theorems 4 ab parenrightbigg 1 and open parenthesis\ begin { a l cf i g period n ∗}2 with openf square(x) = bracket− log 2x, closingone square obtains bracket the following closing parenthesis semicolon Equation: open parenthesis\ tag ∗{$ ( 5 closing 4converses parenthesis ) $} of1 A ..–\ Gl 1 e leqslantinequality q s l a n t A : open\ f r a parenthesisc { A( p-tilde\ tilde sub{ commap} { x-tilde, }\ closingtilde parenthesis{x} ) }{ G divided( \ tilde by G{ openp} parenthesis{ , }\ tilde-ptilde { subx} comma) }\ tilde-xl e q closing s l a n t parenthesis\exp leqslant( \ f ropen a c { parenthesis( b − a plusa b closing) ˆ{ 2 parenthesis}}{ 4 to ab the} power) of ( 2 divided c f by . 4 ab [ sub 2period ] ) ; \\\ tag ∗{$ ( 5 ) $} 1 \ l e q s l a n t \ f r a c { A( \ tilde {p}A(˜p{x˜), }\(tildeb − a)2{x} ) }{ G( \ tilde {p} { , } Since 1 plus x leqslant e to the power1 of x comma, xexp( in R comma)( puttingcf.[2]); x = open parenthesis (4)b minus a \ tilde {x} ) }\ l e q s l a n t \ f r a c6{ ( a6 + b ) ˆ{ 2 }}{ 4 ab } { . } closing parenthesis to the power of 2 divided byG 4(˜ ap, bx˜) sub comma4 oneab can see that the inequality open parenthesis \end{ a l i g n ∗} 2 5 closing parenthesis is A(˜p,x˜) (a + b) 1 6 6 (5) stronger than open parenthesis 4 closing parenthesis for eachG(˜p a,x˜ comma) 4 bab in R. to the power of plus period \noindentAn even strongerSince converse $ 1 of + A endash x \ Gl e inequality q s l a n t can e be ˆ{ obtainedx } applying, x Theorem\ in R , $ putting $ x = \ f r a c { ( b − a ) ˆ{ 2 }}{ 42 a b } { , }$ one can see that the inequality ( 5 ) is C period x (b−a) Since 1 + x 6 e , x ∈ R, putting x = one can see that the inequality ( 5 ) is Theorem .. 5 period .. Let p-tilde sub comma tilde-x sub4ab comma, A open parenthesis p-tilde sub comma tilde-x stronger than ( 4 ) for each a, b ∈ R+. closing\noindent parenthesisstronger comma than G open ( parenthesis 4 ) for eachp-tilde sub $a comma , tilde-x b closing\ in parenthesisR ˆ{ + } .. be. .. $ deﬁned .. as An even stronger converse of A – G inequality can be obtained applying Theorem .. above .. and x sub i in open square bracket a comma b closing square bracket comma C. An0 even less a strongerless b period converse .. We have of comma A −− G inequality can be obtained applying Theorem Theorem 5 . Let p˜ x˜ A(˜p x˜),G(˜p x˜) be deﬁned as above and C.Equation: open parenthesis 6 closing parenthesis, , .., 1 leqslant, A open parenthesis p-tilde sub comma x-tilde x ∈ [a, b], 0 < a < b. We have , closing parenthesis dividedi by G open parenthesis tilde-p sub comma tilde-x closing parenthesis leqslant L open parenthesisTheorem \ aquad comma5 b . closing\quad parenthesisLet $ \ Itilde open parenthesis{p} { , a}\ commatilde b closing{x} parenthesis{ , } A( divided by\ Gtilde to the{p} { , } \ tilde {x} ),G( \ tilde {p} { , }\ tilde {x} ) $ \quad be \quad d e f i n e d \quad as \quad above \quad and power of 2 open parenthesis a comma b closing parenthesisA(˜p,x˜) L :(a, = b T)I( opena, b) parenthesis a comma b closing parenthesis $ x { i }\ in [ a , b1 ] , $ := T (a, b), (6) comma 6 G(˜p x˜) 6 G2(a, b) $ 0 < a < b . $ \quad We, have , where G open parenthesis a comma√ b closing parenthesis : = square root of ab sub comma L open parenthesis b−a 1 bb a comma b closingwhere parenthesisG(a, b) : = := b minusab, L a(a, divided b) := by log b minusI(a, b log)a := sub( commaa )1/(b − Ia open), are parenthesis the a \ begin { a l i g n ∗} log b−log a , e a comma b closing parenthesisgeometric , : l = ogarithmic 1 divided and by e identric parenleftbigg means b , to respectively the power . of b divided by a to the power of a parenrightbigg\ tag ∗{$ ( 1 6 slash ) open $} parenthesis1 \ l e q s b l aminus n t a\ closingf r a c { parenthesisA( \ subtilde comma{p} ....{ are, the}\ tilde {x} ) }{ G ( \ tilde {p} { ,Since}\ , fortildea 6={b,x G} (a, b)) }\< L(la, e q b) s l a< n t I(a,\ bf) r a c<{ AL(a, b), (one can a see , that b the ) I geometric commainequality l ogarithmic ( 6 ) isand stronger identric than means ( 5comma ) . respectively period (Since a comma , b for a ) equal-negationslash}{ G ˆ{ 2 } ( b comma a G, open b parenthesis ) } : a comma= T b closing ( a parenthesis , b less ) L , In addition to the above we quote a converse for G – H inequality , where H = open parenthesis a comma b closingP parenthesis−1 less I open parenthesis a comma b closing parenthesis less A open \end{ a l i g n ∗} H(˜p,x˜) := ( pi/xi) , denotes the generalized harmonic mean . parenthesis a comma b closingTheorem parenthesis6 comma. With one the can notation see that of the Theorem E , we have inequality open parenthesis 6 closing parenthesis is stronger than open parenthesis 5 closing parenthesis period \noindent where$G ( a , b ) : = \ sqrt { ab } { , } L ( a , In addition to the above we quote a converse for G endashG(˜p,x˜) H inequality comma where H = b ) : = \ f r a c { b − a }{\ log1 6 b −6 T (a, \ log b). a } { , } I ( a , H open parenthesis p-tilde sub comma tilde-x closingH(˜ parenthesisp,x˜) : = open parenthesis sum p i slash x sub i closingb ) parenthesis : = to\ thef r a cpower{ 1 of}{ minuse } 1 comma( \ f r denotesa c { b the ˆ{ generalizedb }}{ a ˆ harmonic{ a }} mean) period 1 / ( b 3 . Proofs − Theorema ) 6 period{ , }$ .. With\ h f the i l l notationare the of Theorem E comma we have Since x ∈ [a, b], there is a sequence {λ }, λ ∈ [0, 1], 1 leqslant G open parenthesisProof of p-tilde Theorem sub 3 comma . x-tildei closing parenthesis divided byi H openi parenthesis such that x = λ a + (1 − λ )b, i = 1, 2, .... Hence , tilde-p\noindent sub commageometric tilde-x closing ,i l ogarithmic parenthesisi leqslanti and identric T open parenthesis means a , comma respectively b closing parenthesis . period 3 period .. Proofs \ hspace ∗{\ f i l l } Since , for $ a \ne b , G ( a , b ) < L( Proof of Theorem 3X period .. SinceX x subx i in openX square bracket a commaX b closing square bracket comma pif(xi) − f( pi i) = pif(λia + (1 − λi)b) − f( pi(λia + (1 − λi)b)) therea is, a sequence b ) open< braceI lambda ( a sub i , closing b brace ) comma< A lambda ( sub a i in open, b square ) bracket ,$ 0 comma onecanseethatthe X X λ X 1 closing square bracket comma 6 pi(λif(a) + (1 − λi)f(b)) − f(a pi i + b pi(1 − λi)) \noindentsuch that xinequality sub i = lambda ( 6sub ) i ais plus stronger open parenthesis than ( 1 5 minus ) . lambda sub i closing parenthesis b comma X λ X λ X λ X λ i = 1 comma 2 comma period period= f(a)( periodpi periodi) + f( Henceb)(1 − commapi i) − f(a( pi i) + b(1 − pi i)). \ hspaceLine 1∗{\ sumf pi l i l f} openIn addition parenthesis to x sub the i closing above parenthesis we quote minus a converse f open parenthesis for G −− sumH p inequality i to the power , where P λ P λ of$ Hx i closing = $ parenthesisDenoting = sumpi pi i f:= parenleftbigp, 1 − lambdapi i := subq, iwe a plushave open that 0parenthesis6 p, q 6 11 minus, p + q lambda= 1. sub i closing parenthesisConsequently b parenrightbig , minus f open parenthesis sum p i parenleftbig lambda sub i a plus open parenthesis\noindent 1 minus$ H lambda ( \ subtilde i closing{p} parenthesis{ , }\ btilde parenrightbig{x} ) closing : parenthesis = ( Line\sum 2 leqslantp sum i p X X x i/ parenleftbig x { i lambda} ) ˆ sub{ − i f open1 } parenthesispif, $(xi) denotes− af closing( pi the parenthesisi) 6 generalizedpf(a) + plusqf(b open) − harmonicf parenthesis(pa + qb) mean 1 minus . lambda sub i closing parenthesis f open parenthesis b closing parenthesis parenrightbig minus f open parenthesis a sum p 6 max p[pf(a) + qf(b) − f(pa + qb)] := Tf (a, b), i\ centerline to the power{ ofTheorem lambda i 6 plus . \ bquad sum pWith i open the parenthesis notation 1 minus of Theorem lambda sub E i, closing we have parenthesis} closing parenthesis Line 3and = f openthe proof parenthesis of Theorem a closing 3 is done parenthesis. open parenthesis sum p i to the power of lambda i closing\ [ 1 parenthesis\ l e q s l a n plus t f\ openf r a c parenthesis{ G( b\ closingtilde parenthesis{p} { , open}\ parenthesistilde {x} 1 minus) }{ sumH( p i to the\ tilde power{p} { , } of\ tilde lambda{x i} closing) }\ parenthesisleqslant minus f T open ( parenthesis a , a open b parenthesis ) . \ ] sum p i to the power of lambda i closing parenthesis plus b open parenthesis 1 minus sum p i to the power of lambda i closing parenthesis closing parenthesis period \ centerlineDenoting sum{3 p . i to\quad the powerProofs of lambda} i : = p comma 1 minus sum p i to the power of lambda i : = q comma we have that 0 leqslant p comma q leqslant 1 comma p plus q = 1 period \ hspaceConsequently∗{\ f i l comma l } Proof of Theorem 3 . \quad Since $ x { i }\ in [ a , b ]Line , $ 1 sum there p i f open is a parenthesis sequence x sub $ \{\ i closinglambda parenthesis{ minusi }\} f open parenthesis, \lambda sum p{ i toi the}\ powerin of[ x i 0 closing , parenthesis 1 ] leqslant , $ pf open parenthesis a closing parenthesis plus qf open parenthesis b closing parenthesis minus f open parenthesis pa plus qb closing parenthesis Line 2 leqslant maximum p bracketleftbig pf open\noindent parenthesissuch a closing that parenthesis $ x { plusi } qf= open\ parenthesislambda { b closingi } parenthesisa + ( minus 1 f open− parenthesis \lambda pa{ i } plus) qb b closing , parenthesis i =1 bracketrightbig , 2 : , = T . sub f . open . parenthesis .$Hence, a comma b closing parenthesis comma and the proof of Theorem 3 is done period square \ [ \ begin { a l i g n e d }\sum p i f ( x { i } ) − f ( \sum p i ˆ{ x } i ) = \sum p i f ( \lambda { i } a + ( 1 − \lambda { i } ) b ) − f ( \sum p i ( \lambda { i } a + ( 1 − \lambda { i } ) b ) ) \\ \ l e q s l a n t \sum p i ( \lambda { i } f ( a ) + ( 1 − \lambda { i } ) f ( b ) ) − f ( a \sum p i ˆ{\lambda } i + b \sum p i ( 1 − \lambda { i } )) \\ = f ( a ) ( \sum p i ˆ{\lambda } i ) + f ( b ) ( 1 − \sum p i ˆ{\lambda } i ) − f ( a ( \sum p i ˆ{\lambda } i ) + b ( 1 − \sum p i ˆ{\lambda } i ) ) . \end{ a l i g n e d }\ ]

\noindent Denoting $ \sum p i ˆ{\lambda } i : = p , 1 − \sum p i ˆ{\lambda } i : = q ,$ wehavethat $0 \ leqslant p , q \ leqslant 1 , p + q = 1 .$ Consequently ,

\ [ \ begin { a l i g n e d }\sum p i f ( x { i } ) − f ( \sum p i ˆ{ x } i ) \ leqslant pf ( a ) + qf ( b ) − f ( pa + qb ) \\ \ l e q s l a n t \max{ p } [ pf (a)+qf (b) − f ( pa + qb ) ] : = T { f } ( a , b ) , \end{ a l i g n e d }\ ]

\noindent and the proof of Theorem 3 is done $ . \ square $ 1 1 0 .. S IMI Cacute \noindentProof of Theorem1 1 0 4\quad periodS .. IMI Applying $ \ Theoremsacute{C} 1 and$ 2 with hash x-tilde = 2 comma for each p in open square bracket 0 comma 1 closing square bracket and a comma b in I-tilde we get \ hspaceLine 1∗{\ openf iparenthesis l l } Proof i closing of Theorem parenthesis 4 . pf\ openquad parenthesisApplying a closing Theorems parenthesis 1 and plus 2 qf with open parenthesis $ \# \ tilde {x} =b closing 2 parenthesis ,$ foreach minus f open parenthesis pa plus qb closing parenthesis leqslant D sub f open parenthesis a comma b closing parenthesis comma Line 2 open parenthesis ii closing parenthesis pf open parenthesis a closing \noindent $ p \ in [ 0 , 1 ]$and$a , b \ in \ tilde { I } $ parenthesis plus qf1 open 1 0 parenthesisS IMI C´ b closing parenthesis minus f open parenthesis pa plus qb closing parenthesis we get leqslant S sub f open parenthesisProof of a comma Theorem b closing 4 . parenthesisApplying Theorems period 1 and 2 with #˜x = 2, for each Since the rightp hyphen∈ [0, 1] hand and a, sides b ∈ I˜ ofwe open get parenthesis i closing parenthesis and open parenthesis i i closing parenthesis\ [ \ begin { doa l inot g n edepend d } ( on p i comma )pf(a)+qf(b) we get at once − f ( pa + qb ) \ l e q s l a n t D { f } ( a , b ) , \\ T sub f open parenthesis a comma b( closingi) pf(a parenthesis) + qf(b) − =f( maximumpa + qb) 6 pD bracketleftbigf (a, b), pf open parenthesis a ( ii ) pf ( a )+qf (b) − f ( pa + qb ) closing parenthesis plus qf open parenthesis(ii) bpf closing(a) + qf parenthesis(b) − f(pa + minusqb) 6 fS openf (a, b parenthesis). pa plus qb closing parenthesis\ l e q s l a n t bracketrightbig S { f } leqslant( aD sub , f open b parenthesis ) . \ aend comma{ a l i gb n closing e d }\ ] parenthesis semicolon Equation: square .. T sub f openSince parenthesis the right a- handcomma sides b closing of ( i) parenthesis and ( i i ) = do maximum not depend p bracketleftbig on p, we get at pf once open parenthesis a closing parenthesis plus qf open parenthesis b closing parenthesis minus f open parenthesis pa plus qb closing parenthesis\noindent bracketrightbigSince the rightleqslant− S subhand f open sides parenthesis of ( i a comma) and b ( closing i i ) parenthesis do not depend period on $ p T (a, b) = max p[pf(a) + qf(b) − f(pa + qb)] D (a, b); , $Proof we of get Theorem at once 5 period .. Byf Theorem 3 comma applied with f open parenthesis6 f x closing parenthesis = minus log x comma we ob hyphenTf (a, b) = max p[pf(a) + qf(b) − f(pa + qb)] 6 Sf (a, b). \ begintain { a l i g n ∗} Proof of Theorem 5 . By Theorem 3 , applied with f(x) = − log x, we ob - T 0 leqslant{ f } log( A open a parenthesis , b p-tilde ) = sub\max comma{ p x-tilde} [ closing pf parenthesis ( a divided )+qf by G open parenthesis (b ) − f (tain pa + qb ) ] \ l e q s l a n t D { f } ( a , b ) ; \\ T { f } tilde-p sub comma tilde-xA closing(˜p,x˜) parenthesis leqslant T sub minus log x open parenthesis a comma b closing ( a , b0 )log = \maxT−{ logp }x(a,[ b) = max pfp[ (a)+qflog (pa + qb) − p log a − q log (b)b]. By st andard− f parenthesis = maximum6 pG bracketleftbig(˜p,x˜) 6 log open parenthesis pa plus qb closing parenthesis minus p log a minus q( log pab bracketrightbig + qbargument period ) it ] is easy\ l eto q finds l a n that t the S unique{ f } maximum( a is attained , b at ) . \ tag ∗{$ \ square $} \end{ a l i g n ∗} By st andard argument it is easy to find that the unique maximumb is1 attained at Line 1 b 1 Line 2 p = overbar b sub minus a minus overbar log b minus log a period \ hspaceSince 0∗{\ lessf ai l less l } Proof b comma of we Theorem get 0 less 5 p less. \pquad 1= andb−a commaByTheorem− logb after− log somea. 3 , calculation applied comma with it $ follows f that( x ) = − $ log $x ,$ weob − 0 leqslant log ASince open 0 parenthesis< a < b, we p-tilde get 0 sub< p comma < 1 and x-tilde , after closing some calculation parenthesis , divided it follows by that G open parenthesis tilde-p sub comma tilde-x closingA(˜p,x˜) parenthesisb−a leqslant) log openb log parenthesisb−a log a b minus a divided by log b minus log 0 6 log 6 log ( − log (ab)+ −1, Exponentiating , we obtain a\noindent to the powert aof i n closing parenthesisG(˜p,x˜) minuslog b− loglog a open parenthesis abb−a closing parenthesis plus b log b minus a log the assertion from Theorem 5. a divided by b minus a minus 1 comma 1 \noindent $ 0 Proof\ leqslant of Theorem $ log 6. $\ f rBy a c { changeA( of variable\ tildexi →{p} the{ , proof}\ easilytilde fol{ -x} ) }{ G Exponentiating comma we obtain the assertion from Theorem 5 period squarexi , ( Proof\ tilde of Theorem{p} lows{ 6 from, period}\ ( 6 ..tilde ) By since change{ thenx} of) variable}\ l xe q sub s l a i n right t arrow T { 1 − divided }$ l by o g x sub $ x i sub ( comma a the , proofb ) easily = fol\ hyphenmax{ p } [$ log $( pa + qb ) − p $ l o g $ a − q $ l o g $ b ] . $ 1 1 lows from open parenthesis 6 closing parenthesisA(˜p,x˜) → since then; G(˜p,x˜) → H(˜p x˜) G(˜p x˜) ByA stopen andard parenthesis argument p-tilde sub it comma is easy tilde-x to closing find parenthesis, that the right unique arrow, maximum 1, divided by is H attained open parenthesis at p-tilde sub commaand tilde-x the right closing - hand parenthesis side of (semicolon 6 ) stays Gunaltered open parenthesis under the p-tildetransformation sub comma. tilde-x closing parenthesis\ [ \ begin { righta l i g arrow n e d } 1b divided 1 \\ by G open parenthesisReferences p-tilde sub comma tilde-x closing parenthesis sub comma p = \ overline {\}{ b } { − a } − \ overline {\}{\ log } b − \ log a and the right hyphen[ 1 ] P . hand R . Beesack side of , J open . Peˇcarić, parenthesisOn Jensen 6 closing ’ s inequality parenthesis for convex stays functions unaltered, J . Mathunder . Analthe transfor- . . \end{ a l i g n e d }\ ] mation period squareAppl . 1 10 ( 1 985 ) , 536 – 552 . [ 2 ] I . Budimir , S . S . Dragomir , J . Peˇcarić, Further reverse References results for Jensen ’ s dis crete inequality and applications in Information Theory , J . Inequal . open square bracketPure Appl 1 closing . Math square . 2 ( 1 ) bracket ( 200 1 ) P , Artperiod . 5 . R [ period 3 ] S . S Beesack . Dragomir comma , A converse J period result Peccaronaricacute for Jensen \noindent Since $ 0 < a < b , $ we get $ 0 < p < 1 $ and , after some calculation , it follows that comma On Jensen’ quoteright s discrete inequality s inequality via Grüss for convex inequality functions and applications comma J in period Information Math Theory period, AnalAn . Univ period . Appl period Oradea , Fasc . Mat . 7 ( 1 999 – 2000 ) , 1 78 – 189 . [ 4 ] D . S . Mitrinović, Analytic Inequalities \noindent $ 0 \ leqslant $ log $\ f r a c { A( \ tilde {p} { , }\ tilde {x} ) }{ G 1 10 open parenthesis, Springer 1985 - Verlag closing , New parenthesis York , 1 970 comma . [ 5536 ] S . endash Simić, On 552 a period global upper bound for Jensen ’ s ( \ tilde {p} { , }\ tilde {x} ) }\ leqslant $ log $ ( \ f r a c { b − a }{\ log open square bracketinequality 2 closing, J . Math square . Anal bracket . Appl I . period 343 ( 2008 Budimir ) , 414 comma – 41 9 . S period S period Dragomir comma J b − \ log a }ˆ{ ) } − $ l o g $ ( ab ) + \ f r a c { b \ log b − a period Peccaronaricacute[ 6 ] G . comma H . Hardy Further , J . E reverse . Littlewood results , G for . Pölya Jensen , Inequalities quoteright, s Cambridge dis crete Universityinequality Press , \ log a }{ b − a } − 1 , $ and applicationsCambridge in Information , 1 978 Theory. comma J period Inequal period Pure Appl period Math period 2 open Exponentiating , we obtain the assertion from Theorem $ 5 . \ square $ parenthesis 1 closing parenthesisMatematiˇckiinstitut open parenthesis SANU ( 200Received 1 closing 3 1 1 2 parenthesis 2007 ) Kneza comma Mihaila Art 36 period 5 period open square bracket 3 closing square bracket S periodBeograd S period , Serbia Dragomir comma A converse result for Jensen \ hspace ∗{\ f i l l } Proof of Theorem 6 . \quad By change of variable $ x { i }\rightarrow quoteright s discrete inequality via Grudieresissss s imi inequality c @ mi . ands a − n u . ac . r s \ f rapplications a c { 1 }{ inx Information{ i }} { Theory, }$ comma the proof An period easily Univ period fol − Oradea comma Fasc period Mat period 7 open parenthesis 1 999 endash 2000 closing parenthesis comma 1 78 endash 189 period \noindentopen squarelows bracket from 4 closing ( 6 ) square since bracket then D period S period Mitrinovicacute comma Analytic Inequalities comma Springer hyphen Verlag comma New York comma 1 970 period \ [A(open square\ brackettilde { 5p closing} { square, }\ brackettilde { Sx period} ) Simicacute\rightarrow comma On\ afglobal r a c { upper1 }{ boundH( for Jensen\ tilde {p} { , } quoteright\ tilde {x s} inequality) } comma;G( J period\ Mathtilde period{p} Anal{ , period}\ Appltilde period{x} 343) open\rightarrow parenthesis 2008\ closingf r a c { 1 }{ G parenthesis( \ tilde comma{p} { , }\ tilde {x} ) } { , }\ ] 414 endash 41 9 period open square bracket 6 closing square bracket G period H period Hardy comma J period E period Littlewood comma\noindent G periodand Podieresislya the right comma− hand Inequalities side of comma ( 6 Cambridge ) stays University unaltered Press under comma the Cambridge transformation comma $ .1 978\ periodsquare $ Matematiccaronki institut SANU .. open parenthesis Received 3 1 1 2 2007 closing parenthesis \ centerlineKneza Mihaila{ References 36 } Beograd comma Serbia \noindents s imi c at[ mi 1 period ] P .R. s a-n u Beesack period ac period , J . r sPe\v{c} a r i \ ’{ c} , On Jensen ’ s inequality for convex functions , J . Math . Anal . Appl . 1 10 ( 1 985 ) , 536 −− 552 . [ 2 ] I . Budimir , S . S . Dragomir , J . Pe\v{c} a r i \ ’{ c} , Further reverse results for Jensen ’ s dis crete inequality and applications in Information Theory , J . Inequal . Pure Appl . Math . 2 ( 1 ) ( 200 1 ) , Art . 5 . [ 3 ] S . S . Dragomir , A converse result for Jensen ’ s discrete inequality via Gr\”{u} ss inequality and applications in Information Theory , An . Univ . Oradea , Fasc . Mat . 7 ( 1 999 −− 2000 ) , 1 78 −− 189 . [ 4 ] D . S . Mitrinovi \ ’{ c} , Analytic Inequalities , Springer − Verlag , New York , 1 970 . [ 5 ] S . Simi \ ’{ c} , On a global upper bound for Jensen ’ s inequality , J . Math . Anal . Appl . 343 ( 2008 ) , 414 −− 41 9 .

\noindent [ 6 ]G. H . Hardy , J . E . Littlewood ,G. P\”{o} lya , Inequalities , Cambridge University Press , Cambridge , 1 978 .

Matemati\v{c} ki institut SANU \quad ( Received 3 1 1 2 2007 ) Kneza Mihaila 36

\ centerline {Beograd , Serbia }

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