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ON a NEW CONVERSE of JENSEN ' S INEQUALITY Slavko Simic

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ON a NEW CONVERSE of JENSEN ' S INEQUALITY Slavko Simic PUBLICATIONS DE L quoteright INSTITUT MATH Eacute MATIQUE nnoindentNouvelle seacuteriePUBLICATIONS comma DE tome L 85 ' INSTITUT open parenthesis MATH 99 $ closingnacute parenthesisfEg $ 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 nnoindentON A NEWNouvelle CONVERSE s n 'f OFeg JENSENrie , tome85 quoteright ( S 99 INEQUALITY ) ( 29 ) , 17 −− 1 1 0 n h f i l l DOI : 1 0 . 2 298 / PIM 9991 7 S Slavko Simicacute n centerlineCommunicatedfON by A Stevan NEW CONVERSE Pilipovicacute OF JENSEN ' S INEQUALITY g Abstract periodPUBLICATIONS .. We give another DE L ' INSTITUTglobal upper MATH boundE´ MATIQUE for Jensen quoteright s discrete inequal hyphen n centerline f Slavko Simi n 'f cg g ity which is betterNouvelle than s´erie, 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 n centerline1 period .. IntroductionfCommunicated by Stevan PilipoviSlavkon ' Simi´cf cg g Throughout this paper x-tilde = open braceCommunicated x sub i closing by Stevan brace Pilipovi´c represents a finite sequence of real numbers n centerline f Abstract . nquad We give another global upper bound for Jensen ' s discrete inequal − g 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 n centerline f ity which is better than already existing ones . nquad For instance , we determine a g 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 n centerline fnew converses for generalized A −− G and G −− H inequalities . g bracket 1 closing square bracketThroughout comma open this paper squarex ~ bracket= fxig 4represents closing square a finite bracket sequence of real numbers asserts that : belonging to a fixed closed interval I = [a; b]; a < b andp ~ = fpig; P pi = 1 is a sequence n centerline f1 . nquad Introduction g 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 ] n hspaceThere∗fn are manyf i l lassertsg importantThroughout that inequalities: this paperwhich are $ particularn tilde f casesxg of= Jensennf quoterightx f si g ng $ represents a finite sequence of real numbers inequality such as the weighted A endash G endash H inequality comma Cauchy quoteright s inequality comma nnoindent belongingtoafixed closedX interval $IX x = [ a , b ] , the Ky Fan 0 6 pif(xi) − f( pi i): a and< Hodieresislderb $ and inequalities $ n tilde commafpg etc= periodnf p i ng , nsum 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 $ intervaln tilde ,I Cauchyf commaxg f ' whereupon s. inequalityg$ fis , the convex Ky period An upper globalFan bound and open H¨olderinequalities parenthesis i period , etc .e period comma depending in f and I only closing parenthesis forn hspace Jensen∗fn quoterightf i l l g 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 . nnoindentTheorem 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 minusn [ 0 f openn l e parenthesis q s l awas n t given sumnsum by p i Dragomir top the power i [ 3 ] of f, x i closing ( x parenthesisf i g leqslant) − 1 dividedf ( by 4n opensum parenthesisp i b^f x g minusi ) a closing . n ] 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 Hn parenthesis"fog 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 n hspace2000 Mathematics∗fn f i l l gOne 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 .. generalizednnoindent Adepend hyphen G on hyphen $ n Htilde fpg $ and(cf: $ [1]n tilde; [2]; [3])f:xg f , g$ but only on $ f $ andinequality the interval period $ I , $ whereupon $ f $ is convex . 1 7 n hspace ∗fn f i l l gAn 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 . nnoindent was given by Dragomir [ 3 ] , 1 7 n centerline fTheorem 1 . nquad If $ f $ is a differentiable convex mapping on $ I , $ then we have g n begin f a l i g n ∗g n tag ∗f$ ( 1 ) $g nsum p i f ( x f i g ) − f ( nsum p i ^f x g i ) n l e q s l a n t n f r a c f 1 gf 4 g ( b − a ) ( f ^f nprime g ( b ) − f ^f nprime g ( a ) ) : = D f f g ( a , b ) . nendf a l i g n ∗g There is a number of papers where inequality ( 1 ) is utilized in applications concerning some parts of Analysis , nquad Numerical Analysis , Information Theory etc n begin f a l i g n ∗g (cf.[1],[2],[3]). nn n r u l e f3emgf0.4 pt g nendf a l i g n ∗g n centerline f2000 Mathematics Subject Classification : Primary 26 B 25 , 26 D 20 . g Key words and phrases : Jensen ' s discrete inequality ; nquad global bounds ; nquad generalized A − G − H inequality . n centerline f1 7 g 1 8 .. S IMI Cacute nnoindentIn open square1 8 bracketnquad 5S closing IMI square $ nacute bracketfCg we$ obtained an upper global bound without differentiability restriction n hspaceon f period∗fn f Namely i l l g 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 nnoindentEquation: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 parenthesisn centerline plusfTheorem f open parenthesisIn 2 [. 5 ]n wequad b obtained closingI f parenthesis $ann uppertilde global minusfpg bound 2f f open, g without parenthesis n tilde differentiabilityfx ag plus$ b are divided restriction defined by 2 to as the above , we have that g power of closing parenthesison f: Namely : = ,S we sub proved f open the parenthesis following a comma b closing parenthesis comma n begin f a l i g n ∗g 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 n tagIn∗f many$ ( cases 2 the bound) $gn Ssum sub f openp parenthesis i f a ( comma x bf closingi g parenthesis) − isf better ( thannsum D subp f open i ^f x g i ) n leqslant f ( a ) + f ( b ) − 2 f ( n f r a c f a parenthesis a comma b closing parenthesis period .. For instance comma for ) X X a + b +f open b gf parenthesis2 g^f ) xg closing: parenthesis =pif(x S ) −ff =(f minusg 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 parenthesisnendf a l i g x n closing∗g 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 commannoindent 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); paren- thesisn hspace cup∗fn openf i parenthesis l l g In many 3 comma cases plus the infinity bound closing $ S parenthesisf f g period( a , b )$ isbetterthan $ D f f g ( a , b ) .
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