Geeralzed Covex Fuctos o Fractal Sets ad Two Related Iequaltes Huxa Mo, X Su ad Dogya Yu 3,,3School of Scece, Bejg Uversty of Posts ad Telecommucatos, Bejg,00876, Cha, Correspodece should be addressed to Hu-xa Mo; huxmo@bupteduc I the paper, we troduce the geeralzed covex fucto o fractal sets R (0 of real le umbers ad study the propertes of the geeralzed covex fucto Based o these propertes, we establsh the geeralzed Jese s equalty ad geeralzed Hermte- Hadamard s equalty Furthermore, some applcatos are gve Itroducto Let f : I R R For ay x, x I ad [0,], f the followg equalty f ( x( x f( x ( f( x holds, the f s called a covex fucto o I The covexty of fuctos play a sgfcat role may felds, for example bologcal system, ecoomy, optmzato ad so o [-] Ad may mportat equaltes are establshed for the class of covex fuctos For example, the Jese's equalty ad Hermte-Hadamard s equalty are the best ow results the lterature, whch ca be stated as follows Jese's equalty [3]: Assume that f s a covex fucto o [ ab, ] The for ay x [ ab, ] ad [0,] (,,, wth, we have f x f( x Hermte-Hadamard s equalty [4]: Let f be a covex fucto o [ ab, ] wth a b If f s tegral o [ ab,, ] the ab b f( a f( b f f( x dx b a a I recet years, the fractal has receved sgfcatly remarable atteto from scetsts ad egeers I the sese of Madelbrot, a fractal set s the oe whose Hausdorff dmeso strctly exceeds the topologcal dmeso [5-9] May researchers studed the propertes of fuctos o fractal space ad costructed may ds of fractoal calculus by usg dfferet approaches (see [0-4] Partcularly, [3], Yag stated the aalyss of local fractoal fuctos o fractal space systematcally, whch cludes local fractoal calculus, the mootocty of fucto ad so o Ispred by these vestgatos, we wll troduce the geeralzed covex fucto o fractal sets ad establsh the geeralzed Jese's equalty ad geeralzed Hermte- Hadamard s equalty related to geeralzed covex fucto We shall focus our atteto o the covexty sce a fucto f s cocave f ad oly f f s covex So, every result for the covex fucto ca be easly re-stated terms of cocave fuctos
The artcle s orgazed as follows: I Secto, we state the operatos wth real le umber o fractal sets ad gve the deftos of the local fractoal dervatves ad local fractoal tegral I Secto 3, we troduce the defto of the geeralzed covex fucto o fractal sets ad study the propertes of the geeralzed covex fuctos I Secto 4, we establsh the geeralzed Jese s equalty ad geeralzed Hermte- Hadamard s equalty o fractal sets I Secto 5, some applcatos are gve o fractal sets by meas of the geeralzed Jese s equalty Prelmares Recall the set R of real le umbers ad use the Gao-Yag-Kag s dea to descrbe the deftos of the local fractoal dervatve ad local fractoal tegral Recetly, the theory of Yag s fractoal sets [3] was troduced as follows For 0, we have the followg -type set of elemet sets: Z : The -type set of the teger are defed as the set {0,,,,, } Q : The -type set of the ratoal umbers are defed as the set { m p/ q : pz, q 0} J : The -type set of the rratoal umbers are defed as the set { m p/ q : pz, q 0} R : The -type set of the real le umbers are defed as the set R Q J If a, b ad c belog to the set R of real le umbers, the ( a b ad ab belog to the set R ; ( a b b a ( ab ( b a ; (3 a ( b c ( ab c ; (4 a b b a ( ab ( ba ; (5 a ( b c ( a b c ; (6 a ( b c a b a c ; (7 a 0 0 a a ad a a a Let us ow state some deftos about the local fractoal calculus o R Defto [3] A o-dfferetable fucto f : RR, x f( x s called to be local fractoal cotuous at x 0, f for ay 0, there exsts 0, such that f( x f( x holds for xx0, where, R If f ( x s local fractoal cotuous o the terval ( ab,, we deote f ( x C ( a, b Defto [3] The local fractoal dervatve of f ( x of order at x x0 s defed 0
by 3 f d f( x ( f( x f( x ( x lm, ( 0 0 x x dx xx 0 ( x x 0 0 where ( f ( x f( x0 ( ( f( x f( x0 tmes (( If there exsts f ( x Dx Dx f( x for ay x I R, the we deote f D( ( I, where 0,, Defto 3 [3] The local fractoal tegral of the fucto f ( x of order s defed by ( aib f( x b f(( t dt ( a a N lm f ( tj( tj, ( a t 0 wth tj tj tjad t max{ tj j,,, N }, where [ tj, tj ], j 0,, N t at t t t b s a partto of the terval [ ab, ] ad 0 N N Here, t follows that aia for ay x [ ab, ], there exsts ( a j 0 f( x 0 f a b ad I ( x f( x, the t s deoted by I f( x I f( x f a b If ( ( a b b a ( f ( x Ix [ a, b] Lemma [3] (Geeralzed local fractoal Taylor theorem Suppose that ( f ( x C ( I, for terval I R, 0,, 0 Ad let x0 [ ab, ] The for ay x I, there exsts at least oe pot, whch les betwee the pots x ad x, such that 0 ( (( f ( x0 f ( ( f( x ( xx0 ( xx0 0 ( a ( ( a Remar Whe I R s a ope terval ( ab,, Yag [3] has gve the proof for the geeralzed local fractoal Taylor theorem I fact, usg the geeralzed local fractoal Lagrage s theorem ad followg the proof of the class Taylor theorem, we ca show that for ay terval I R, the formula s also true 3 Geeralzed covex fuctos From a aalytcal pot of vew, we have the followg defto Defto 3 Let f : I R R For ay x, x I ad [0,], f the followg equalty f ( x ( x f( x ( f( x holds, the f s called a geeralzed covex fucto o I Defto 3 Let f : I R For ay x x I ad [0,], f the followg equalty f ( x ( x f( x ( f( x holds, the f s called a geeralzed strctly covex fucto o I R
It follows mmedately, from the gve deftos, that a geeralzed strctly covex fucto s also geeralzed covex But, the coverse s ot true Ad f these two equaltes s reversed, the f s called a geeralzed cocave fucto or geeralzed strctly cocave fucto, respectvely Here are two basc examples of geeralzed strctly covex fuctos: p ( f( x x, x 0, p ; ( f ( x E ( x, x R, where x E ( x ( s the Mttag-Leffer fucto 0 Note that the lear fucto f ( x a x b, x R s geeralzed covex ad also geeralzed cocave We shall focus our atteto o the covexty sce a fucto f s cocave f ad oly f f s covex So, every result for the covex fucto ca be easly re-stated terms of cocave fuctos I the followg, we wll study the propertes of the geeralzed covex fuctos Theorem 3 Let f : I R The f s a geeralzed covex fucto f ad oly f the equalty f( x f( x f ( x3 f( x ( x x ( x3 x holds, for ay x, x, x3 I wth 3 x3 x Proof I fact, tae, x3 x the x x( x3 Ad by the geeralzed covexty of f, we get x3 x x x f ( x f ( x( x3 f ( x ( f ( x3 f ( x f ( x3 x3x x3x From the above formula, t s easy to see that f( x f( x f ( x3 f( x ( x x ( x x 3 Reversely, for ay two pots x, x 3 ( x x 3 o I R, we tae x x( x3 for x3 x (0, The x x x3 ad Usg the above verse process, we have x x So, f s a covex fucto o I R 3 f ( x ( x f( x ( f( x 3 3 I the same way, t ca be show that f s a geeralzed covex fucto o I R f ad oly f f( x f( x f ( x3 f( x f( x3 f( x, ( x x ( x x ( x x 3 3 4
for ay x, x, x3 I wth x x x 3 Theorem 3 Let f D ( I, the the followg codtos are equvalet ( f s a geeralzed covex fucto o I, ( ( f s a creasg fucto o I, (3 for ay x, x I, ( f ( x f ( x f( x ( x x ( Proof ( Let x, x I wth x x Ad tae h 0 whch s small eough such that xhx, h I Sce x h x x x h, the usg Theorem 3 we have f ( x f( xh f( x f( x f( x h f( x ( a ( a ( a h ( x x h 5 Sce f D ( I, the let h 0, t follows that ( f( x f( x ( f ( x ( a f ( x ( x x ( So, f s creasg I ( 3 Tae x, x I Wthout loss of geeralty, we ca assume that x x Sce ( f s creasg the terval I, the applyg the geeralzed local fractoal Taylor theorem, we have ( ( f ( f ( x f( x f( x ( x x ( x x, ( a ( a where ( x, x That s to say ( f ( x f ( x f( x ( x x ( a (3 For ay x, x I, we let x 3 x ( x, where 0 It s easy to see that xx3 ( ( x x ad x x3 ( x x The from the thrd codto, we have ( ( f ( x3 f ( x3 f( x f( x3 ( xx3 f( x3 ( ( xx, ( a ( a ad ( ( f ( x3 f ( x3 f ( x f( x3 ( x x3 f( x3 ( x x ( a ( a At the above two formulas, multply ad (, respectvely, the we obta f ( x ( f( x f( x3 f( x( x So, f s a geeralzed covex fucto o I Corollary 3 Let f D ( a, b The f s a geeralzed covex fucto (or a geeralzed cocave fucto f ad oly f ( ( f ( x 0( or f ( x 0, for ay x ( ab,
4 Some equaltes Theorem 4 (Geeralzed Jese s equalty Assume that f s a geeralzed covex fucto o [ ab, ] The for ay x [ ab, ] ad [0,] (,,, wth, we have f x f( x Proof Whe, the equalty s obvously true Assume that for the equalty s also true The for ay x, x,, [, ] x ab ad 0,,, wth have If ( f x f( x x, x,, x, x [ a, b] ad 0 for,,, wth, we, the oe sets up,,,, It s easy to see Thus, f( xx x x xx x f ( x ( f( xx x f( x ( [ f( x f( x f( x] f( x ( f ( x f( x f( x f( x f( x So, the mathematcal ducto gves the proof of Theorem 4 x Corollary 4 Let f D [ a, b] ad [ ab, ] ad [0,] (,, wth ( f ( x 0 for ay x [ ab, ], we have The for ay f x f( x Usg the geeralzed Jese s equalty ad the covexty of fuctos, we ca also get some tegral equaltes I [3], Yag establshed the geeralzed Cauchy-Schwatz's equalty by the estmate p q a b a b, where a, b 0, pq, ad p q p q 6
Now, va the geeralzed Jese's equalty, we wll gve aother proof for the geeralzed Cauchy-Schwarz's equalty Corollary 4 (Geeralzed Cauchy-Schwarz's equalty Let a 0, b 0,,,, The we have a b a b Proof Tae Tae f ( x x It s easy to see that ( f ( x 0 for ay x ( ab, b b, x a The 0 (,,, wth b Thus, by Jese's equalty ( f x f( x, we have The above formula ca be reduced to whch mples that Thus we have b a b a b b b b b a a b ( b b a a b, a b a b Theorem 4 (Geeralzed Hermte-Hadmard s equalty Let geeralzed covex fucto o [ ab, ] wth a b The f x I a b be a ( ( x [, ] 7 ab ( ( f( a f( b f aib f( x ( b a
Proof Let x ab y The 8 ab a b ab f ( x( dx f( ab y( dy ab ab Furthermore, whe x, b, abx a, Ad by the covexty of, f we have a b f( abx f( x f Thus b a f( x( dx ab a b b ab b ab f ( x( dx f( x( dx a b[ f ( abx f( x]( dx a b f ( dx a b ( ba f (4 For aother part, we frst ote that f f s a geeralzed covex fucto, the, for t [0,], t yelds ad By addg these equaltes we have f ( ta ( t b t f ( a ( t f ( b, f (( ta tb ( t f( a t f( b f ( ta( t b f(( t atb t f( a ( t f( b ( t f( a t f( b f( a f( b The, tegratg the resultg equalty wth respect to t over [0,], we obta So, It s easy to see that ad [ f ( ta ( t b f (( t a tb]( dt ( 0 ( f( a f( b( dt ( 0 ( [ f ( ta ( t b f (( t a tb]( dt (, 0 aib f x ( ( ba f ( a f( b f a f b dt 0 ( ( ( ( ( (
( ( f ( a f( b aib f( x ( b a Combg the equaltes (4 ad (4, we have ab ( ( f( a f( b f aib f( x ( b a Note that, t wll be reduced to the class Hermte- Hadmard equalty f 5 Applcatos of geeralzed Jese s equalty Usg the geeralzed Jese's equalty, we ca get some equaltes (4 3 3 Example 5 Let a 0, b 0 ad a b The a b 3 Proof Let f ( x x, x (0, It s easy to see that f s a geeralzed covex fucto So, ab f( a f( b f That s 3 3 3 ( ab a b 8 Thus, we coclude that ab Example 5 Let x, y R The x y E ( E ( x E ( y, Where E ( x 0 x s the Mttag-Leffer fucto ( Proof Tae f ( x E ( x ( It s easy to see ( E ( x E ( x 0 So, the geeralzed Jese's equalty gves x y E ( E ( x E ( y Example 53 (Power Mea Iequalty Let a, a, a 0 ad 0 s t or st 0 Deote r r r r a a a Sr, rr, The Ss St Ad Ss St f ad oly f a a a Proof Case I: 0 s t Tae ( ts f( x x, x 0 The 9
0 ( f ( x x 0 t ( s ( ts t ( ( s By the geeralzed Jese s equalty, we have That s s s s s s s a a a f( a f( a f( a f s s s ( ts s ( t s s ( t s s ( t s ( ( ( a a a a a a From the above formula, t s easy to see So, we have S S Case II: st 0 s s s s s t t t t a a a a a a t Let b a ad apply the case for 0 t s, we ca get the cocluso Example 54 If abc,, 0 ad abc, the fd the mmum of Soluto Note that 0 abc,, Let formula we have 0 0 0 a b c a b c d x dx 0 f( x x, x(0, x ( x ( ( (, The, va the 8 9 ( (0 ( (0 f x x x 3 ( 0 (8 x x (9 x x By the geeralzed Jese's equalty, 0 0 abc f 3 3 [ f ( a f ( b f ( c ] 3 a b c 3 a b c 0 0 0
0 0 So, the mmum s, whe a b c 3 9 3 3 Example 55 If abcd,,, 0 ad c d ( a b, the show that 3 a, a c b d 3 3 3 b Proof Let x x, y ( ac, y ( bd By the geeralzed c d Cauchy-Schwartz equalty, we have 3 3 a b ( ac bd c d ( x x ( y y ( x y x y ( a b ( a b ( c d ac bd Cacelg ac bd o both sdes, we get the desred result 5 Cocluso I the paper, we troduce the defto of geeralzed covex fucto o fractal sets Based o the defto, we study the propertes of the geeralzed covex fuctos ad establsh two mportat equaltes: the geeralzed Jese s equalty ad geeralzed Hermte-Hadamard s equalty At last, we also gve some applcatos for these equaltes o fractal sets Acowledgmets The authors would le to express ther grattude to the revewers for ther very valuable commets Ad, ths wor s supported by the Natoal Natural Scece Foudato of Cha (No 604 Refereces [] J R Joatha ad P A Mattew, Jeses Iequalty Predcts Effects of Evrometal Varato, Treds Ecology & Evoluto, vol 4, o 9, pp36-366, 999 [] M Gralatt ad J T Lamaa, Jese s Iequalty, Parameter Ucertaty, ad Multperod Ivestmet, Revew of Asset Prcg Studes, vol, o, pp-34, 0 [3] PRBeesacadJPeˇ car c, O Jese s equalty for covex fuctos, Joural of Mathematcal Aalyss ad Applcato, vol 0, pp 536 55, 985 [4] D S Mtrovc ad I B Lacovc, Hermte ad Covexty, Aequatoes Mathematcae, vol 8, pp 9-3, 985 [5] K M Kolwaar ad A D Gagal, Local Fractoal Calculus: a Calculus for Fractal Space-tme, I: Fractals, Sprger, Lodo, 999 [9] A K Golmahaeh ad D Baleau, O a New Measure o Fractals, Joural of Iequaltesad Applcatos, vol 5, o, pp -9, 03
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