Research Article Jensen Functionals on Time Scales for Several Variables

Transcription

1 Hdaw Publshg Corporato Iteratoal Joural of Aalyss, Artcle ID , 14 pages Research Artcle Jese Fuctoals o Tme Scales for Several Varables Matloob Awar, 1 Raba Bb, 1 Mart Boher, 2 ad Josp PeIarT 3 1 School of Natural Sceces, Natoal Uversty of Sceces ad Techology, H-12, Islamabad 44000, Paksta 2 Departmet of Mathematcs ad Statstcs, Mssour Uversty of Scece ad Techology, Rolla, MO , USA 3 Faculty of Textle Techology, Uversty of Zagreb, Perottjeva 6, Zagreb, Croata Correspodece should be addressed to Raba Bb; emaorr@gmal.com Receved 16 November 2013; Accepted 21 February 2014; Publshed 10 Aprl 2014 Academc Edtor: Baruch Cahlo Copyrght 2014 Matloob Awar et al. Ths s a ope access artcle dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. We defe Jese fuctoals ad cocered geeralzed meas for several varables o tme scales. We derve propertes of Jese fuctoals ad apply them to geeralzed meas. I ths settg, we obta geeralzatos, refemets, ad coversos of may remarkable equaltes. 1. Itroducto Jese sequaltyswellkowaalyssadmayother areas of mathematcs. Most of the classcal equaltes ca be obtaed by usg the Jese equalty. For tme scale theory,jese sequaltyforoevarablesobtaedby Agarwal et al. 1, ad ow there are varous extesos ad geeralzatos of t gve by may researchers (see 2 8. I 3, t s show that the Jese equalty for oe varable holds for tme scale tegrals cludg the Cauchy delta, Cauchy abla, damod-α, Rema, Lebesgue, multple Rema, ad multple Lebesgue tegrals. Further, 4, we gve propertes ad applcatos of Jese fuctoals o tme scales for oe varable. I ths paper, we obta the Jese equalty for several varables ad deduce Jese fuctoals. We dscuss several propertes ad applcatos of Jese fuctoals. I the sequel,wegvealltheresultsforlebesguedeltategrals. For other tme scale tegrals, as metoed above, all those results ca be obtaed a smlar way. These results geeralze the results gve 4 for oe varable. Now, we gve a bref troducto of tme scale tegrals; for a detaled troducto we refer to 1, A tme scale T s a arbtrary closed subset of R, ad tme scale calculus provdes ufcato ad exteso of classcal results. For example, whe T = R, the tme scale tegral s a ordary tegral, ad whe T = Z, the tme scale tegral becomes a sum. I 10, Chapter 5, the Lebesgue tegral s troduced: let a, b T be a tme scale terval defed by a, b = {t T :a t<b}, (1 where a, b T wth a b.letμ be the Lebesgue - measure o a, b. Supposef : R s a μ - measurable fucto. The the Lebesgue -tegral of f o a, b s deoted by fdμ, f (t dμ (t, or f (t t. (2 All theorems of the geeral Lebesgue tegrato theory, cludg the Lebesgue domated covergece theorem, hold also for Lebesgue -tegrals o T. Now, we gve some propertes of Lebesgue -tegrals ad state Jese s equalty ad Hölder s equalty for Lebesgue -tegrals. Throughout ths paper, a, b deotes a tme scale terval otherwse s specfed.

2 2 Iteratoal Joural of Aalyss Theorem 1 (see 3, Theorem 3.2. If f ad g are -tegrable fuctos o a, b,the (αf + βg dμ =α fdμ +β gdμ f (t 0 t a, b mples fdμ 0. α, β R, Theorem 2 (see 3, Theorem4.2.Assume Φ C(I, R s covex, where I R s a terval. Suppose f: Is -tegrable. Moreover, let p: R be oegatve ad -tegrable such that >0.The (3 Φ( pfdμ p(φ fdμ. (4 Theorem 3 (see 3, Theorem6.2.For p =1,defeq = p/(p 1.Letw, f, g be oegatve fuctos such that wf p, wg q, wfg are -tegrable o a, b.ifp>1,the 1/p 1/q wfgdμ ( wf p dμ ( wg q dμ. (5 If 0 < p < 1 ad wg q dμ > 0,orfp < 0 ad wf p dμ >0,the(5 s reversed. Remark 4. Theorem 1 recalls that the Lebesgue -tegral s a sotoc lear fuctoal (see 13. So we ca also use the approach of sotoc lear fuctoals wheever results are kow for sotoc lear fuctoals. I the ext secto, we gve Jese equalty o tme scales for several varables ad defe Jese fuctoals. I Secto 3, we vestgate propertes of Jese fuctoals ad some of ts cosequeces regardg superaddtvty ad mootocty. I Secto 4, we apply these results to weghted geeral meas, defed o tme scales, ad gve may applcatos. Fally Secto 5, we gve applcatos to Hölder s equalty o tme scales. = 1,2,...,,are-tegrable o a, b such that f(t = (f 1 (t,...,f (t K for all t.moreover,let p : R be oegatve ad -tegrable such that >0.The Φ( pfdμ pφ (f dμ. (7 Proof. Suppose Φ s covex o K R. Therefore, for every pot x 0 K, there exsts a pot R (see 13, Theorem 1.31 such that Φ (x Φ(x 0, x x 0. (8 Let =( 1,...,.By(8, we get pφ (f dμ Φ( pfdμ p{φ(f Φ( pfdμ / } dμ = p, f ( pfdμ / dμ p =1 (f ( pf dμ / dμ = =0, adhecetheproofscompleted. Remark 6. By usg the fact that the tme scale tegral s a sotoc lear fuctoal, Theorem 5 ca also be obtaedby usg Theorem 1 ad 13, Theorem 2.6. Defto 7. Assume Φ C(K,R, wherek R s closed ad covex. Suppose f, = 1,2,...,,are-tegrable o a, b such that f(t = (f 1 (t,...,f (t K for all t a, b. Moreover, let p : a, b R be oegatve ad -tegrable such that >0. The oe defes the Jese fuctoal o tme scales for several varables by (9 2. Jese Iequalty ad Jese Fuctoals Let f(t = (f 1 (t,...,f (t be a -tuple of fuctos such that f 1,...,f are -tegrable o a, b. The fdμ deotes the -tuple: J (Φ, f,p= pφ (f dμ Φ( pfdμ (10 ( f 1 dμ,..., f dμ. (6 That s, -tegral acts o each compoet of f. Theorem 5 (Jese equalty. Assume Φ C(K, R s covex, where K R s closed ad covex. Suppose f, Remark 8. By Theorem 5,thefollowgstatemetsareobvous. If Φ s covex, the whle f Φ s cocave, the J (Φ, f,p 0 (11 J (Φ, f,p 0. (12

3 Iteratoal Joural of Aalyss 3 Example 9. Let = {1,2,...,}, f 1 ( = x 1,...,f ( = x,adp( = p, = 1,2,..., (10. The the Jese fuctoal (10becomes Proof. Let Φ be covex. Because the tme scales tegral s lear (see Theorem 1, t follows from Defto 7 that J (Φ, f,p+q J (Φ, X, p = =1 p Φ(x P Φ( =1 p x, (13 P where X = (x 1, x 2,...,x wth x = (x 1,x 2,...,x, p = (p 1,...,p,adP = =1 p >0.Somepropertesofthe Jese fuctoal J are vestgated 14, 15. Example 10. If a, b s a real terval, the Jese s fuctoal (10becomes p (t Φ(f 1 (t,f 2 (t,...,f (tdμ(t p (t dμ (t Φ( p (t f 1 (t dμ (t p (t dμ (t, p (t f 2 (t dμ (t,..., p (t dμ (t = (p + q Φ (f dμ (p + q dμ Φ( (p + q fdμ (p + q dμ = pφ (f dμ + qφ (f dμ ( + Φ( pfdμ + qfdμ + pφ (f dμ + qφ (f dμ Φ( pfdμ (17 p (t f (t dμ (t. p (t dμ (t (14 Φ( qfdμ = J (Φ, f,p+j (Φ, f,q. 3. Propertes of Jese Fuctoals I the followg theorem, we gve our ma result cocerg the propertes of the Jese fuctoal (10. Theorem 11. Assume Φ C(K, R, wherek R s closed ad covex. Suppose f, = 1,2,...,,are-tegrable o a, b such that f(t = (f 1 (t,...,f (t K for all t a, b. Moreover, let p, q : a, b R be oegatve ad - tegrable such that >0ad >0.IfΦ s covex, the J (Φ, f, s superaddtve; that s, J (Φ, f,p+q J (Φ, f,p + J (Φ, f,q, (15 If p q,wehavep q 0.Now,becauseJese sfuctoal s superaddtve ad oegatve, we have J (Φ, f,p=j (Φ, f,p q+q J (Φ, f,p q+j (Φ, f,q J (Φ, f,q. (18 O the other had, f Φ s cocave, the the reversed equaltes of (15 ad(16 ca be obtaed a smlar way. Corollary 12. Let Φ, f, p, q satsfy the hypotheses of Theorem 11. Further, supposethereexstoegatvecostats m ad M such that ad J (Φ, f, s creasg; that s, p qwth > mples Mq (t p(t mq(t t a, b, M > >m. (19 J (Φ, f,p J (Φ, f,q. (16 Moreover, f Φ s cocave, the J (Φ, f, s subaddtve ad decreasg; that s, (15 ad (16 hold reverse order. If Φ s covex, the MJ (Φ, f,q J (Φ, f,p mj (Φ, f,q, (20 whle f Φ s cocave, the the equaltes (20 hold reverse order.

4 4 Iteratoal Joural of Aalyss Proof. By usg (10, we have J (Φ, f,mq=mj (Φ, f,q, J (Φ, f,mq=mj (Φ, f,q. (21 Now the result follows from the secod property of Theorem 11. Corollary 13. Let Φ, f, p satsfy the hypotheses of Theorem 11. Further, assume that p attas ts mmum value ad ts maxmum value o ts doma. If Φ s covex, the where max t p (t J (Φ, f J (Φ, f,p J (Φ, f = m t p (t J (Φ, f, Φ (f dμ ( Φ( fdμ. (22 (23 Moreover, f Φ s cocave, the the equaltes (22 hold reverse order. Proof. Let p atta ts mmum ad maxmum values o ts doma a, b.the Let max p (t p(t m p (t. (24 t t p (t = max p (t, t By usg (10, we have p(t = m p (t. (25 t J (Φ, f, p = max t p (t J (Φ, f, J (Φ, f,p=m t p (t J (Φ, f. (26 Now the result follows from the secod property of Theorem 11. Example 14. Let the fuctoal J (Φ, X, p be defed as Example 9. Letq = (q 1,...,q wth q 0 ad =1 q = Q >0.IfΦ s covex, the Theorem 11 mples J (Φ, X, s superaddtve; that s, J (Φ, X, p + q J (Φ, X, p + J (Φ, X, q, (27 ad J (Φ, X, s creasg; that s, f p q such that P >Q, the J (Φ, X, p J (Φ, X, q. (28 Moreover, f Φ s cocave, the the equaltes (27 ad (28 hold reverse order. If p attas ts mmum ad maxmum values o ts doma, the Corollary 13 yelds max {p } J (Φ, X J (Φ, X, p m {p } J (Φ, X, 1 1 (29 where J (Φ, X = =1 Φ(x Φ( =1 x, (30 f Φ s covex. Further, the equaltes (29 hold reverse order f Φ s cocave. 4. Applcatos to Weghted Geeralzed Meas I the sequel, I R s a terval ad K R s closed ad covex. Defto 15. Assume χ C(I, R s strctly mootoe ad φ : K I s a fucto of varables. Suppose f, = 1, 2,...,,are -tegrable o a, b such that f(t = (f 1 (t,...,f (t K for all t. Moreover, let p : a, b R be oegatve ad -tegrable such that pχ(φ(f s -tegrable ad >0. The oe defes the weghted geeralzed mea o tme scales by M (χ, φ (f,p=χ 1 ( pχ (φ (fdμ. (31 Theorem 16. Assume χ, ψ C(I, R, = 1,2,...,,are strctly mootoe ad φ : K I R s a fucto of varables. Suppose f : a, b I, = 1,2,...,,aretegrable such that f(t = (f 1 (t,...,f (t K for all t a, b. Moreover,let p, q : a, b R be oegatve ad - tegrable such that pχ(φ(f, qχ(φ(f, pψ (f, qψ (f, = 1,2,...,,are-tegrable ad >0, > 0.IfH defed by H(s 1,...,s =χ φ(ψ 1 1 (s 1,...,ψ 1 (s (32 s covex, the the fuctoal χ (M (χ, φ (f,p χ φ(m (ψ 1,f 1,p,..., M (ψ,f,p (33

5 Iteratoal Joural of Aalyss 5 s superaddtve, that s, (p + q dμ χ (M (χ, φ (f,p+q χ φ(m (ψ 1,f 1,p+q,..., M (ψ,f,p+q = χ (M (χ, φ (f,p χ φ(m (ψ 1,f 1,p,..., M (ψ,f, p. Now, all clams follow mmedately from Theorem 11. (36 χ (M (χ, φ (f,p χ φ(m (ψ 1,f 1,p,..., M (ψ,f, p + χ (M (χ, φ (f,q χ φ(m (ψ 1,f 1,q,..., M (ψ,f, q, (34 ad creasg; that s, p qwth > mples χ (M (χ, φ (f,p χ φ(m (ψ 1,f 1,p,..., M (ψ,f, p χ (M (χ, φ (f,q χ φ(m (ψ 1,f 1,q,..., M (ψ,f,q. (35 Moreover, f H s cotuous ad cocave, the (33 s subaddtve ad decreasg; that s, (34 ad (35 hold reverse order. Proof. The fuctoal defed (33s obtaed by replacg Φ wth H ad f wth ψ (f, = 1,2,...,,theJese fuctoal (10 adlettgψ(f =(ψ 1 (f 1,...,ψ (f ; that s, J (H, Ψ (f,p = pχ φ (f 1,...,f dμ H( pψ 1 (f 1 dμ,..., pψ (f dμ = χ(m (χ, φ (f,p χ φ(m (ψ 1,f 1,p,...,M (ψ,f,p Corollary 17. Let H, φ, f, p, χ, f,adψ, =1,...,,satsfy the hypothess of Theorem 16. Further, assume that p attas ts mmum value ad ts maxmum value o ts doma. If H s covex, the max p (t ( t where χ(m (χ, φ (f χ φ (M (ψ 1,f 1,..., M (ψ,f χ (M (χ, φ (f,p χ φ (M (ψ 1,f 1,p,...,M (ψ,f,p m p (t ( t χ(m (χ, φ (f χ φ (M (ψ 1,f 1,..., M (ψ,f, (37 M (χ, φ (f =χ 1 ( χ(φ(fdμ. (38 Moreover, f H s cocave, the the equaltes (37 hold reverse order. Proof. The proof s omtted as t s smlar to the proof of Corollary 13. Remark 18. If we take the dscrete form of the weghted geeralzedmea(31 wth =1,theweobta the quasarthmetc mea. Namely, let ψ : I R R be cotuous ad strctly mootoe, a =(a 1,...,a wth a k I, k = 1,...,,adw =(w 1,...,w wth w k 0ad k=1 w k =1. The the quasarthmetc mea of a wth weght w s defed by M =ψ 1 ( k=1 w k ψ(a k. (39 Now the followg examples coect the quasarthmetc mea (39 ad the propertes of Jese fuctoals. Example 19 (see 16, Corollary 3. Let w ad ψ be defed as Remark 18 ad let ψ be strctly creasg ad strctly

6 6 Iteratoal Joural of Aalyss covex wth cotuous dervatves of secod order such that ψ /ψ s cocave. Further, let X, p, x, =1,...,, be defed as Example 9,adq =(q 1,...,q wth q 0, =1,...,, ad =1 q =Q >0.The,Φ M (x =ψ 1 ( k=1 w kψ(x k s a covex fucto (see 17,Theorem1,page197.Heceby Theorem 11,the fuctoal Also, by Corollary 12,wehave max {p } J 1 (Φ M, X J (Φ M, X, p m {p } J 1 (Φ M, X, (49 J (Φ M, X, p = =1 s superaddtve, that s, p Φ M (x P Φ M ( =1 p x (40 P J (Φ M, X, p + q J (Φ M, X, p+j (Φ M, X, q, (41 ad creasg; that s, f p q such that P >Q,the J (Φ M, X, p J (Φ M, X, q. (42 Also, by Corollary 12,wehave max {p } J (Φ M, X J (Φ M, X, p 1 m {p } J (Φ M, X, 1 where J (Φ M, X = =1 (43 Φ M (x Φ M ( =1 x. (44 Example 20 (see 16, Corollary 4. Cosder (39, but wth dfferet codtos o ψ ad w.namely,f ( w 1for =1,...,; ( ψ:r + R + ; ( lm x 0 ψ(x = or lm x ψ(x =, the we defe M =ψ 1 ( k=1 w k ψ (a k. (45 Let X, p, x, = 1,...,, be defed as Example 9 ad q = (q 1,...,q wth q 0 ad =1 q = Q > 0.Let ψ be strctly creasg ad strctly covex wth cotuous dervatves of secod order such that ψ/ψ s covex. The Φ M (x =ψ 1 ( k=1 w kψ(x k s a covex fucto (see 17, Theorem 2, page 197. Hece, by Theorem 11,thefuctoal J (Φ M, X, p = =1 s superaddtve, that s, p Φ M (x k P Φ M ( =1 p x k P (46 J (Φ M, X, p + q J (Φ M, X, p+j (Φ M, X, q, (47 ad creasg; that s, f p q,the J (Φ M, X, p J (Φ M, X, q. (48 where J (Φ M, X = =1 Φ M (x k Φ M ( =1 x k. (50 Example 21 (see 16, Corollary 5. For a real-valued fucto f defed o terval a, b,a th order dvded dfferece of f at dstct pots x 0,...,x a, b s defed recursvely by x f=f(x, =0,...,, x 0,...,x f= x 1,...,x f x 0,...,x 1 f x x 0. (51 Further, f s -covex o a, b, 0, f ad oly f, for all choces of +1dstct pots a, b, x 0,...,x f 0. (52 Let X, p, x, = 1,...,, be defed as Example 9 ad q = (q 1,...,q,wthq 0ad =1 q =Q >0.Letf:I R be ( + 1-covex, where I R s a closed ad bouded terval. The by Theorem 11,forΦ G (x =x 1,...,x f,the fuctoal J (Φ G, X, p = =1 s superaddtve, that s, p Φ G (x P Φ G ( =1 p x (53 P J (Φ G, X, p + q J (Φ G, X, p + J (Φ G, X, q, (54 ad creasg; that s, f p q such that P >Q,the Also, by Corollary 12,wehave where J (Φ G, X, p J (Φ G, X, q. (55 max 1 {p } J (Φ G, X J (Φ G, X, p m 1 {p } J (Φ G, X, J (Φ G, X = =1 (56 Φ G (x Φ G ( =1 x. (57 Corollary 22. Assume χ, ψ 1,adψ 2 C 2 (I, R are strctly mootoe. Suppose f 1,f 2 : a, b I are -tegrable such that f 1 (t + f 2 (t I for all t a, b ad p, q : a, b R are oegatve ad -tegrable such that pχ(f 1 +f 2,

7 Iteratoal Joural of Aalyss 7 qχ(f 1 +f 2, pψ (f,adqψ (f, = 1,2,are-tegrable ad >0, >0.Further,let E= ψ 1 ψ1, F = ψ 2 ψ2 If ψ 1, ψ 2,adχ are postve ad ψ the the fuctoal, G = χ. (58 χ 1, ψ 2,adχ are egatve, χ (M (χ, f 1 +f 2,p χ (M (ψ 1,f 1,p+M (ψ 2,f 2, p s superaddtve, that s, (p + q dμ χ (M (χ, f 1 +f 2,p+q χ(m (ψ 1,f 1,p+q +M (ψ 2,f 2, p + q χ (M (χ, f 1 +f 2,p χ (M (ψ 1,f 1,p+M (ψ 2,f 2,p (59 + χ (M (χ, f 1 +f 2,q χ (M (ψ 1,f 1,q+M (ψ 2,f 2, q, (60 ad creasg; that s, f p q such that >,the χ (M (ψ 1,f 1,p+M (ψ 2,f 2,p m t p (t ( χ (M (χ, f 1 +f 2 χ (M (ψ 1,f 1 +M (ψ 2,f 2. (62 Moreover, f ψ 1, ψ 2, χ, ψ 1, ψ 2,adχ are all postve, the the equaltes (60, (61,ad(62 are reversed f ad oly f G(x+y E(x+F(y. Proof. Let =2 Theorem 16. Bysettgφ(x,y= x+y, we have H(s 1,s 2 =χ(ψ 1 1 (s 1 +ψ 1 2 (s 2. (63 If ψ 1, ψ 2,adχ are postve ad ψ 1, ψ 2,adχ are egatve, the H s covex f ad oly f G(x + y E(x + F(y (see 18. If ψ 1, ψ 2, χ, ψ 1, ψ 2,adχ are all postve, the H s cocave f ad oly f G(x+y E(x+F(y(see 18. Now, all clams follow mmedately fromtheorem 16. Corollary 23. Assume χ, ψ 1,adψ 2 C 2 (I, R are strctly mootoe. Suppose f 1,f 2 : a, b I are -tegrable such that f 1 (tf 2 (t I for all t ad p, q : a, b R are oegatve ad -tegrable such that pχ(f 1 f 2, qχ(f 1 f 2, pψ (f,adqψ (f, =1,2,are-tegrable ad >0, >0.Further,let A (t = ψ 1 (t ψ1 (t +tψ 1 (t, B(t = ψ2 (t ψ2 (t +tψ 2 (t, C (t = χ (t χ (t +tχ (t. (64 If ψ 1, ψ 2,adχ arepostveada, B,adC are egatve, the the fuctoal χ (M (χ, f 1 +f 2,p χ (M (ψ 1,f 1,p+M (ψ 2,f 2,p χ (M (χ, f 1 +f 2,q χ (M (ψ 1,f 1,q+M (ψ 2,f 2, q, (61 χ (M (χ, f 1 f 2,p χ (M (ψ 1,f 1,p M (ψ 2,f 2,p s superaddtve, that s, (p + q dμ χ (M (χ, f 1 f 2,p+q χ(m (ψ 1,f 1,p+q M (ψ 2,f 2,p+q (65 f ad oly f G(x + y E(x + F(y.Ifp attas ts mmum ad maxmum values o ts doma a, b,the(61 yelds max t p (t ( χ (M (χ, f 1 +f 2 χ (M (ψ 1,f 1 +M (ψ 2,f 2 χ (M (χ, f 1 +f 2,p χ (M (χ, f 1 f 2,p χ (M (ψ 1,f 1,p M (ψ 2,f 2,p + χ (M (χ, f 1 f 2,q χ (M (ψ 1,f 1,q M (ψ 2,f 2,q, (66

8 8 Iteratoal Joural of Aalyss ad creasg; that s, f p q such that >,the f 1 f 2, q f 1 f 2, pfω 1, qfω 1, pf 2,adqf 2 are -tegrable ad >0, >0.Thethefuctoal χ (M (χ, f 1 f 2,p χ (M (ψ 1,f 1,p M (ψ 2,f 2,p χ (M (χ, f 1 f 2,q χ (M (ψ 1,f 1,q M (ψ 2,f 2, q, (67 f ad oly f C(x y A(x + B(y.Ifp attas ts mmum ad maxmum values o ts doma a, b,the(67 yelds max t p (t ( χ (M (χ, f 1 f 2 χ (M (ψ 1,f 1 M (ψ 2,f 2 χ (M (χ, f 1 f 2,p χ (M (ψ 1,f 1,p M (ψ 2,f 2,p m t p (t ( χ (M (χ, f 1 f 2 χ (M (ψ 1,f 1 M (ψ 2,f 2. (68 If ψ 1,ψ 2,χ,A,B,adC are all postve, the the equaltes (66, (67, ad(68 are reversed f ad oly f C(x y A(x + B(y. Proof. Let =2Theorem 16.Bysettgφ(x, y = x y,we have H(s 1,s 2 =χ(ψ 1 1 (s 1 ψ 1 2 (s 2. (69 If ψ 1, ψ 2,adχ arepostveada, B, adc are egatve, the H s covex f ad oly f C(x y A(x + B(y.Ifψ 1, ψ 2, χ, A, B, adc are all postve, the H s cocave f ad oly f C(x y A(x+B(y (see 18. Now, all clams follow mmedately from Theorem 16. Corollary 24. Let, ω, R be such that (a <0<ω,,orω, <0<; (b <ω, <0,or <0<ω<,orω<0< <,for 1/ + 1/; (c <ω<0<,or< <0<ω,for1/ + 1/. Suppose f 1,f 2 : a, b R are -tegrable ad p, q : a, b R are oegatve ad -tegrable such that p p f 1 f 2 dμ ( pfω 1 dμ s superaddtve, that s, (p + q f 1 f 2 dμ ( pf 2 dμ 1/ (70 (p + q dμ ( (p + q fω 1 dμ (p + q dμ p f 1 f 2 dμ ( pfω 1 dμ + q f 1 f 2 dμ ( (p + q f 2 dμ 1/ (p + q dμ ( pf 2 dμ 1/ ( qfω 1 dμ ( qf 2 dμ 1/, (71 ad creasg; that s, f p q such that >,the p f 1 f 2 dμ ( pfω 1 dμ ( pf 2 dμ 1/

9 Iteratoal Joural of Aalyss 9 q f 1 f 2 dμ ( qfω 1 dμ ( qf 2 dμ 1/. (72 If p attas ts mmum ad maxmum values o ts doma, the max p (t f 1 t f 2 dμ (( fω 1 dμ p f 1 f 2 dμ ( f 2 dμ 1/ ( pfω 1 dμ ( pf 2 dμ 1/ m p (t f 1 t f 2 dμ (( fω 1 dμ ( f 2 dμ 1/. (73 Moreover, the equaltes (71, (72, ad(73 are reversed provded that (a ω, >>0,for1/ + 1/; (b ω, <<0,for1/ + 1/. Proof. Let =2 Theorem 16. By settg φ(x, y = x y, χ(t = t, ψ 1 (t = t ω,adψ 2 (t = t,wehave H(s 1,s 2 =χ(ψ 1 1 (s 1 ψ 1 2 (s 2 = (s 1 s 1/ 2. (74 Now, H s covex f ad oly f d 2 H 0, whch mples ω ( ω 1 0, ( 1 0, 3 ω ( 1 1 ω 1 0, (75 ad these are satsfed f, ω,ad satsfy codtos (a, (b, ad (c. H s cocave f ad oly f d 2 H 0, ad ths mples ω ( ω 1 0, ( 1 0, 3 ω ( 1 1 ω 1 0. (76 These are satsfed f, ω, ad satsfy codtos (a ad (b. Now, all clams follow mmedately from Theorem 16. Corollary 25. Let, ω, R be such that, ω, > 0,, ω, =1ad (a <1<ω,,orω, <1<; (b <ω, <1,or <1<ω<,orω<1< <,for 1/ log 1/log ω+1/log ; (c <ω<1<, or< <1<ω,for1/ log 1/ log ω+1/log. Suppose f 1,f 2 : a, b R are -tegrable ad p, q : a, b R are oegatve ad -tegrable such that p f 1+f 2, q f 1+f 2, pω f 1, qω f 1, p f 2,adq f 2 are -tegrable ad >0, >0.Thethefuctoal p f 1+f 2 dμ log ω ( p ωf 1 dμ / +log ( p f 2 dμ / s superaddtve, that s, (p + q f 1+f 2 dμ (p + q dμ ( log ω ( (p+q ωf 1 dμ / (p+ p f 1+f 2 dμ (77 log ( (p+q f 2 dμ / (p+ log ω ( p ωf 1 dμ / +log ( p f 2 dμ / + q f 1+f 2 dμ log ω ( q ωf 1 dμ / +log ( q f 2 dμ /, (78

10 10 Iteratoal Joural of Aalyss ad creasg; that s, f p q such that >,the f 2, q(f 1 +f 2, pf ω 1, qfω 1, pf 2,adqf 2 are -tegrable ad >0, >0.Thethefuctoal p f 1+f 2 dμ log ω ( p ωf 1 dμ / +log ( p f 2 dμ / q f 1+f 2 dμ log ω ( q ωf 1 dμ / +log ( q f 2 dμ /. (79 If p attas ts mmum ad maxmum values o ts doma, the max p (t f 1+f 2 dμ ( t log ω ( ωf 1 dμ /(+log ( f 2 dμ /( p f 1+f 2 dμ log ω ( p ωf 1 dμ / +log ( p f 2 dμ / m p (t f 1+f 2 dμ ( t log ω ( ωf 1 dμ /(+log ( f 2 dμ /(. (80 Moreover, the equaltes (78, (79, ad(80 are reversed provded that (a ω, >>1,for1/ log 1/log ω+1/log ; (b ω, <<0,for1/ log 1/log ω+1/log. Proof. Let =2 Theorem 16. By settg φ(x, y = x + y, χ(t = t, ψ 1 (t = ω t,adψ 2 (t = t,wehave 1/ log ω 1/ log H(s 1,s 2 =(s1 s2 log. (81 Now, the proof s smlar to the proof of Corollary 24. Corollary 26. Let, ω, R be such that (a 0<ω, <1,forallf 1,f 2 >0; (b 0< ω<1,forf 2 (((ω (1 /(( (1 ωf 1 0; (c 0<ω <1,for((( ω(1 /(( (1 ωf 1 f 2 0. Suppose f 1,f 2 : a, b R are -tegrable ad p, q : a, b R are oegatve ad -tegrable such that p(f 1 + p (f 1 +f 2 dμ ( pfω 1 dμ s superaddtve, that s, (p + q (f 1 +f 2 dμ +( pf 2 dμ 1/ (82 (p + q dμ ( (p + q fω 1 dμ (p + q dμ p (f 1 +f 2 dμ ( pfω 1 dμ +( (p + q f 2 dμ 1/ (p + q dμ +( pf 2 dμ 1/ + q (f 1 +f 2 dμ ( qfω 1 dμ +( qf 2 dμ 1/, (83 ad creasg; that s, f p q such that >,the p (f 1 +f 2 dμ ( pfω 1 dμ q (f 1 +f 2 dμ ( qfω 1 dμ +( pf 2 dμ 1/ +( qf 2 dμ 1/. (84

11 Iteratoal Joural of Aalyss 11 If p attas ts mmum ad maxmum values o ts doma, the ad q cos(f, =1,2,are-tegrable ad >0, >0.Thethefuctoal max p (t (f 1 +f 2 dμ t (( pfω 1 dμ +( pf 2 dμ 1/ p (f 1 +f 2 dμ ( pfω 1 dμ m p (t (f 1 +f 2 dμ t +( pf 2 dμ 1/ (( pfω 1 dμ +( pf 2 dμ 1/. (85 Moreover, the equaltes (83, (84, ad(85 are reversed provded that (a 1< ω,,forallf 1,f 2 >0; (b 1< ω,for0 f 2 (((ω ( 1/(( (ω 1f 1 ; (c 1<ω,forf 2 ((( ω( 1/(( (ω 1f 1 0. Proof. Let =2 Theorem 16. By settg φ(x, y = x + y, χ(t = t, ψ 1 (t = t ω,adψ 2 (t = t,wehave H(s 1,s 2 =(s 1 +s 1/ 2. (86 Now, the proof s smlar to the proof of Corollary 22, wth some extra cosderatos of the deftos of E, F, ad G. Corollary 27. Suppose f 1,f 2 : a, b 0, π/4 are - tegrable. Moreover, let p, q : a, b R be oegatve ad -tegrable such that p cos(f 1 +f 2, q cos(f 1 +f 2, p cos(f, cos arccos ( p cos (f 1dμ + arccos ( p cos (f 2dμ p cos (f 1 +f 2 dμ (87 s subaddtve, that s, (p + q dμ cos arccos ( (p + q cos (f 1dμ (p + q dμ + arccos( (p + q cos (f 2dμ (p + q dμ (p + q cos (f 1 +f 2 dμ cos arccos ( p cos (f 1dμ p cos (f 1 +f 2 dμ + + arccos( p cos (f 2dμ cos arccos ( q cos (f 1dμ q cos (f 1 +f 2 dμ, + arccos( q cos (f 2dμ (88 ad decreasg; that s, f p q such that >,the cos arccos ( p cos (f 1dμ

12 12 Iteratoal Joural of Aalyss + arccos ( p cos (f 2dμ p cos (f 1 +f 2 dμ cos arccos ( q cos (f 1dμ q cos (f 1 +f 2 dμ. + arccos ( q cos (f 2dμ (89 If p attas ts mmum ad maxmum values o ts doma, the max p (t ( cos arccos ( cos (f 1dμ t cos (f 1 +f 2 dμ + arccos ( cos (f 2dμ cos arccos ( p cos (f 1dμ p cos (f 1 +f 2 dμ + arccos ( p cos (f 2dμ m p (t ( cos arccos ( cos (f 1dμ t cos (f 1 +f 2 dμ. + arccos( cos (f 2dμ (90 Proof. Let =2 Theorem 16. By settg φ(x, y = x + y ad χ(t = ψ 1 (t = ψ 2 (t = cos(t,wehave H (s 1,s 2 = cos (arccos ( s 1 + arccos ( s 2. (91 Now, the proof s smlar to the proof of Corollary Applcatos to Hölder s Iequalty Suppose f, = 1,2,...,,areoegatve-tegrable fuctos o a, b such that =1 fα s -tegrable, where α 0, = 1,...,,aresuchthat =1 α =1.The,byusg Theorem 3 (Hölder s equalty o tme scales, we have f α =1 dμ =1 ( If =1 α =A >0,the(92mples or ( f α /A =1 f α /A =1 dμ dμ A =1 ( =1 ( f dμ α. (92 f dμ α /A (93 f dμ α. (94 I ths secto, we dscuss propertes of the fuctoal, deduced from the Hölder equalty (93,defed the followg way. Defto 28. Suppose f = (f 1,...,f s such that f, = 1,...,,are oegatve -tegrable fuctos o a, b.let α =(α 1,...,α be such that α 0ad =1 α =A >0. The oe defes the fuctoal H by H (f, α = =1 ( f dμ α ( =1 fα /A dμ A. (95 Theorem 29. Let α =(α 1,...,α ad β =(β 1,...,β be real -tuples wth α 0, β 0 ad =1 α = A > 0, =1 β =B >0.Supposef, = 1,...,,areoegatve -tegrable o a, b such that =1 fα /A ad =1 fβ /B are -tegrable. The H (f, α + β H (f, α H (f, β, (96 ad H (f,,μ s creasg; that s, f α β such that A > B,the Proof. By Defto 28,wehave H (f, α + β = H (f, α H (f, β. (97 =1 ( f dμ α +β ( =1 f(α +β /(A +B dμ A +B, (98

13 Iteratoal Joural of Aalyss 13 where ( f (α +β /(A +B =1 = ( dμ ( A +B A /(A +B f α /A =1 ( f α /A =1 B /(A +B f β /B =1 dμ A ( dμ A +B f β /B =1 Now, by combg (98ad(99, we have H (f, α + β dμ B. =1 ( f dμ α =1 ( f dμ β ( =1 fα /A dμ A ( =1 fβ /B dμ B = H (f, α H (f, β. If α β,theα β 0, ad therefore H (f, α = H (f,(α β+β Ths completes the proof. H (f, α β H (f, β H (f, β. (99 (100 (101 Corollary 30. Let f ad α satsfy the hypothess of Theorem 29.The =1 f dμ ( =1 f1/ dμ max 1 {α } H (f, α =1 f dμ ( =1 f1/ dμ Proof. Let α max =(max 1 {α },...,max 1 {α }, α m =(m 1 {α },...,m 1 {α }. m 1 {α }. (102 (103 By Defto 28,wehave H (f, α max = =1 f dμ ( =1 f1/ dμ H (f, α m = =1 f dμ ( =1 f1/ dμ max 1 {α } m 1 {α },. (104 Sce α max α α m, the result follows from the secod property of Theorem 29. Corollary 31. Let f, α, ad β satsfy the hypothess of Theorem 29 wth A =B =1. If there exst costats M> 1>msuch that Mβ α mβ,the H (f,mβ H (f, α H (f,mβ. (105 Proof. By Defto 28,wehave H (f,mβ =MH (f, β, H (f,mβ =mh (f, β. (106 Now the result follows from the secod property of Theorem 29. Remark 32. Some results for sotoc lear fuctoals relatedtotheresultsgvethspapercabefoud16. Coflct of Iterests The authors declare that there s o coflct of terests regardg the publcato of ths paper. Refereces 1 R. Agarwal, M. Boher, ad A. Peterso, Iequaltes o tme scales: a survey, Mathematcal Iequaltes ad Applcatos,vol. 4, o. 4, pp , M.R.SdAmm,R.A.C.Ferrera,adD.F.M.Torres, Damod-α Jese s equalty o tme scales, Joural of Iequaltes ad Applcatos, vol.2008,artcleid576876,13 pages, M.Awar,R.Bb,M.Boher,ad J.Pečarć, Itegral equaltes o tme scales va the theory of sotoc lear fuctoals, Abstract ad Appled Aalyss, vol.2011,artcleid483595,16 pages, M. Awar, R. Bb, M. Boher, ad J. Pecarc, Jese s fuctoals o tme scales, Joural of Fucto Spaces ad Applcatos, vol.2012,artcleid384045,17pages, J. Barć, M. Matć, ad J. Pečarć, O the bouds for the ormalzed jese fuctoal ad jese-steffese equalty, Mathematcal Iequaltes ad Applcatos, vol.12,o.2,pp , C. Du, Hermte-Hadamard equalty o tme scales, Joural of Iequaltes ad Applcatos,vol.2008,ArtcleID287947, 24 pages, U. M. Özka, M. Z. Sarkaya, ad H. Yldrm, Extesos of certa tegral equaltes o tme scales, Appled Mathematcs Letters,vol.21,o.10,pp ,2008.

14 14 Iteratoal Joural of Aalyss 8 F. H. Wog, C. C. Yeh, ad W. C. La, A exteso of Jese s equalty o tme scales, Advaces Dyamcal Systems ad Applcatos,vol.1,o.1,pp , M. Boher ad A. Peterso, Dyamc Equatos o Tme Scales: A Itroducto wth Applcatos, Brkhäuser, Bosto, Mass, USA, M. Boher ad A. Peterso, Advaces Dyamc Equatos o Tme Scales: A Itroducto Wth Applcatos, Brkhäuser, Bosto, Mass, USA, M. Boher ad G. S. Guseov, Multple tegrato o tme scales, Dyamc Systems ad Applcatos, vol.14,o.3-4,pp , M. Boher ad G. S. Guseov, Multple Lebesgue tegrato o tme scales, Advaces Dfferece Equatos, vol. 2006, ArtcleID26391,12pages, J.E.Pečarć, F. Proscha, ad Y. L. Tog, Covex Fuctos, PartalOrdergs,adStatstcalApplcatos,vol.187ofMathematcs Scece ad Egeerg, Academc Press, Bosto, Mass, USA, S. S. Dragomr, J. Peĉarć, ad L. E. Persso, Propertes of some fuctoals related to Jese s equalty, Acta Mathematca Hugarca,vol.70,o.1-2,pp , S. S. Dragomr, Bouds for the ormalsed jese fuctoal, Bullet of the Australa Mathematcal Socety,vol.74,o.3,pp , M. Krć, N. Lovrčevć, ad J. Pečarć, O the propertes of Mcshae s fuctoal ad ther applcatos, Perodca Mathematca Hugarca,vol.66,o.2,pp , D. S. Mtrovć, J. E. Pečarć, ada. M. Fk, Classcal ad New Iequaltes Aalyss, Kluwer Academc, Lodo, UK, E. Beck, Über Uglechuge vo der Form f(m φ (x; α,m φ (y; α M (χ, f, (x, y ; α, Publkacje Elektrotehčkog fakulteta. Serja: Matematk, o , 14 pages, 1970.

15 Advaces Operatos Research Hdaw Publshg Corporato Advaces Decso Sceces Hdaw Publshg Corporato Joural of Appled Mathematcs Algebra Hdaw Publshg Corporato Hdaw Publshg Corporato Joural of Probablty ad Statstcs The Scetfc World Joural Hdaw Publshg Corporato Hdaw Publshg Corporato Iteratoal Joural of Dfferetal Equatos Hdaw Publshg Corporato Submt your mauscrpts at Iteratoal Joural of Advaces Combatorcs Hdaw Publshg Corporato Mathematcal Physcs Hdaw Publshg Corporato Joural of Complex Aalyss Hdaw Publshg Corporato Iteratoal Joural of Mathematcs ad Mathematcal Sceces Mathematcal Problems Egeerg Joural of Mathematcs Hdaw Publshg Corporato Hdaw Publshg Corporato Hdaw Publshg Corporato Dscrete Mathematcs Joural of Hdaw Publshg Corporato Dscrete Dyamcs Nature ad Socety Joural of Fucto Spaces Hdaw Publshg Corporato Abstract ad Appled Aalyss Hdaw Publshg Corporato Hdaw Publshg Corporato Iteratoal Joural of Joural of Stochastc Aalyss Optmzato Hdaw Publshg Corporato Hdaw Publshg Corporato