ON VAN DE LUNE ALZER S INEQUALITY

Transcription

1 Joual o Mathematical Iequalities Volume, Numbe ), ON VAN DE LUNE ALZER S INEQUALITY S. ABRAMOVICH, J. BARIĆ, M. MATIĆ AND J. PEČARIĆ commuicated by N. Elezović) Abstact. I this pape it is show that the iequality kow i the liteatue as Alze s iequality 993), has aleady bee kow sice 975. ad is due to Ja va de Lue. A eview o dieet methods i povig Va de Lue - Alze s iequality ad geealizatios i a seveal diectios, is give. It is show how some esults ad poos ca be coected, eied ad exteded. New esults, ispied by the geealizatio o Va de Lue - Alze s iequality o iceasig covex sequeces peseted by N. Elezović ad J. Pečaić, ae obtaied.. Itoductio I 964. H. Mic ad L. Sathe i [24] poved that, o N the iequality + <!)..) + )!) + holds. I 988. J. S. Matis, i [23], gave aothe lowe boud o the atio!) / + )!) + om.): Let be a positive eal umbe ad let be a atual umbe. The + ) i=!)..2) + )!) + + i= So, H. Alze came to the idea to compae the let-had sides o.) ad.2) ad, i 993 i [2], he poved the ext theoem. THEOREM. I is a positive eal umbe ad i is a positive itege, the + + ) i= + i=!) + )!) + Mathematics subject classiicatio 2000): 26A5, 26D5, 26D20. Key wods ad phases: sequeces, covex uctios, iequalities...3) c D l,zageb Pape JMI

2 564 S. ABRAMOVICH, J.BARIĆ, M.MATIĆ AND J. PEČARIĆ Sice the, the let-had side o the iequality.3) is called Alze s iequality. Alze s poo uses iteestig techiques, but its complexity have ivoked the iteest o seveal mathematicias. The ist easy poo o Alze s iequality is due to J. Sádo who used i his poo Cauchy mea value theoem ad mathematical iductio, see [33]. Also,J.Sádo used i [36] the method o Lagage mea value theoem ad mathematical iductio. The secod elemetay poo is give by J. S. Ume i [38] usig dieetiatio ad iductio. Ad, ially, C.-P. Che ad F. Qi, i [6],peseted two othe simple poos o Alze s iequality usig Lagage s mea value theoem, mootoicity ad covexity o uctios, ad mathematical iductio. Howeve, i 975, i [], J. va de Lue stated Poblem 399., which was solved by seveal mathematicias, ad which easily implicates Alze s iequality. The pupose o this pape is to show that what is called sice 993 "Alze s iequality" is the esult o Ja va de Lue s wok,kow aleady sice 975. Accodig to that, i this pape the let-had side o the iequality.3) will be called Va de Lue - Alze s iequality. We also give some geealizatios o Alze s iequality as well as coectios ad eiemets o alteative poos o oe o the iequalities i a aticle by H. Alze [2]). 2. Ja va de Lue s esults The iequality, which we ecogize as Alze s iequality, has aleady bee kow at least sice 975 ad is due to J. va de Lue. I this sectio we peset J. va de Lue s Poblem 399, see [, p. 254]), it s coectio with Alze s iequality ad ou coclusios i the COMMENTS. PROBLEM 399. Fo N ad s R let σ s) := k s, U s) := s σ s), L s) := s σ s), k= whee σ 0 s) =0. Pove that i s is positive, U s) is deceasig i ad L s) is iceasig i. The mathematicias F. J. Baig, R. Doobos, A. A. Jages, J. H. va Lit, J. va de Lue ad G. R. Veldkamp gave solutios o Ja va de Lue s Poblem 399. F. J. M. Baig, J. va de Lue, G. R. Veldkamp ad R. Doobos this poo ca be see i []) used mathematical iductio ad J. H. va Lit showed that the poblem is a special case o a moe geeal situatio i the ollowig way: Let be iceasig ad covex o [0, ]. Let us coside S = k= ) k 2.) ad s = k=0 ) k. 2.2)

3 ON VAN DE LUNE -ALZER S INEQUALITY 565 Usig covexity o o [0, ] J. va de Lit poved that S ) N is a deceasig ad s ) N is a iceasig sequece, i.e. S + S, 2.3) ad s s ) By applyig the esult to the uctio g deied by gx) = x) it is easy to see that 2.3) ad 2.4) also hold i is iceasig ad cocave. By applyig the esult to x) =x s, s > 0), the assetio o the Poblem 399 ollows. COMMENTS. I the cosideed uctio is stictly iceasig ad covex o stictly iceasig ad cocave o [0, ] the S is stictly deceasig ad s is stictly iceasig sequece. Now applyig J. H. va Lit s esults o uctio x) =x s, s > 0, we obtai that U is stictly iceasig ad L is stictly deceasig uctio i N. The act that U s) is stictly deceasig i is equivalet to i.e. U s) > U + s), s s + 2 s + + s ) > + ) s s + 2 s ) s ), s + 2 s + + s s > s + 2 s ) s + ) s, + ) s + ) i s + ) s < i=, + i s + < whee s is positive eal umbe. I 993. Host Alze poved + i= + ) i= + i s i= + ) i= + i= i s s, 2.5), 2.6) whee N ad is positive eal umbe. It is obvious that Ja va de Lue s iequality 2.5) diectly implies iequality 2.6) i.e. 2.5) holds with stict iequality " < " i place o " ". Povig iequality 2.6),J.Sádo, [33]), adj.s.ume,[38]), also came to coclusio that 2.6) is tue o stict iequality.

4 566 S. ABRAMOVICH, J.BARIĆ, M.MATIĆ AND J. PEČARIĆ We omulate J. H. va Lit s esults i the ollowig theoem: THEOREM 2. Let be a iceasig ad covex o a iceasig ad cocave uctio o [0, ].LetS ad s be deied by 2.) ad 2.2), espectively. The S is a deceasig sequece ad s is a iceasig sequece i.e. ad + + k=0 k= ) k + ) k + k=0 k= ) k 2.7) ) k. 2.8) + I is stictly iceasig ad covex uctio o stictly iceasig ad cocave uctio o [0, ] the S is stictly deceasig ad s is stictly iceasig sequece ad 2.7) ad 2.8) holds with stict iequality. Applyig Theoem 2 to x) =x s, s > 0, we get ollowig coollaies: COROLLARY. Let σ s) = k s,o N ad σ 0 s) =0.The k= is stictly deceasig uctio i ad is stictly iceasig uctio i. U s) = s σ s), L s) = s σ s), COROLLARY 2. Let x) =x s,whee s R,s> 0. The o N it holds + < + ) k= + k s k= k s s. 3. Alteative poos I 995. J. Sádo, i a shot pape [33], gave a alteative poo o H. Alze s iequality + ) + i=, 3.) + i=

5 ON VAN DE LUNE -ALZER S INEQUALITY 567 based o mathematical iductio ad Cauchy s mea value theoem o dieetial calculus. I act, he poved eve shape statemet, discoveed by Ja va de Lue i 975, that 3.) holds with stict iequality. I 996 J.S. Ume gave aothe elemetay poo o iequality 3.), see [38]), usig iductio ad dieetiatio. Howeve, his poo ca be modiied, i the ollowig way, to get stict iequality i 3.). We give J.S. Ume s Lemma ad it s poo with ou coectios. LEMMA. I is a positive eal umbe, the Poo. Fo x [0, ],deie < + x) [ x + x) +], 0 < x. 3.2) x) = + x) [ x + x) +]. The uctio is cotiuous o [0, ] ad 0) =0. To pove iequality 3.2) it suices to show x) > 0,o 0 < x <. Dieetiatio o yields x) = + x) { [ x + x) +] + + x)[ + ) x) ] }. Fo x [0, ],let sset gx) = [ x + x) +] + + x)[ + ) x) ]. The uctio g is cotiuous o [0, ] ad g0) =0. Nowwehave g x) = + ){ x) + [ + x) x) x) ] }. Sice x) < ad x) < + x) x) o 0 < x <, it ollows g x) > 0, o > 0 ad 0 < x <. Theeoe gx) > g0) which implies x) > 0, o 0 < x <. Now, 3.) ca be easily poved o stict iequality usig mathematical iductio ad Lemma see [38]). I C.-P. Che ad F. Qi showed, i [6], thatj.sádo s ad J. S. Ume s poos o 3.) ca be completed i othe ways usig Lagage s mea value theoem, mootoicity ad covexity o uctio s, ad mathematical iductio. 4. Geealizatios o Va de Lue - Alze s iequality I 999 F. Qi poved the ext theoem. THEOREM 3. Let ad m be atual umbes, k a oegative itege. The +k + k + m + k < i=k+, 4.) +m+k +m i=k+ whee is ay give positive eal umbe. The lowe boud is best possible.

6 568 S. ABRAMOVICH, J.BARIĆ, M.MATIĆ AND J. PEČARIĆ Theoem 3 is poved i [26] applyig mathematical iductio ad Cauchy mea value theoem. Some uthe esults elated to this ca be oud i [32] ad [9]. I J. S. Ume, i [39], showed how the esult o H. Alze ca be exteded usig suitable mappig. His maesult cotaied i the ollowig theoem) is poved usig two lemmas i which we made some impovemets. Let us quote the ist lemma. LEMMA 2. Let a, b, c ad d be eal umbes satisyig < a, c, 0 < b, d <, 0 < ab, I x) := c )a x + dab) x,the < 2 c + d) ad ac ab) d. < x), o all x [0, ). COMMENT. By caeul ispectio o Ume s poo o the above Lemma we see that the sig " < "i< 2 c + d) ca be eplaced with " " ad still it ca be poved that x) > 0) =c )+d > oallx 0, + ), which is cucial o the the est o Ume s esults i [39]. Fom Lemma 2 ollows the ext lemma. LEMMA 3. Let ϕ : 0, ) 0, ) be a uctio such that ϕ is stictly iceasig o 0, ), 4.2) ad o all ϕ exists o 0, ), 4.3) ϕ is stictly iceasig o 0, ), 4.4) ϕx) ϕx + ) ϕx + ), o all x 0, ), 4.5) ϕx + 2) [ ] ϕu + 2) ϕv+2) ϕv+) [ ] ϕu) ϕu + 2) ϕv) ϕv+), 4.6) ϕu + ) ϕu + ) ϕu + ) u, v 0, ). The ϕv+) < [ϕv+2) ϕv+)] { } { ϕu+2) ϕu) +ϕv) ϕu+) ϕu+) ϕu+2) }, 4.7) ϕu+) o all u, v 0, ) ad > 0. COMMENT. Cosideig the chages we made i Lemma 2 assetio 4.4) is chaged to ϕ is iceasig o 0, ), i.e., the uctio ϕ is covex o 0, ).

7 ON VAN DE LUNE -ALZER S INEQUALITY 569 THEOREM 4. Let ϕ : 0, ) 0, ) be a uctio satisyig coditios 4.2), 4.3), 4.4), 4.5) ad { } ϕ+m+k+2) ϕ+m+2) ϕ+m+) { ϕ+m+k) ϕ+m+k+) ϕ+m+k+) ϕ+m+k+2) } ϕ+m) ϕ+m+), ϕ+m+k+) o all N,m, k N {0} ad ϕ) 2ϕ2).The ϕ + k) ϕ + m + k) < ϕ) +k i=k+ +m+k ϕ+m) i=k+ [ϕi)] [ϕi)] 4.8), 4.9) o all, m N,k N 0ad > 0. Applyig the above theoem to the uctio ϕx) =a x,oall x 0, ),J.S. Ume poved the ollowig coollay. COROLLARY 3. Let, m N,k N {0},> 0 ad a 2.The a m < +k a a i i=k+ +m+k a +m a i i=k+. 4.0) I the ext coollay J.S. Ume gave a geealizatio o H. Alze s iequality. COROLLARY 4. I p = o p 2 the ) p + k < + m + k whee, m N,k N {0} ad > 0. +k p i p i=k+ +m+k +m) p i p i=k+, 4.) I F. Qi, i [27], peseted the iequality which geealizes Alze s esult, as well as oe esult poved by J.-C. Kuag ad iequality 4.). Namely, i 999, J.-C. Kuag, i [20], poved the ollowig iequality k= ) k > + + k= ) k > x)dx, 4.2) + 0 o a stictly iceasig covex o cocave) uctio o 0, ]. Motivated by iequalities i 4.) ad 4.2), cosideig covexity, F. Qi poved the ext theoem.

8 570 S. ABRAMOVICH, J.BARIĆ, M.MATIĆ AND J. PEČARIĆ THEOREM 5. Let be a stictly iceasig covex o cocave) uctio i 0, ]. The the sequece +k ) i +k is deceasig i ad k ad has a lowe boud i=k+ t)dt, that is 0 +k i=k+ ) i > +k+ + k + i=k+ ) i + k + > 0 t)dt, 4.3) whee k is a oegative itege ad is a atual umbe. Applyig Theoem 5 to x) =x,o > 0adk = 0 it ollows i.e ) < i= + < + ) < + ) i= + i= i=, + ) i= + i=, ad that is Alze s iequality with " < " istead o " ". Futhemoe, applyig Theoem 5 o x) =x,o > 0 it ollows i.e. +k i=k+ +k+ i=k+ + k) > + ) + k + ), 4.4) +k i=k+ +k+m i=k+ + k) > + m) + k + m), which is equivalet to iequality 4.). Fok = 0 iequality 4.3) becomes equivalet to 4.2). COMMENT. Notice that the let-had iequality i 4.2) is equivalet to J. H. va Lit s esult i.e. to iequality 2.7) i Theoem 2 o stictly iceasig covex o cocave) uctio.,

9 ON VAN DE LUNE -ALZER S INEQUALITY 57 I 200 F. Qi, i [28], poved a algebaic iequality which is a itegal aalogue o the ollowig iequality + k + m + k < +m +k i=k+ +m+k i=k+, 4.5) poved i [26]. A extesio o this Qi s esult ca be oud i pape [4]). THEOREM 6. Let b > a > 0 ad δ > 0 be eal umbes. The o ay give positive R we have [ b + δ a b + a + ] b a b + δ) + a + The lowe boud i 4.6) is best possible. The iequality 4.6) ca be ewitte as b b + δ < b a b+δ a b x dx x dx a b+δ a > b b + δ. 4.6). 4.7) Fo a = k, b = + k ad δ = m, iequality4.7) is itegal aalogue o the 4.5). Iequality 4.7) was geealized by B. Gavea ad I. Gavea i [8], toai- equality o liea positive uctioals. Usig a completely dieet uexpected appoach, I. Gavea impoved Va de Lue - Alza s iequality ad some elated iequalities poved i [4]. Gavea used Bestei ad Bestei - Stacu opeatos to get his may ew esults [7]. Byusig Bestei polyomials o degee he impoved iequality 2.3) o iceasig covex uctios. He got the ollowig iequalities: [ k 6 + ) mi k=0,, k + +, k + ] ; + k=0 ) k + ) k k=0 ]. [ k 6 + ) max k=0,, k + +, k + ; Usig Bestei-Stacu type opeatos Gavea poved some geeal iequalities om which he got 0 6 k=0 ) k + + [, + ; 2 ) k + k=0 ] [ ]), + ;

10 572 S. ABRAMOVICH, J.BARIĆ, M.MATIĆ AND J. PEČARIĆ as well as [ ] β + β 2 + β) + β + ) + β, β + + β + ; ) i + β + ) i + β + β + + β + i= i= [ ] β + + β 2 + β) + β + ) + β +, ; o β 0, whee [x, x 2 ; ] := x 2) x ) ; [x, x 2, x 3; ; ] := [x 2, x 3 ; ] [x, x 2 ; ]. x 2 x x 3 x Also by usig iequalities esultig om Bestei-Stacu opeatos he got as a special case the kow iequality poved by F. Qi ad B-N. Guo i [30]: k= ak a ) + + k= ak which holds o a iceasig sequece a ) N, a [0, ], such that a a + )) N is also a iceasig sequece, ad o iceasig covex uctio o [0, ]. a + 5. Applicatios to sequeces I 998 N. Elezović adj.pečaić, i [6], showed that Alze s iequality is satisied o a lage class o iceasig covex sequeces. They gave a geealizatio o 3.) cotaied i the ext theoem. THEOREM 7. o 0,a 0 = 0, R +,the I the sequece a ), o positive eal umbes satisies ) [ ) ] + a+2 a +2 a +, 5.) a + a + a + a a + a + a i i= + a a i i= ). 5.2) N. Elezović adj.pečaić poved Theoem 7 usig mathematical iductio ad Lemma. I [6] they also gave a simpliicatio o poo o Lemma as well as the ollowig examples ad coollaies o Theoem 7 o sequeces o positive eal umbes.

11 ON VAN DE LUNE -ALZER S INEQUALITY 573 COROLLARY 5. Let the sequece a ) o positive eal umbes satisy The 5.2) holds. a ) a 2 +, a a 2 5.3) a 2a + + a +2 0,. 5.4) EXAMPLE. The sequece a = satisies 5.3) ad 5.4). Hece, Theoem 7 geealizes Alze s iequality. COROLLARY 6. Fo each stictly iceasig sequece a ) o positive eal umbes thee exist a > 0 such that 5.2) holds. EXAMPLE 2. The sequece a = 2 satisies 5.3) ad 5.4). Theeoe we have ) 2i ) 2 + i=. 2 ) + 2i ) EXAMPLE 3. The sequece a = k )+, k > 0, satisies 5.4). Futhe, 5.3) is equivalet to kk + ). 5.5) Theeoe, 5.2) holds o this sequece wheeve 5.5) is valid. EXAMPLE 4. The sequece a = a, a >, satisies 5.4). Futhe, 5.3) is equivalet to a ) a As i pevious example, o each > 0theeexista > owhich5.6) is valid. i= COMMENTS. Iequality 5.2) is equivalet to + a i i= a i i= a + a + a a I we egad the let ad ight sides o the last iequality as membes o a eal sequece A ) we ca coclude that sequece A ) is deceasig i.e. N) p N) A +p A. Hece, iequality 5.2) is equivalet to +m a i i= a +m a +m a i i= a a,

12 574 S. ABRAMOVICH, J.BARIĆ, M.MATIĆ AND J. PEČARIĆ i.e. a a +m a +m a i i= +m a a i i= NEW RESULTS. Followig the easoig o N. Elezović ad J.Pečaić esults peseted above [6]) we exted the esults obtaied thee o x) =x, x 0, 0 to a iceasig x) whee x x) is covex. Istead o dealig with i= a + i a + i= a i a + + we deal with ) ai ) + a a ) i= i= REMARK. Iequality 5.8) is equivalet to i= Theeoe, i a i > 0, a i ) > 0 i=,...., > 0, a > 0, =, 2, ) a + ) ai ). 5.8) a + ) + a i ) a ) a ) a + ). 5.9) a + a + ) the iequality a a ) 0 5.0) a + a + ) is a ecessay coditio o 5.8) to hold. THEOREM 8. Let a > 0, =,... ad let the uctio x) > 0 be deied o [0, ), ad satisies 5.0). I [ a +2 ) a+2 + a ] a ) 5.) a + ) a + a + a + ) o 0, a 0 = 0, the 5.8) holds. Poo. The poo is by iductio ad ollows the steps o the poo o Theoem 7. Fo = iequality 5.8) is equal to 5.) o = 0. Fo a geeal iequality 5.8) is equivalet to 5.9) ad theeoe the iductio hypothesisis equivalet to which meas that + a i ) i= +2 a i ) i= a + 2 a + ) a + a + ) a a ) a + 2 a + ) a + a + ) a a ) + a +2)

13 ON VAN DE LUNE -ALZER S INEQUALITY 575 ad hece it is suiciet to pove that a + 2 a + ) a + a + ) a a ) + a +2) which is equivalet to 5.). a +2 2 a +2 ) a +2 a +2 ) a + a + ) 5.2) THEOREM 9. Let x) be a positive iceasig uctio o [0, ) such that x x) is covex ad ) A xa) 2 A), x x ) Let the sequece a > 0, =,..., a 0 = 0 satisy a +2 a + a + a, 5.4) ad The 5.8) holds. a 2 a ) ) a a 2 ) Poo. Iequality 5.5) is equivalet to 5.) o = 0. Let us deote The it ollows om 5.4) that w > 0. I 2thew adthe a +2 a + w = a +2 a ) [ a +2 ) a+2 + a ] a ) a +2). a + ) a + a + a + ) a + ) Theeoe, i this case 5.) holds ad om Theoem 8 we get that 5.8) holds. I 0 w, iequality 5.4) is equivalet to a w) a ) As x) 0 ad is iceasig we get that [ a +2 ) a+2 + a ] a ) a + ) a + a + a + ) a [ + + w)) w + w) a ] + w)) a + ) a + ) ) + w)a + ) + w) a + ) = a + ) a + )) 2 w + w a + a + )) + ) + w a + w) a + w)). 5.8)

14 576 S. ABRAMOVICH, J.BARIĆ, M.MATIĆ AND J. PEČARIĆ As x x) is covex, we get that w + w a + a + )) + + w a + w) a + w))) w a + + w + w ) w a + + w + w + w )) + w = a ) + + w a+. + w Isetig 5.9) i 5.8) we get that a +2 ) a + ) [ a+2 a + + a a ) a + a + ) + w)a +) + w) a + ) a + a + ) a + )) 2 + w) = + w) a +) a + ) +w 2. a + ) ] ) a+ + w 5.9) 5.20) As 0 w, theeoe + w 2. The, om 5.3) we get that ) a+ + w) a + ) 2 a + ). + w 5.2) Togethe with 5.20) we get that 5.) holds. Hece 5.8) is poved. COROLLARY 7. Let x) =x, 0, x 0. Theeoe x x) =x + is covex o x 0, so 5.2) holds with equality ad iequality 5.8) becomes equal to iequality 5.7). COROLLARY 8. Let log x) be a covex uctio. The 5.2) holds ad as x) is iceasig, also x) is covexadtheeoe 5.8) holds. I 2000 F. Qi ad L. Debath, i [29], usig mathematical iductio ad Cauchy mea value theoem, poved ollowig esults. THEOREM 0. Let ad m be atual umbes. Suppose {a, a 2, } is a positive ad iceasig sequece satisyig ak+2 ), 5.22) k + 2)a k+2 k + )a k+ k + )a k+ ka k a k+ o ay give positive eal umbe ad k N. The we have the iequality a a +m The lowe boud o 5.23) is best possible. +m a i i= +m a i i=. 5.23)

15 ON VAN DE LUNE -ALZER S INEQUALITY 577 COROLLARY 9. Let ad m be atual umbes. Suppose a = {a, a 2, } is a positive ad iceasig sequece satisyig a 2 k+ a ka k+2, 5.24) { a k+ a k k + a 2 k+ a max, k + 2 }, k N. 5.25) ka k+2 a k+ a k+2 The, o ay give positive eal umbe, we have iequality 5.23). The lowe boud o 5.23) is best possible. Applyig Coollay 9 to a =k +, k + 2, ) iequality 4.) ollows. I 2002 Z. Xu ad D. Xu, i [40], gave some ew esults elated to Alze s ad Mati s iequality. We will peset the esults elated to Alze s iequality. THEOREM. Let a ) N be a stictly iceasig positive sequece, ad let m be a atual umbe ad be a positive eal umbe. I a a ) a + a a a, 2, 5.26) the a a +m < +m a i i= +m a i i= The lowe boud i 5.27) is best possible.,. 5.27) COMMENTS. Notice that the iequality 5.27) is equal to 5.23) with dieet coditios. Coditio 5.26) ca be itepeted i the ollowig way: The iequality is equivalet to a a a ) ) a + a a+ ), a a which meas that the sequece Futhemoe, iequality { a+ a ) } N a + a a a is iceasig i. is equivalet to a a + a 2, which meas that the sequece a ) N is logaithmic cocave o 2.

16 578 S. ABRAMOVICH, J.BARIĆ, M.MATIĆ AND J. PEČARIĆ As a cosequece o Theoem Z. Xu ad D. Xu easily poved that the iequality + k + m + k < +m i + k) i= +m i= i + k), 5.28) is valid o ay oegative eal umbe k ad ot oly o k beig oegative itege like it was peseted i above cases. I F. Qi, B.-N. Guo ad L. Debath, i [3], usig mathematical iductio, poved iequality 5.27), metioed above, with dieet coditios. We quote thei esult ad coollay. THEOREM 2. Let ad m be atual umbes. Suppose a i ) +m i= is a iceasig, logaithmically covex, ad positive sequece. Deote the powe mea P ) o ay give positive eal umbe by The the sequece is { } +m Pi) a i i= P ) = ) a i. 5.29) i= is deceasig o ay give positive eal umbe, that P ) P +m ) a 5.30) a +m The lowe boud i 5.30) is the best possible. Cosideig that the expoetial uctios a xα ad a αx, o give costats α ad a >, ae logaithmically covex o [0, ), as a coollay o Theoem 2 it ollows. COROLLARY 0. Let α ad a > be two costats. Fo ay give eal umbe the ollowig iequalities hold a +k)α a +m+k)α a α+k a α+m+k +m +m +k a iα i=k+ +m+k a iα i=k+ +k a α i=k+ +m+k a α i=k+, 5.3), 5.32) whee a m ae atual umbes, ad k is a oegative itege. The lowe bouds i 5.3) ad 5.32) ae the best possible.

17 ON VAN DE LUNE -ALZER S INEQUALITY A iequality o Va de Lue - Alze o egative powes The iequality o Va de Lue - Alze o egative powes was poved by H. Alze i [3]. The esults which we will peset hee oe ew poos ad extesios. I C.-P. Che ad F. Qi, i [7], poved that Va de Lue - Alze s iequality is valid o all eal umbes ot oly o > 0 ). We ow quote thei esult. THEOREM 3. Let be a atual umbe. The o all eal umbes it holds + < + i= + i= <. 6.) Both bouds ae best possible. Theoem 3 is poved by usig mathematical iductio ad Jese s iequality. J. Sádo, i [34], gave a elegat poo o iequality 6.) usig Cauchy s mea value theoem istead o Jese s iequality. Fo some uthe esults o this topic the eade is also eeed to the papes [0], [] ad [2] witte by C.-P. Che ad F. Qi. I C.-P. Che ad F. Qi, i [9], studyig mootoicity popety o geealized logaithmic meas deied by poved the ollowig theoem. [ b p p+)b a)], p, 0; b a L p a, b) = l b l a, p = ; ) b b b a e a a, p = 0, THEOREM 4. Let c > b > a > 0 be eal umbes. The the uctio ) = L a, b) L a, c) 6.2) is stictly deceasig with, ). The ollowig coollay is staightowad. COROLLARY. R The lowe boud i 6.3) is best possible. Let c > b > a > 0 be eal umbes. The o ay eal umbe b c < L a, b) L a, c). 6.3)

18 580 S. ABRAMOVICH, J.BARIĆ, M.MATIĆ AND J. PEČARIĆ Fo c = b + δ, δ > 0, Coollay gives a extesio o itegal vesio o Va de Lue - Alze s iequality i.e. iequality b b + δ < b a b+δ a b x dx x dx a b+δ which is valid o all eal umbes. I C.-P. Che ad F. Qi, i [8], poved that Theoem 4 ca be geealized as ollows. THEOREM 5. Let c > b > a ad be eal umbes, ad let be a positive, twice dieetiable uctio ad satisy t) > 0 ad l t)) 0 o a, + ). The sup x [a,b] x) sup x [a,c] x) < a b b a a c c a a x)dx x)dx, <, 6.4) o all eal. Both bouds i 6.4) ae best possible. O the othe had, i 994., C. E. M. Peace ad J. Pečaić, i [25], poved the ollowig theoem which geealizes Theoem 4. THEOREM 6. Let a i,b i, i =, 2), be positive umbes satisyig a max, a ) 2 b max, b ) 2. a 2 a b 2 b The the uctio G deied by G) = L a, a 2 ) 6.5) L b, b 2 ) is odeceasig. We quote the poo: Poo. Powe itegal meas o ode p ae deied by [ b a M p ; a, b) = [ exp b a b a ] p t) p dt, p 0, b a ] log t)dt, p = 0. I et) =t, e x,y t) =xt + y t), the the geealized logaithmic meas have the two itegal epesetatios L p x, y) =M p e; x, y), L p x, y) =M p e x,y ;, 0).

19 ON VAN DE LUNE -ALZER S INEQUALITY 58 Thus G) = M e a,a 2 ;0, ) M e b,b 2 ;0, ) ad ou esult is a simple cosequece o the ollowig esult see [22]). Let ad g be positive ad itegable uctios o [a, b]. I the maps x gx),x x) ae mootoic i the same sese, the the uctio F deied by F) = M ;a,b) M g;a,b) gx) is odeceasig. I ou case t) =e a,a 2 t) ad gt) =e b,b 2 t).ib = b 2 the the deomiato i 6.5) is idepedet o ad the claim educes to the well-kow esult o the odeceasig chaacte o L.Ib b 2, the sice the deomiato i the deiitio o G is ivaiat ude the itechage o b ad b 2, we may without loss o geeality suppose that b > b 2. Simila symmety i the umeato o 6.5) allows us to assume a a 2, so that we ca suppose that a a 2 b b 2 >. I b > b 2 the uctio g is iceasig ad sice a ) b 2 a a 2 b = b e b,b 2 t)) 2, 6.5) tells us that g is odeceasig, cocludig the poo. The same assetio as oe that C. E. M. Peace ad J. Pečaić gave i Theoem 6 was obtaied by A.-J. Li, X.-M. Wag ad C.-P. Che, i [2], i Extedig the Ky Fa iequality to seveal geeal itegal oms, they obtaied the L ollowig theoems o mootoic popeties o the uctio sa,b) L sα a,α b) with α, a, b 0, + ) ad s R. THEOREM 7. Let α s) = b a b x s dx a α x) s dx s = L s a, b) L s α a, α b), s, + ) ad α be a positive umbe. The α s) is a stictly iceasig uctio o [a, b] 0, α 2 ], ad is a stictly deceasig uctio o [a, b] [ α 2, α). THEOREM 8. Let s) = b b a a d d c c x s dx x s dx s = L sa, b) L s c, d), s, + ) ad a, b, c, d be positive umbes. The s) is a stictly iceasig uctio o ad < bc, o a stictly deceasig uctio o ad > bc.

20 582 S. ABRAMOVICH, J.BARIĆ, M.MATIĆ AND J. PEČARIĆ Poos o Theoem 7 ad Theoem 8 ae doe usig aalogous method as oe that C.-P. Che ad F. Qi used i [9] to pove Theoem 4. I J. Sádo, i [37], poved the ollowig theoem. THEOREM 9. Suppose that : 0, ] R is a stictly deceasig, covex o cocave) uctio. The oe has the iequality + + i= ) i > + i= i ). 6.6) COMMENTS. Notice that the evesed sig iequality, whe is stictly iceasig ad cocave o covex uctio, was poved by J.-C. Kuag, i [20], i 999, ad, moeove, was kow sice 975. thaks to Ja va de Lue s wok [, p. 254]). J. Sádo s poo is based o the method o J.-C. Kuag, [20]. Applyig Theoem 9 to the uctio x) = x s = x s, which is covex ad stictly deceasig, we get + + i= + i ) s > ) s, i which is equivalet to let-had side o 6.) o = s, s > 0. i= 7. Applicatio i guessig theoy I 998 S. S. Dagomi ad J. va de Hoek, i [4], poved a aalytic iequality which has impotat applicatios to the estimatio o the momets o guessig mappigs. To pove thei maesult they stat with sequeces S p ) ad G p ) deied i the ollowig way: S p ) = j= j p ad G p ) = S p) p+, whee p is positive eal umbe ad is atual umbe. The the ext theoem holds. THEOREM 20. Let p,p R.The ) The lowe boud o G p ) is G p ) 2) The sequece G p ) is oiceasig, i.e. + ) p + ) p+, o all ; 7.) p+ G p + ) G p ), o all. 7.2)

21 ON VAN DE LUNE -ALZER S INEQUALITY 583 COMMENTS. Note that sequece G p ) is deied i the same way as the uctio U s) i J. va de Lue s Poblem 399. Moeove, iequality 7.) is idetical to Alze s iequality 3.)eplace p with ). Namely, accodig to the deiitio o G p ) ad the act that + i p = i p + + ) p, iequality 7.) is equivalet to i= + ) p+ i p p+ i= i= i p i= + ) p + ) p+ p+, p+ i p p+ + ) p, i= ) + ) p+ i p p+ i p + + ) p, i= i= + + ) p+ i p p+ i p, i= + i= + ) i= + i p i= o p R, p. Futhe, iequality 7.2) is equivalet to J. va de Lue s statemet that uctio U s) is stictly deceasig i N. J. Sádo i his pape [35] 999.) also poit out two thigs. Fist, that 7.) is actually Alze s iequality poved also i his pape [33] o p > 0 ad o stict iequality ad secod, that 7.2) is equivalet to 7.). S. S. Dagomi ad J. va de Hoek poved Theoem 20 usig the ollowig lemma, i p p, LEMMA 4. Fo p,p R ad we have + 2) p [ p+ + + ) p] + ) 2p+, 7.3) ad thei maesult is poved usig Theoem 20. I C.-P. Che, F. Qi, P. Ceoe ad S. S. Dagomi, i [3], amog othe esults, peseted the ollowig theoem. THEOREM 2. Let be a { iceasig ad covex o cocave) uctio deied o [0, ]. The the sequece ) } { i deceases ad ) } i i= N i=0 N

22 584 S. ABRAMOVICH, J.BARIĆ, M.MATIĆ AND J. PEČARIĆ iceases, ad + i=0 i ) i= i ) i=0 i= i ). ) i t)dt ) COMMENTS. The ist iequality i 7.4) is equivalet to Kuag s iequality 4.2), moeove it is equivalet to iequality 2.7) whichwaspovedbyj. H. valit i Theoem 2. The last iequality i 7.4) is equivalet to iequality 2.8) which is also kow sice 975. Applyig Theoem 2 to x) =x o x [0, ] ad > 0 the authos o [3] poved the ollowig coollay. COROLLARY 2. Let N. The o all eal umbe > 0, it ollows i= + + i=. 7.5) + + i= i= The ight had iequality i 7.5) is Va de Lue - Alze s iequality. I I. BetićadJ.Pečaić, i [5], geealized iequality 7.2) om Theoem 20. They used ollowig uctio. i) i= F, p, a) = ), whee i) =i + a) p. Obviously, F, p, 0) =G p ). By obtaiig the same esult as S. S. Dagomi ad J. va de Hoek gave i [4] ad [5], with F istead o G,I. Betić ad J.Pečaić obtaied the best estimates o some iequalities i metioed papes. Geealizig iequality 7.2), they peseted the ollowig theoem. THEOREM 22. Let 2 be a itege ad p,a be eal umbes. Let s deie i + a) p i= F, p, a)= + a) p. The F +, p, a) F, p, a) o each p,a ad o each itege 2.

23 ON VAN DE LUNE -ALZER S INEQUALITY 585 COMMENTS. Iequality is equivalet to i.e. F +, p, a) F, p, a) + i + a) p i= + ) + + a) p + a + a + i + a) p i= + a) p, + ) i + a) p i= + i + a) p i= p, 7.6) o each itege 2 ad o eal umbes p, a. Notice that iequality 7.6) is the geealizatio o F. Qi s iequality 4.) because ow it is poved o all eal umbes k ot oly o oegative iteges k. The outlie o the poo o Theoem 22 goes as ollows: It eed s to be show that F, p, a) F +, p, a) 0op, a ad 2. Wehave F, p, a) F +, p, a) i + a) p + i + a) p i= = + a) p i= + ) + + a) p ) = i + a) p + a) p + ) + + a) p + i= = F, p, a) + ) + + a)p + a) p ) a) p. So, we have to pove that F, p, a) + + a) p + ) + + a) p + a) p, which is, by deiitio o uctio F, p, a), equivaletto i + a) p i= + a) p + + a) p + ) + + a) p + a) p. 7.7) Iequality 7.7) was poved i [5], usig mathematicaliductio, covexityo uctio x) =x + a) p,o p adx a ad applyig Jese s iequality.

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25 ON VAN DE LUNE -ALZER S INEQUALITY 587 [34] J. SÁNDOR, O a iequality o Alze, II, O. Math. Mag. 2003), o. 2, [35] J. SÁNDOR, Commets o a iequality o the sum o powes o positive iteges, RGMIA Reseach Repot Collectio 2 999), o. 2. [36] J. SÁNDOR, O a iequality o Beett, Geeal Mathematics Sibiu) 3 995), o. 3 4, [37] J.SÁNDOR, O a iequality o Alze o egative powes, RGMIA Reseach Repot Collectio ), o. 4. [38] J.S. UME, A elemetay poo o H. Alze s iequality, Math. Japo ), o. 3, [39] J.S. UME, A iequality o a positive eal uctio, Math. Iequal. Appl ), o. 4, [40] Z. XU, D. XU, A geeal om o Alze s iequality, Computes ad Mathematics with Applicatios ), [4] S.-L. ZHANG, C.-P. CHEN, F. QI, Cotiuous aalogue o Alze s iequality, Tamkag J. Math ), o. 2, Received Novembe 6, 2007) S. Abamovich Depatmet o Mathematics Uivesity o Haia Haia, 3905 Isael abamos@math.haia.ac.il J. Baić FESB Uivesity o Split Rudea - Boškovića b.b., 2000 Split Coatia jbaic@esb.h M. Matić Depatmet o Mathematics Faculty o Natual Scieces, Mathematics ad Educatio Uivesity o Split Teslia 2, 2000 Split Coatia mmatic@pmst.h J. Pečaić Faculty o Textile Techology Uivesity o Zageb Pieottijeva 6, 0000 Zageb pecaic@elemet.h Joual o Mathematical Iequalities jmi@ele-math.com