Internatonal Mathematcs and Mathematcal Scences Volume 009, Artcle ID 43857, 14 pages do:10.1155/009/43857 Research Artcle An Extenson of Stolarsky Means to the Multvarable Case Slavko Smc Mathematcal Insttute SANU, Kneza Mhala 36, 11000 Belgrade, Serba Correspondence should be addressed to Slavko Smc, ssmc@turng.m.sanu.ac.rs Receved 10 July 009; Accepted 3 September 009 Recommended by Feng Q We gve an extenson of well-known Stolarsky means to the multvarable case n a smple and applcable way. Some basc nequaltes concernng ths matter are also establshed wth applcatons n Analyss and Probablty Theory. Copyrght q 009 Slavko Smc. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. 1. Introducton There s a huge amount of papers nvestgatng propertes of the so-called Stolarsky or extended two-parametrc mean value, defned for postve values of x, y, as ) r x s y s) ) 1/ s r E r,s x, y : s x r y r), 1.1 rs r s x y ) / 0. E means can be contnuously extended on the doman { r, s; x, y ) r, s R; x, y R } 1.
Internatonal Mathematcs and Mathematcal Scences by the followng: r x s y s) ) 1/ s r rs r s / 0; ) E r,s x, y s x r y r) exp 1 s xs log x y s log y x s y s x s y s s log x log y ) ), r s / 0; ) 1/s, s/ 0, r 0; 1.3 xy, r s 0; x, y x>0, and n ths form are ntroduced by Keneth Stolarsky n 1. Most of the classcal two-varable means are specal cases of the class E. For example, E 1, x y / s the arthmetc mean, E 0,0 xy s the geometrc mean, E 0,1 x y / log x log y s the logarthmc mean, E 1,1 x x /y y 1/ x y /e s the dentrc mean, and so forth. More generally, the rth power mean x r y r / 1/r s equal to E r,r. Recently, several papers are produced tryng to defne an extenson of the class E to n, n > varables. Unfortunately, ths s done n a hghly artfcal mode cf. 4, wthout a practcal background. Here s an llustraton of ths pont; recently Merkowsk 4 has proposed the followng generalzaton of the Stolarsky mean E r,s to several varables: E r,s X : [ L X s ] 1/ s r L X r, r/ s, 1.4 where X x 1,...,x n s an n-tuple of postve numbers and L X s : n 1! E n 1 n x su 1 du 1 du n 1. 1.5 The symbol E n 1 stands for the Eucldean smplex whch s defned by E n 1 : { u 1,...,u n 1 : u 0, 1 n 1; u 1 u n 1 1}. 1.6 In ths paper, we gve another attempt to generalze Stolarsky means to the multvarable case n a smple and applcable way. The proposed task can be accomplshed by foundng a weghted varant of the class E, wherefrom the mentoned generalzaton follows naturally. In the sequel, we wll need notons of the weghted geometrc mean G G p, q; x, y and weghted rth power mean S r S r p, q; x, y, defned by G : x p y q ; S r : px r qy r) 1/r, 1.7
Internatonal Mathematcs and Mathematcal Scences 3 where p, q, x, y R ; p q 1; r R/{0}. 1.8 Note that S r r > G r for x / y, r / 0, and lm r 0 S r G. 1.1. Weghted Stolarsky Means We ntroduce here a class W of weghted two-parameters means whch ncludes the Stolarsky class E as a partcular case. Namely, for p, q, x, y R,p q 1,rs r s x y / 0, we defne ) ) r S s s G s 1/ s r ) r px s qy s x ps y qs 1/ s r W W r,s p, q; x, y : s S r r G r s px r qy r x pr y qr. 1.9 Varous propertes concernng the means W can be establshed; some of them are the followng: W r,s ) Ws,r ) ; W r,s ) Wr,s q, p; y, x ) ; Wr,s p, q; y, x ) xywr,s p, q; x 1,y 1) ; 1.10 W ar,as ) Wr,s p, q; x a,y a)) 1/a, a / 0. Note that 1 W r,s, 1 ) r ) ; x, y x s y s s )1/ s r xy s x r y r ) r xy r x s y s) ) 1/ s r xr y r) E r, s; x, y ). s 1.11 In the same manner, we get W r,s 3, 1 ) x 3 ; x3,y 3 s y s ) 1/ s r )) ; E r, s; x, y x r y r 3 W r,s 4, 1 ) 3x s 4 ; x4,y 4 xy ) s ) y s 1/ s r E 1.1 3x r xy ) )). r r, s; x, y y r The weghted means from the class W can be extended contnuously to the doman D { r, s; x, y ) r, s R; x, y R }. 1.13
4 Internatonal Mathematcs and Mathematcal Scences Ths extenson s gven by W r,s ) r s px s qy s x ps y qs px r qy r x pr y qr ) 1/ s r, rs r s x y ) / 0; ) pxs qy s x ps y qs 1/s pqs log x/y ), s x y ) / 0, r 0; ) ) exp s pxs log x qy s log y p log x q log y x ps y qs px s qy s x ps y qs, s x y ) / 0, r s; x p 1 /3 y q 1 /3, x/ y, r s 0; x, x y. 1.14 Note that those means are homogeneous of order 1, that s, W r,s p, q; tx, ty tw r,s p, q; x, y, t>0, symmetrc n r, s, W r,s p, q; x, y W s,r p, q; x, y but are not symmetrc n x, y unless p q 1/. 1.. Multvarable Case A natural generalzaton of weghted Stolarsky means to the multvarable case gves r p x s x p s p x r x p ) s ) ) r ) 1/ s r, rs s r / 0; ) p x s p s 1/s x s p log x ), r 0, s/ 0; p log x W r,s p; x exp p x s s log x ) ) p s p log x x ) p x s p s, r s / 0; x p log 3 x ) 3 p log x exp 3 p log x ) ), r s 0, p log x 1.15 where x x 1,x,...,x n R n, n, p s an arbtrary postve weght sequence assocated wth x and W r,s p; x 0 afor x 0 a,a,...,a. We also wrte, nstead of n 1, n 1.
Internatonal Mathematcs and Mathematcal Scences 5 The above formulae are obtaned by an approprate lmt process, mplyng contnuty. For example, applyng t s 1 s log t s log t s3 6 log3 t o s 3) s 0, 1.16 we get ) W 0,0 p; x lm W s,0 p; x lm p x s p s x s 0 s 0 s p log x ) p log x lm s 0 s p log x ) ) p log x p s ) ) ) s s p log x p log 3 x p log 3 x 6 ) ) p p s log x s log ) p x ) s 3 log 3 ) ) p x o s 3)) 6 1/s 1/s p log 3 x ) 3 p log x lm 1 s 0 3 p log x ) )s 1 o 1 p log x p log 3 x ) 3 p log x exp 3 p log x ) ). p log x 1/s 1.17 Remark 1.1. Analogously to the former consderatons, one can defne a class of Stolarsky means n n varables E r,s x; n as E r,s x; n : W nr,ns p 0, x, 1.18 where p 0 {1/n} n 1.
6 Internatonal Mathematcs and Mathematcal Scences Therefore, E r,s x; n r n 1 xns s n 1 xnr n n 1 xs n n 1 xr ) 1/n s r, rs r s / 0. 1.19 Detals are left to the readers.. Results The followng basc asserton s of mportance. Proposton.1. The expressons W r,s p; x are actual means, that s, for arbtrary weght sequence p one has mn{x 1,x...,x n } W r,s p; x max{x 1,x,...,x n }..1 Our man result s contaned n the followng. Proposton.. The means W r,s p, x are monotone ncreasng n both varables r and s. Passng to the contnuous varable case, we get the followng defnton of the class W r,s p, x. Assumng that all ntegrals exst, W r,s p, x r p t x s t dt exp s p t log x t dt )) ) 1/ s r s p t x r t dt exp r p t log x t dt )), rs s r / 0; p t x s t dt exp s p t log x t dt ) ) 1/s s p t log x t dt p t log x t dt ), r 0, s/ 0; p t x s exp s t log x t dt p t log x t dt ) exp s p t log x t dt ) ), p t xs t dt exp s p t log x t dt ) p t log 3 x t dt p t log x t dt ) 3 exp 3 p t log x t dt p t log x t dt ) ), r s 0, r s / 0;. where x t s a postve ntegrable functon and p t s a nonnegatve functon wth p t dt 1.
Internatonal Mathematcs and Mathematcal Scences 7 From our former consderatons, a very applcable asserton follows. Proposton.3. W r,s p, x s monotone ncreasng n ether r or s. 3. Applcatons 3.1. Applcatons n Analyss As an llustraton of the above, we gve the followng proposton. Proposton 3.1. The functon w s, defned by ) 1/s 1 πs Γ 1 s e γs, s/ 0; w s : exp γ 4ξ 3 ), s 0, π 3.1 s monotone ncreasng for s 1,. In partcular, for s 1, 1, one has Γ 1 s e γs Γ 1 s e γs πs sn πs 1 πs 4 144, 3. where Γ,ξ,γ stands for the Gamma functon, Zeta functon, and Euler s constant, respectvely. 3.. Applcatons n Probablty Theory For a random varable X and an arbtrary probablty dstrbuton wth support on,, t s well known that Ee X e EX. 3.3 Denotng the central moment of order k by μ k μ k X : E X EX k, we mprove ths nequalty to the followng propsostons. Proposton 3.. For an arbtrary probablty law wth support on R, one has Ee X 1 μ ) )) μ3 exp e EX. 3μ 3.4
8 Internatonal Mathematcs and Mathematcal Scences Proposton 3.3. One also has that Ee sx e sex ) 1/s 3.5 s σ X / s monotone ncreasng n s. 3.3. Shfted Stolarsky Means Especally nterestng s studyng the shfted Stolarsky means E, defned by E r,s x, y ) : lm p 0 W r,s p, q; x, y ). 3.6 Ther analytc contnuaton to the whole r, s plane s gven by r x s y s 1 s log x/y ))) ) 1/ s r s x r y r 1 r log x/y ))), rs r s x y ) / 0; ))) x s y s 1/s 1 s log x/y Er,s ) s log x/y ), s x y ) / 0, r 0; x, y x s exp s y s) log x sy s log y log x/y ) ) x s y s 1 s log x/y )), s x y ) / 0, r s; x 1/3 y /3, r s 0; x, x y. 3.7 Man results concernng the means E are contaned n the followng propostons. Proposton 3.4. Means E r,s x, y are monotone ncreasng n ether r or s for each fxed x, y R. Proposton 3.5. Means E r,s x, y are monotone ncreasng n ether x or y for each r, s R. A well known result of Q 5 states that the means E r,s x, y are logarthmcally concave for each fxed x, y > 0andr, s 0, ; also, they are logarthmcally convex for r, s, 0. Accordng to ths, we propose the followng proposton. Open Queston Is there any compact nterval I,I R such that the means E r,s x, y are logarthmcally convex concave for r, s I and each x, y R?
Internatonal Mathematcs and Mathematcal Scences 9 A partal answer to ths problem s gven n what follows. Proposton 3.6. On any nterval I whch ncludes zero and r, s I, ) E r,s x, y are not logarthmcally convex concave); ) W r,s p, q; x, y are logarthmcally convex concave) f and only f p q 1/. 4. Proofs For the proof of Proposton.1, we apply the followng asserton on Jensen functonals J f p, x from 6. Theorem 4.1. Let f, g : I R be twce contnuously dfferentable functons. Assume that g s strctly convex and φ s a contnuous and strctly monotonc functon on I. Then the expresson )) Jn p, x; f φ 1 ) J n p, x; g n 4.1 represents a mean value of the numbers x 1,...,x n, that s, )) Jn p, x; f mn{x 1,...,x n } φ 1 ) max{x 1,...,x n } J n p, x; g 4. f and only f the relaton f t φ t g t 4.3 holds for each t I. Recall that the Jensen functonal J n p, x; f s defned on an nterval I,I R by ) n n J n p, x; f : p f x f p x ), 4.4 1 1 where f : I R, x x 1,x,...,x n I n, and p {p } n 1 s a postve weght sequence. The famous Jensen s nequalty asserts that J n p, x; f ) 0, 4.5 whenever f s a strctly convex functon on I, wth the equalty case f and only f x 1 x x n.
10 Internatonal Mathematcs and Mathematcal Scences Proof of Proposton.1. Defne the auxlary functon h s x by h s x : e sx sx 1 s, s/ 0; x, s 0. 4.6 Snce e sx 1, s/ 0; h s x s x, s 0, 4.7 h s x e sx, s R, we conclude that h s x s a contnuously twce dfferentable convex functon on R. Denotng f t : h s t, g t : h r t, we realze that the condton 4.3 of Theorem 4.1 s fulflled wth φ t e s r t. Hence, applyng Theorem 4.1, we obtan that log W r,s p,e x represents a mean value, whch s equvalent to the asserton of Proposton.1. Proof of Proposton.. We prove frst a global theorem concernng log-convexty of the Jensen s functonal wth a parameter, whch can be very usable cf. 7. Theorem 4.. Let f s x be a twce contnuously dfferentable functon n x wth a parameter s. If f s x s log-convex n s for s I : a, b ; x K : c, d, then the Jensen functonal J f w, x; s J s : w f s x f s w x ), 4.8 s log-convex n s for s I, x K, 1,,...,wherew {w } s any postve weght sequence. At the begnnng, we need some prelmnary lemmas. Lemma 4.3. A postve functon f s log-convex on I f and only f the relaton ) s t f s u f uw f t w 0 4.9 holds for each real u, w and s, t I. Ths asserton s nothng more than the dscrmnant test for the nonnegatvty of second-order polynomals. Other well known assertons are the followng cf 8, pages 74, 97-98 lemmas.
Internatonal Mathematcs and Mathematcal Scences 11 Lemma 4.4 Jensen s nequalty. If g x s twce contnuously dfferentable and g x 0 on K, then g x s convex on K and the nequalty ) w g x g w x 0 4.10 holds for each x K, 1,,...,and any postve weght sequence {w }, w 1. Lemma 4.5. For a convex f, the expresson f s f r s r 4.11 s ncreasng n both varables. Proof of Theorem 4.. Consder the functon F x defned as F x F u, v, s, t; x : u f s x uvf s t / x v f t x, 4.1 where u, v R; s, t I are real parameters ndependent of the varable x K. Snce F x u f s x uvf s t / x v f t x, 4.13 and by assumng f s x s log-convex n s, t follows from Lemma 4.3 that F x 0, x K. Therefore, by Lemma 4.4, weget ) w F x F w x 0, x K, 4.14 whch s equvalent to ) s t u J s uvj v J t 0. 4.15 Accordng to Lemma 4.3 agan, ths s possble only f J s s log-convex and the proof s done. Now, the proof of Proposton. easly follows. From the above, we see that h s x s twce contnuously dfferentable and that h s x s a log-convex functon for each real s, x.
1 Internatonal Mathematcs and Mathematcal Scences Applyng Theorem 4., we conclude that the form w e sx e s w x, s/ 0, s Φ h w, x; s Φ s : w x w x, s 0, 4.16 s log-convex n s. By Lemma 4.5,wthf s log Φ s, we fnd out that log Φ s log Φ r s r log ) Φ s 1/ s r 4.17 Φ r s monotone ncreasng ether n s or r. Therefore, by changng varable x log x, we fnally obtan the proof of Proposton.. Proof of Proposton.3. The asserton of Proposton.3 follows from Proposton. by the standard argument cf. 8, pages 131 134. Detals are left to the reader. Proof of Proposton 3.1. The proof follows puttng x t t, p t e t,t 0, and applyng Proposton..wthr 0. Correspondng ntegrals are 0 e t log t γ; 0 e t log t γ π 6 ; 0 e t log 3 t γ 3 γπ ξ 3, 4.18 wth Γ 1 s Γ 1 s πs sn πs. 4.19 Proof of Proposton 3.. By Proposton.3, weget W 0,1 p,e x W 0,0 p,e x, 4.0 that s, Ee X e EX μ / ) EX 3 EX 3 exp. 4.1 3μ Usng the dentty EX 3 EX 3 μ 3 3μ EX, we obtan the proof of Proposton 3.. Proof of Proposton 3.3. Ths asserton s straghtforward consequence of the fact that W 0,s p,e x s monotone ncreasng n s.
Internatonal Mathematcs and Mathematcal Scences 13 Proof of Proposton 3.4. Drect consequence of Proposton.. Proof of Proposton 3.5. Ths s left as an easy exercse to the readers. Proof of Proposton 3.6. We prove only part. The proof of goes along the same lnes. Suppose that 0 a, b : I and that E r,s p, q; x, y are log-convex concave for r, s I and any fxed x, y R. Then there should be an s, s > 0 such that F s ) : W0,s ) W0, s ) W0,0 )) 4. s of constant sgn for each x, y > 0. Substtutng x/y s : e w,w R, after some calculatons, we get that the above s equvalent to the asserton that F p, q; w s of constant sgn, where F p, q; w ) : pe w q e pw e /3 1 p w pe w q e pw). 4.3 Developng n power seres n w,weget F p, q; w ) 1 160 pq 1 p ) p ) 1 p ) w 5 O w 6). 4.4 Therefore, F p, q; w can be of constant sgn for each w R only f p 1/ q. Suppose now that I s of the form I : 0,a or I : a, 0,a>0. Then there should be an s, s / 0,s I such that W 0,0 ) W0,s ) W0,s )) 4.5 s of constant sgn for each x, y R. Proceedng as before, ths s equvalent to the asserton that G p, q; w s of constant sgn wth G p, q; w ) : p 3 q 3 w 6 e /3 p 1 w pe w q e pw) pe w q e pw) 4. 4.6 However, G p, q; w ) 405 p4 q 4 1 p ) 1 q ) q p ) w 11 O w 1). 4.7 Hence, we conclude that G p, q; w can be of constant sgn for suffcently small w, w R only f p q 1/. Combnng ths wth Feng Q theorem, the asserton from Proposton 3.6 follows.
14 Internatonal Mathematcs and Mathematcal Scences References 1 K. B. Stolarsky, Generalzatons of the logarthmc mean, Mathematcs Magazne, vol. 48, no., pp. 87 9, 1975. E. B. Leach and M. C. Sholander, Multvarable extended mean values, Mathematcal Analyss and Applcatons, vol. 104, no., pp. 390 407, 1984. 3 Z. Páles, Inequaltes for dfferences of powers, Mathematcal Analyss and Applcatons, vol. 131, no. 1, pp. 71 81, 1988. 4 J. K. Merkowsk, Extendng means of two varables to several varables, Inequaltes n Pure and Appled Mathematcs, vol. 5, no. 3, artcle 65, pp. 1 9, 004. 5 F. Q, Logarthmc convexty of extended mean values, Proceedngs of the Amercan Mathematcal Socety, vol. 130, no. 6, pp. 1787 1796, 00. 6 S. Smc, Means nvolvng Jensen functonals, submtted to Internatonal Mathematcs and Mathematcal Scences. 7 S. Smc, On logarthmc convexty for dfferences of power means, Inequaltes and Applcatons, vol. 007, Artcle ID 037359, 8 pages, 007. 8 G. H. Hardy, J. E. Lttlewood, and G. Pólya, Inequaltes, Cambrdge Unversty Press, Cambrdge, UK, 1978.
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