LÉVY KHINTCHINE REPRESENTATION OF TOADER QI MEAN FENG QI

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1 LÉVY KHINTCHINE REPRESENTATION OF TOADER QI MEAN FENG QI Institute of Mthemtics, Henn Polytechnic University, Jiozuo City, Henn Province, 454, Chin; College of Mthemtics, Inner Mongoli University for Ntionlities, Tonglio City, Inner Mongoli Autonomous Region, 843, Chin; Deprtment of Mthemtics, College of Science, Tinjin Polytechnic University, Tinjin City, 3387, Chin BAI-NI GUO School of Mthemtics nd Informtics, Henn Polytechnic University, Jiozuo City, Henn Province, 454, Chin Abstrct. In the pper, by virtue of Lévy Khintchine representtion nd n lterntive integrl representtion for the weighted geometric men, the uthors estblish Lévy Khintchine representtion nd n lterntive integrl representtion for the Toder Qi men. Moreover, the uthors lso collect n probbilistic interprettion nd pplictions in engineering of the Toder Qi men.. Preliminries nd min results In this section, we prepre some necessry knowledge nd stte our min results. We orgnize this section s three subsections... Toder Qi men. For, b > nd q, denote /q M q, b = q θ b q sin θ d θ. It is esy to see tht M q, b = / θ q /q b sin θ d θ = / q θ b q sin θ d θ /q. E-mil ddresses: qifeng68@gmil.com, qifeng68@hotmil.com, qifeng68@qq.com, bi.ni.guo@gmil.com, bi.ni.guo@hotmil.com. Mthemtics Subject Clssifiction. Primry 44A5; Secondry 6E6, 3E, 33C, 6G5. Key words nd phrses. Lévy Khintchine representtion; integrl representtion; Bernstein function; Stieltjes function; Toder Qi men; weighted geometric men; Bessel function of the first kind; probbilistic interprettion; probbilistic interprettion; ppliction in engineering; inequlity. This pper ws typeset using AMS-LATEX.

2 F. QI AND B.-N. GUO In 37, it ws remrked tht M, b = lim q M q, b = G, b = b, but it ws not known how to determine ny men M q for q. In 38, p. 38, Section 5, the connection M q, b = G, b J i q ln /q. b ws underlined, where i = is the imginry unit, the Bessel function of the first kind J ν z cn be defined 9, p. 7,.. by k k+ν z J ν =, k!γν + k + nd the clssicl Euler gmm function Γz cn be defined by Γz = lim n k= n k= n!n z z + k, z C \ {,,,... }. In 3, Lemm. nd 4, Lemm., the reltion q M q, b = G, b I ln /q. b ws estblished, where the modified Bessel functions of the first kind I ν z cn be defined by k+ν z I ν z =, rg z <, ν C. k!γν + k + Since k= J iz = I z = k z, k! the formuls. nd. re essentilly the sme one. With the help of., the men M q, b, the modified Bessel function of the first kind I, nd the rithmetic-geometric men M, b, which cn be defined 5, 5, 6 by M, b = / k= θ + b sin θ d θ, were bounded in 3, 4, 4, 4, 4 successfully. We remrk tht the men M q, b cn be trced bck to 5, 6, 37, 38 nd the closely relted references therein. See lso, pp In the ppers 4, 4, 4, the men M, b ws denoted by T Q, b nd ws nmed the Toder Qi men fter Romnin mthemticin, Gheorghe Toder, nd the first uthor of this pper. For the ske of convenience, we still dopt in this pper the nottion nd terminology. In Section 4 of this pper, we will collect probbilistic interprettion nd pplictions in engineering of the Toder Qi men T Q, b.

3 LÉVY KHINTCHINE REPRESENTATION OF TOADER QI MEAN 3.. Lévy Khintchine representtion. For stting our min results, we need to recll the following notion nd conclusions. Definition. 39, Chpter IV. An infinitely differentible function f on n intervl I is sid to be completely monotonic on I if it stisfies n f n t for x I nd n N, where N stnds for the set of ll positive integers. Definition. 4, Definition.. An infinitely differentible function f : I, is clled Bernstein function on n intervl I if f t is completely monotonic on I. Proposition. 36, Theorem 3.. A function f :,, is Bernstein function if nd only if it dmits the representtion fx = α + βx + e xt d µt,.3 where α, β nd µ is positive mesure stisfying min{, t} d µt <. In prticulr, the triplet α, β, µ determines f uniquely nd vice vers. The integrl representtion.3 for fx is clled the Lévy-Khintchine representtion of fx. The representing mesure µ nd the chrcteristic triplet α, β, µ re often respectively clled the Lévy mesure nd the Lévy triplet of the Bernstein function fx. It ws pointed out in 36, Chpter 3 tht the notion of the Bernstein functions originted from the potentil theory nd is useful to hrmonic nlysis, probbility, sttistics, nd integrl trnsforms. Definition.3 36, Chpter, Definition.. A Stieltjes function is function f :,, which cn be written in the form fx = x + b + d µs,.4 s + x where, b re nonnegtive constnts nd µ is nonnegtive mesure on, such tht d µs <. + s The integrl ppering in.4 is lso clled the Stieltjes trnsform of the mesure µ. For more informtion on this kind of functions trnsforms, plese refer to the monogrphs 36, Chpter, 39, Chpter VIII nd the closely relted references therein..3. Min results. The ims of this pper re to estblish n integrl representtion nd the Lévy Khintchine representtion of the Toder Qi men T Qx+, x+b for b > > nd to verify tht the Toder Qi men T Qx, x + b for b > > is Bernstein functions nd the divided difference of T Qx, x + b is Stieltjes function on,. Our min results cn be stted s the following theorems. Theorem.. For b > >, the Toder Qi men T Qx +, x + b for x > hs the integrl representtion T Qx +, x + b = T Q, b + x + b h, b; t t t + x d t.5

4 4 F. QI AND B.-N. GUO nd the Lévy Khintchine representtion where nd T Qx +, x + b = T Q, b + x + b h, b; t = H, b; s = F λ, s = / / u sin θ t θ b t sin θ d θ, sin θ F θ, b s d θ, λ H, b; s e s e xs d s,.6 s t, b, s,, λ e su d u, λ, s,,.7 u re positive. Consequently, the Toder Qi men T Qx, x + b is Bernstein function of x on,, the divided difference T Qx, x + b T Q, b x is Stieltjes function of x on,. Corollry.. For b > > nd x > q, we hve T Qx + q, x + b q = M q, b q + x + nd T Qx + q, x + b q = M q, b q + x + bq q Consequently, the divided difference b q h q, b q ; t t q T Qx, x + b q q M q, b q x q, q is Stieltjes function of x on,.. Lemms For proving our min results, we need the following lemms. t + x d t H q, b q ; s e qs e xs d s. s Lemm.. Let λ, nd b > >. The principl brnch of the weighted geometric men for z C \ b, cn be represented by G,b;λ z λ b λ + sinλ = z G,b;λ z = + z λ b + z λ b t λ b t λ d t, z ; t + z t λ + λb λ b λ, z =. Proof. This follows from letting n = in 9, Theorems 3. nd 4.6, rerrnging, nd tking the limit z. See lso 3, Remrk 4.3, 7, eq.., 8, eq..,, p. 78, Section 4, eq. 4.,, Section 4, eq. 4. nd the closely relted references therein..8.

5 LÉVY KHINTCHINE REPRESENTATION OF TOADER QI MEAN 5 Lemm.. Let λ, nd b > >. The principl brnch of the weighted geometric men G,b;λ z for z C \ b, hs the Lévy Khintchine representtion G,b;λ z = λ b λ + z + sinλ b where F λ, s is defined by.7. F λ, b s e s e zs d s,. s Proof. This is slight reformultion of those results in 5, p. 3, Lemm 3.8,, Lemm., 3, Theorem 3., nd 3, Theorem. nd the closely relted references therein. Proof of Theorem.. Applying. gives 3. Proofs of min results / T Qx +, x + b = x + θ x + b sin θ d θ = / θ b sin θ + x + sin θ b t θ b t sin θ x t t + x d t d θ = T Q, b + x + x / sin θ b t θ b t sin θ t t + x d t d θ = T Q, b + x + x b / sin θ t θ b t sin θ d θ t t + x d t. The integrl representtion.5 is thus proved. Applying. yields / T Qx +, x + b = x + θ x + b sin θ d θ = / θ b sin θ + x + sin θ F θ, b s b e s e xs d s d θ s b / = T Q, b + x + sin θ F θ, b s e s e xs d s d θ s = T Q, b + x + b / sin θ F θ, b s d θ e s e xs d s. s The Lévy Khintchine representtion.6 is thus proved. Since F λ, s is positive on,, compring the representtion.6 with.3 redily revels tht the Toder Qi men T Qx, x + b is Bernstein function of x on,. Reformulting.5 s T Qx +, x + b T Q, b x = + b h, b; t t t + x d t nd compring with.4 immeditely shows tht the divided difference.8 is Stieltjes function. The proof of Theorem. is complete. Proof of Corollry.. This follows from replcing nd b in Theorem. respectively by q nd b q for q.

6 6 F. QI AND B.-N. GUO 4. An interprettion nd n ppliction In this section, we collect probbilistic interprettion nd pplictions in engineering of the Toder Qi men T Q, b. For more detiled informtion, plese refer to the closely relted posts t the ReserchGte nd the pper 7, An probbilistic interprettion. It is not difficult to see tht n= / / T Q, e 4z = e 4z sin x d x = e z e z x d x = e z z n / e ix + e ix n k z n! n d x = e z k = k k! k= k= e z zk k! For the probbilistic interprettion, let N nd N form pir of independent identiclly distributed rndom vribles with Poisson distribution prmeter z. Then P {N = N } = P {N = N = k} k= k= = P {N = k}p {N = k} = e k= z zk k! = T Q, e 4z. The formuls. nd. imply directly the following probbilistic interprettion. If N nd N re two independent Poisson distributed rndom vribles both with the sme prmeter z for z >, then I z is the probbility tht N = N. A simple proof is P {N = N } = P {N = N = k} = P {N = k}p {N = k} = I z. k= 4.. Applictions in engineering. In 7, Section nd 8, Section II, the men nd the men squre over one period T of continuous nd periodic signl xt were defined by κx = xt d t nd κ x = x t d t. T T It is esy to see tht T T Q, b = k= θ b sin θ d θ = κ θ b sin θ, where T = nd xt = t b sin t. In 7, the root of the men squre vlues of smpled periodic signls ws clculted by using non-integer number of smples per period. This implies tht the Toder Qi men T Q, b cn be pplied to engineering. 5. Remrks Finlly we give severl remrks on some relted thing, including severl inequlities for bounding the Toder Qi men T Q, b. Remrk 5.. Lemms. nd. re equivlent to ech other. For the outline to prove this equivlence, plese refer to 9, Remrk 4. nd 3, Theorem.. T.

7 LÉVY KHINTCHINE REPRESENTATION OF TOADER QI MEAN 7 Remrk 5.. By the rithmetic-geometric-hrmonic men inequlity, we hve θ + b sin θ θ b sin θ. Integrting on ll sides on, gives A, b = + b > T Q, b > + b b θ + sin θ b = H, b nd q Aq, b q > M q, b > q H q, b q, where A, b nd H, b denote respectively the rithmetic nd hrmonic mens. Compring the geometric men G, b nd the Toder Qi men T Q, b, which men is bigger? The nswer is A, b + G, b A, b + G, b T Q, b < < 3 did in 3, Remrk 4. nd 4, Remrk. Consequently, we hve A q, b q + G q, b q /q A q, b q + G q, b q /q M q, b < <, q. 3 Remrk 5.3. In, Theorem., it ws derived tht λ + λb λ b λ < sinλ λ b + λb λ ln b for b > > nd λ,. Replcing λ by θ nd integrting over, on both sides of 5. rrives t / / θ + b sin θ d θ θ b sin θ d θ < / sin θ θ b + b sin θ θ ln b d θ which cn be simplified s T Q, b A, b > where b = = / = ln b / / / = / sin θ sin θ θ d θ ln b / 5. θ b + b sin θ θ ln b d θ / sin θ b sin θ θ d θ = J sin θ θ d θ = / θ + / sin θ b sin θ θ d θ sin + θ θ θ d θ = θ + d θ = b ln b, θ d θ θ θ d θ / / sin θ sin θ d θ =,

8 8 F. QI AND B.-N. GUO nd = / = / / sin θ / sin θ d θ = sin + θ θ d θ θ θ d θ = / θ d θ = 4 θ d θ = 4 J, / by virtue of the formul sin θ θ d θ = / + θ θ d θ = sin + θ + θ / z x nx d x = n J nz d θ θ d θ = 4 J in 3, p. 45, item 8. In conclusion, we obtin b T Q, b > A, b J ln b, b > >. Consequently, it follows tht M q, b > A q, b q b q q qj ln b /q, b > >, q. Remrk 5.4. The ppers, 4,,, 4, 6, 7, 8, 9, 3, 4, 7, 8, 9, 3, 3, 3, 33, 34, 35 belong to the sme series for estblishing integrl representtions of severl mthemticl mens. Acknowledgements. The uthors re grteful to Professor Jochim Domst t the Stte University of Applied Sciences in Elbl g of Polnd for his posts t the ReserchGte where he provides the probbilistic interprettions the Toder Qi men T Q, b. The uthors re grteful to Dr. Vier Čerňnová t the Slovk University of Technology in Slovk for her posts t the ReserchGte where she supplies pplictions in engineering of the Toder Qi men T Q, b by recommending the ppers 7, 8. The uthors re thnkful to librrin, Erik Andirkó, t the Librry, Institute of Mthemtics, University of Debrecen, Hungry for her sending n electronic copy of the pper 5 by e-mil. References Á. Besenyei, On complete monotonicity of some functions relted to mens, Mth. Inequl. Appl. 6 3, no., 33 39; Avilble online t P. S. Bullen, Hndbook of Mens nd Their Inequlities, Mthemtics nd its Applictions, Volume 56, Kluwer Acdemic Publishers, Dordrecht-Boston-London, 3. 3 I. S. Grdshteyn nd I. M. Ryzhik, Tble of Integrls, Series, nd Products, Trnslted from the Russin, Trnsltion edited nd with prefce by Dniel Zwillinger nd Victor Moll, Eighth edition, Revised from the seventh edition, Elsevier/Acdemic Press, Amsterdm, 5; Avilble online t B B.-N. Guo nd F. Qi, On the degree of the weighted geometric men s complete Bernstein function, Afr. Mt. 6 5, no. 7, 53 6; Avilble online t

9 LÉVY KHINTCHINE REPRESENTATION OF TOADER QI MEAN 9 5 H. Hruki, New chrcteriztions of the rithmetic-geometric men of Guss nd other well-known men vlues, Publ. Mth. Debrecen 38 99, no. 3-4, H. Hruki nd T. M. Rssis, New chrcteriztions of some men-vlues, J. Mth. Anl. Appl. 996, no., ; Avilble online t 7 G. E. Mog nd E. P. Ribeiro, Men nd RMS clcultions for smpled periodic signls with non-integer number of smples per period pplied to AC energy systems, In: Congresso Ibero-Ltino-Americno de Métodos Computcionis em Engenhri XXV Iberin Ltin-Americn Congress on Computtionl Methods in Engineering, Recife, Pernmbuco, Brzil, November, 4. 8 G. E. Mog nd E. P. Ribeiro, One cycle AC RMS clcultions for power qulity monitoring under frequency devition, EEE Xplore 4 th Interntionl Conference on Hrmonics nd Qulity of Power IEEE Ct. No.4EX95 4, 7 75; Avilble online t 9 F. W. J. Olver, D. W. Lozier, R. F. Boisvert, nd C. W. Clrk eds., NIST Hndbook of Mthemticl Functions, Cmbridge University Press, New York, ; Avilble online t F. Qi, Bounding the difference nd rtio between the weighted rithmetic nd geometric mens, Int. J. Anl. Appl. 3 7, no., F. Qi, Complete monotonicity of logrithmic men, RGMIA Res. Rep. Coll. 7, no., Art. 8; Avilble online t F. Qi, Limit formuls for rtios between derivtives of the gmm nd digmm functions t their singulrities, Filomt 7 3, no. 4, 6 64; Avilble online t 3 F. Qi nd V. Čerňnová, Some discussions on kind of improper integrls, Int. J. Anl. Appl. 6, no., 9. 4 F. Qi nd S.-X. Chen, Complete monotonicity of the logrithmic men, Mth. Inequl. Appl. 7, no. 4, ; Avilble online t 5 F. Qi nd B.-N. Guo, Explicit nd recursive formuls, integrl representtions, nd properties of the lrge Schröder numbers, Krgujevc J. Mth. 4 7, no., 4. 6 F. Qi nd B.-N. Guo, The reciprocl of the geometric men of mny positive numbers is Stieltjes trnsform, J. Comput. Appl. Mth. 3 7, 65 7; Avilble online t F. Qi nd B.-N. Guo, The reciprocl of the weighted geometric men is Stieltjes function, Bol. Soc. Mt. Mex , in press; Avilble online t 8 F. Qi nd B.-N. Guo, The reciprocl of the weighted geometric men is Stieltjes trnsform, ReserchGte Working Pper 6, vilble online t 9 F. Qi nd B.-N. Guo, The reciprocl of the weighted geometric men of mny positive numbers is Stieltjes function, ReserchGte Working Pper 6, vilble online t F. Qi, X.-T. Shi, nd B.-N. Guo, Integrl representtions of the lrge nd little Schröder numbers, ReserchGte Working Pper 6, vilble online t F. Qi, X.-T. Shi, nd B.-N. Guo, Some properties of the Schröder numbers, Indin J. Pure Appl Mth. 47 6, no. 4, 77 73; Avilble online t F. Qi, X.-T. Shi, nd B.-N. Guo, Some properties of the Schröder numbers, ReserchGte Working Pper 6, vilble online t 3 F. Qi, X.-T. Shi, F.-F. Liu, nd Z.-H. Yng, A double inequlity for n integrl men in terms of the exponentil nd logrithmic mens, Period. Mth. Hungr. 7 7, in press; Avilble online t 7/s F. Qi, X.-T. Shi, F.-F. Liu, nd Z.-H. Yng, A double inequlity for n integrl men in terms of the exponentil nd logrithmic mens, ReserchGte Reserch 5, vilble online t 5 F. Qi nd A. Sofo, An lterntive nd united proof of double inequlity for bounding the rithmetic-geometric men, Politehn. Univ. Buchrest Sci. Bull. Ser. A Appl. Mth. Phys. 7 9, no. 3, F. Qi nd A. Sofo, An lterntive nd united proof of double inequlity for bounding the rithmetic-geometric men, vilble online t 7 F. Qi, X.-J. Zhng, nd W.-H. Li, A new proof of the geometric-rithmetic men inequlity by Cuchy s integrl formul, rxiv preprint 3, vilble online t 8 F. Qi, X.-J. Zhng, nd W.-H. Li, An elementry proof of the weighted geometric men being Bernstein function, Politehn. Univ. Buchrest Sci. Bull. Ser. A Appl. Mth. Phys. 77 5, no.,

10 F. QI AND B.-N. GUO 9 F. Qi, X.-J. Zhng, nd W.-H. Li, An integrl representtion for the weighted geometric men nd its pplictions, Act Mth. Sin. Engl. Ser. 3 4, no., 6 68; Avilble online t s F. Qi, X.-J. Zhng, nd W.-H. Li, Integrl representtions of the weighted geometric men nd the logrithmic men, rxiv preprint 3, vilble online t 3 F. Qi, X.-J. Zhng, nd W.-H. Li, Lévy-Khintchine representtion of the geometric men of mny positive numbers nd pplictions, Mth. Inequl. Appl. 7 4, no., 79 79; Avilble online t org/.753/mi F. Qi, X.-J. Zhng, nd W.-H. Li, Lévy-Khintchine representtions of the weighted geometric men nd the logrithmic men, Mediterr. J. Mth. 4, no., 35 37; Avilble online t 7/s9-3-3-z. 33 F. Qi, X.-J. Zhng, nd W.-H. Li, Some Bernstein functions nd integrl representtions concerning hrmonic nd geometric mens, rxiv preprint 3, vilble online t 34 F. Qi, X.-J. Zhng, nd W.-H. Li, The geometric men is Bernstein function, rxiv preprint 3, vilble online t 35 F. Qi, X.-J. Zhng, nd W.-H. Li, The hrmonic nd geometric mens re Bernstein functions, Bol. Soc. Mt. Mex. 3 6, in press; Avilble online t 36 R. L. Schilling, R. Song, nd Z. Vondrček, Bernstein Functions Theory nd Applictions, nd ed., de Gruyter Studies in Mthemtics 37, Wlter de Gruyter, Berlin, Germny,. 37 G. Toder, Some men vlues relted to the rithmetic-geometric men, J. Mth. Anl. Appl , no., ; Avilble online t 38 G. Toder nd T. M. Rssis, New properties of some men vlues, J. Mth. Anl. Appl , no., ; Avilble online t 39 D. V. Widder, The Lplce Trnsform, Princeton Mthemticl Series 6, Princeton University Press, Princeton, N. J., Z.-H. Yng nd Y.-M. Chu, A shrp lower bound for Toder-Qi men with pplictions, J. Funct. Spces 6, Art. ID 4656, 5 pges; Avilble online t 4 Z.-H. Yng nd Y.-M. Chu, On pproximting the modified Bessel function of the first kind nd Toder-Qi men, J. Inequl. Appl. 6, 6:4, pges; Avilble online t 4 Z.-H. Yng, Y.-M. Chu, nd Y.-Q. Song, Shrp bounds for Toder-Qi men in terms of logrithmic nd identric mens, Mth. Inequl. Appl. 9 6, no., 7 73; Avilble online t mi-9-5. c 7 by the uthors; licensee Preprints, Bsel, Switzerlnd. This rticle is n open ccess rticle distributed under the terms nd conditions of the Cretive Commons Attribution CC BY license URL: URL: