GENERALIZED ABSTRACTED MEAN VALUES

GENERALIZED ABSTRACTED MEAN VALUES FENG QI Abstrct. In this rticle, the uthor introduces the generlized bstrcted men vlues which etend the concepts of most mens with two vribles, nd reserches their bsic properties nd monotonicities. 1. Introduction The simplest nd clssicl mens re the rithmetic men, the geometric men, nd the hrmonic men. For positive sequence = ( 1,..., n ), they re defined respectively by (1.1) A n () = 1 n i, G n () = n n i, H n () = n For positive function f defined on [, y], the integrl nlogues of (1.1) re given by (1.2) (1.3) (1.4) It is well-known tht A(f) = 1 f(t) dt, y ( 1 y ) G(f) = ep ln f(t) dt, y H(f) = y. dt f(t). 1 i (1.5) A n () G n () H n (), A(f) G(f) H(f) re clled the rithmetic men geometric men hrmonic men inequlities. These clssicl mens hve been generlized, etended nd refined in mny different directions. The study of vrious mens hs rich literture, for detils, plese refer to [1, 2], [4] [8] nd [19], especilly to [9], nd so on. Some men vlues lso hve pplictions in medicine [3, 18]. 1991 Mthemtics Subject Clssifiction. Primry 26A48; Secondry 26D15. Key words nd phrses. Generlized bstrcted men vlues, bsic property, monotonicity. The uthor ws prtilly supported by NSF of Henn Province, The People s Republic of Chin. This pper is typeset using AMS-LATEX. 1

2 FENG QI Recently, the uthor [9] introduced the generlized weighted men vlues M p,f (r, s;, y) with two prmeters r nd s, which re defined by (1.6) (1.7) M p,f (r, s;, y) = M p,f (r, r;, y) = ep ( p(u)f s (u)du p(u)f r (u)du) 1/(s r), (r s)( y) 0; ( p(u)f ) r (u) ln f(u)du p(u)f, y 0; r (u)du M p,f (r, s;, ) = f(), where, y, r, s R, p(u) 0 is nonnegtive nd integrble function nd f(u) positive nd integrble function on the intervl between nd y. It ws shown in [9, 17] tht M p,f (r, s;, y) increses with both r nd s nd hs the sme monotonicities s f in both nd y. Sufficient conditions in order tht (1.8) (1.9) M p1,f (r, s;, y) M p2,f (r, s;, y), M p,f1 (r, s;, y) M p,f2 (r, s;, y) were lso given in [9]. It is cler tht M p,f (r, 0;, y) = M [r] (f; p;, y). For the definition of M [r] (f; p;, y), plese see [6]. Remrk 1. As concrete pplictions of the monotonicities nd properties of the generlized weighted men vlues M p,f (r, s;, y), some monotonicity results nd inequlities of the gmm nd incomplete gmm functions re presented in [10]. Moreover, n inequlity between the etended men vlues E(r, s;, y) nd the generlized weighted men vlues M p,f (r, s;, y) for conve function f is given in [14], which generlizes the well-known Hermite-Hdmrd inequlity. The min purposes of this pper re to estblish the definitions of the generlized bstrcted men vlues, to reserch their bsic properties, nd to prove their monotonicities. In Section 2, we introduce some definitions of men vlues nd study their bsic properties. In Section 3, the monotonicities of the generlized bstrcted men vlues, nd the like, re proved. 2. Definitions nd Bsic Properties Definition 1. Let p be defined, positive nd integrble function on [, y] for, y R, f rel-vlued nd monotonic function on [α, β]. If g is function vlued on [α, β] nd f g integrble on [, y], the qusi-rithmetic non-symmetricl men of g is defined by (2.1) M f (g; p;, y) = f 1 ( where f 1 is the inverse function of f. p(t)f(g(t)) dt p(t) dt ), For g(t) = t, f(t) = t r 1, p(t) = 1, the men M f (g; p;, y) reduces to the etended logrithmic mens S r (, y); for p(t) = t r 1, g(t) = f(t) = t, to the one-prmeter men J r (, y); for p(t) = f (t), g(t) = t, to the bstrcted men M f (, y); for g(t) = t, p(t) = t r 1, f(t) = t s r, to the etended men vlues E(r, s;, y); for f(t) = t r, to the weighted men of order r of the function g with weight p on [, y]. If we replce p(t) by p(t)f r (t), f(t) by t s r, g(t) by f(t) in

GENERALIZED ABSTRACTED MEAN VALUES 3 (2.1), then we get the generlized weighted men vlues M p,f (r, s;, y). Hence, from M f (g; p;, y) we cn deduce most of the two vrible mens. Lemm 1 ([13]). Suppose tht f nd g re integrble, nd g is non-negtive, on [, b], nd tht the rtio f(t)/g(t) hs finitely mny removble discontinuity points. Then there eists t lest one point θ (, b) such tht (2.2) f(t)dt f(t) = lim g(t)dt t θ g(t). We cll Lemm 1 the revised Cuchy s men vlue theorem in integrl form. Proof. Since f(t)/g(t) hs finitely mny removble discontinuity points, without loss of generlity, suppose it is continuous on [, b]. Furthermore, using g(t) 0, from the men vlue theorem for integrls, there eists t lest one point θ (, b) stisfying ( ) f(t) f(t)dt = g(t)dt = f(θ) (2.3) g(t)dt. g(t) g(θ) Lemm 1 follows. Theorem 1. The men M f (g; p;, y) hs the following properties: (2.4) α M f (g; p;, y) β, M f (g; p;, y) = M f (g; p; y, ), where α = inf g(t) nd β = sup g(t). t [,y] t [,y] Proof. This follows from Lemm 1 nd stndrd rguments. Definition 2. For sequence of positive numbers = ( 1,..., n ) nd positive weights p = (p 1,..., p n ), the generlized weighted men vlues of numbers with two prmeters r nd s is defined s (2.5) (2.6) 1/(r s) p i r i M n (p; ; r, s) =, r s 0; p i s i p i r i ln i M n (p; ; r, r) = ep. p i r i For s = 0 we obtin the weighted men M [r] n (; p) of order r which is defined in [2, 5, 6, 7] nd introduced bove; for s = 0, r = 1, the weighted hrmonic men; for s = 0, r = 0, the weighted geometric men; nd for s = 0, r = 1, the weighted rithmetic men. The men M n (p; ; r, s) hs some bsic properties similr to those of M p,f (r, s;, y), for instnce

4 FENG QI Theorem 2. The men M n (p; ; r, s) is continuous function with respect to (r, s) R 2 nd hs the following properties: (2.7) M s r n m M n (p; ; r, s) M, M n (p; ; r, s) = M n (p; ; s, r), (p; ; r, s) = M s t n where m = min 1 i n { i}, M = m 1 i n { i}. (p; ; t, s) M t r (p; ; r, t), Proof. For n rbitrry sequence b = (b 1,..., b n ) nd positive sequence c = (c 1,..., c n ), the following elementry inequlities [6, p. 204] re well-known (2.8) { bi } min 1 i n c i b i c i m 1 i n n { bi c i }. This implies the inequlity property. The other properties follow from stndrd rguments. Definition 3. Let f 1 nd f 2 be rel-vlued functions such tht the rtio f 1 /f 2 is monotone on the closed intervl [α, β]. If = ( 1,..., n ) is sequence of rel numbers from [α, β] nd p = (p 1,..., p n ) sequence of positive numbers, the generlized bstrcted men vlues of numbers with respect to functions f 1 nd f 2, with weights p, is defined by ( ) 1 p f1 i f 1 ( i ) (2.9) M n (p; ; f 1, f 2 ) = f 2, p i f 2 ( i ) where (f 1 /f 2 ) 1 is the inverse function of f 1 /f 2. The integrl nlogue of Definition 3 is given by Definition 4. Let p be positive integrble function defined on [, y],, y R, f 1 nd f 2 rel-vlued functions nd the rtio f 1 /f 2 monotone on the intervl [α, β]. In ddition, let g be defined on [, y] nd vlued on [α, β], nd f i g integrble on [, y] for i = 1, 2. The generlized bstrcted men vlues of function g with respect to functions f 1 nd f 2 nd with weight p is defined s ( ) 1 ( f1 M(p; g; f 1, f 2 ;, y) = p(t)f ) 1(g(t)) dt (2.10) p(t)f, 2(g(t)) dt where (f 1 /f 2 ) 1 is the inverse function of f 1 /f 2. f 2 Remrk 2. Set f 2 1 in Definition 4, then we cn obtin Definition 1 esily. Replcing f by f 1 /f 2, p(t) by p(t)f 2 (g(t)) in Definition 1, we rrive t Definition 4 directly. Anlogously, formul (2.9) is equivlent to M f (; p), see [6, p. 77]. Definition 1 nd Definition 4 re equivlent to ech other. Similrly, so re Definition 3 nd the qusi-rithmetic non-symmetricl men M f (; p) of numbers = ( 1,..., n ) with weights p = (p 1,..., p n ).

GENERALIZED ABSTRACTED MEAN VALUES 5 Lemm 2. Suppose the rtio f 1 /f 2 is monotonic on given intervl. Then ( ) 1 ( ) 1 ( ) f1 f2 1 (2.11) () =, where (f 1 /f 2 ) 1 is the inverse function of f 1 /f 2. f 2 Proof. This is direct consequence of the definition of n inverse function. Theorem 3. The mens M n (p; ; f 1, f 2 ) nd M(p; g; f 1, f 2 ;, y) hve the following properties: (i) Under the conditions of Definition 3, we hve (2.12) f 1 m M n (p; ; f 1, f 2 ) M, M n (p; ; f 1, f 2 ) = M n (p; ; f 2, f 1 ), where m = min { i}, M = m { i}; 1 i n 1 i n (ii) Under the conditions of Definition 4, we hve (2.13) where α = α M(p; g; f 1, f 2 ;, y) β, M(p; g; f 1, f 2 ;, y) = M(p; g; f 1, f 2 ; y, ), M(p; g; f 1, f 2 ;, y) = M(p; g; f 2, f 1 ;, y), inf g(t) nd β = sup g(t). t [,y] t [,y] Proof. These follow from inequlity (2.8), Lemm 1, Lemm 2, nd stndrd rguments. 3. Monotonicities Lemm 3 ([16]). Assume tht the derivtive of second order f (t) eists on R. If f(t) is n incresing (or conve) function on R, then the rithmetic men of function f(t), 1 s f(t) dt, r s, (3.1) φ(r, s) = s r r f(r), r = s, is lso incresing (or conve, respectively) with both r nd s on R. Proof. Direct clcultion yields (3.2) φ(r, s) s = 1 [ (s r) 2 (s r)f(s) s r ] f(t)dt, s r f(t)dt 2 φ(r, s) s 2 = (s r)2 f (3.3) (s) 2(s r)f(s) + 2 ϕ(r, s) (s r) 3 (s r) 3, ϕ(r, s) (3.4) = (s r) 2 f (s). s In the cse of f (t) 0, we hve φ(r, s)/ s 0, thus φ(r, s) increses in both r nd s, since φ(r, s) = φ(s, r). In the cse of f (t) 0, ϕ(r, s) increses with s. Since ϕ(r, r) = 0, we hve 2 φ(r, s)/ s 2 0. Therefore φ(r, s) is conve with respect to either r or s, since φ(r, s) = φ(s, r). This completes the proof.

6 FENG QI Theorem 4. The men M n (p; ; r, s) of numbers = ( 1,..., n ) with weights p = (p 1,..., p n ) nd two prmeters r nd s is incresing in both r nd s. Proof. Set N n = ln M n, then we hve (3.5) (3.6) N n (p; ; r, s) = 1 r r s s N n (p; ; r, r) = p i t i ln i dt, r s 0; p i t i p i r i ln i. p i r i By Cuchy s inequlity, direct clcultion rrives t p i t i ln i p i t i (ln i) 2 n ( ) 2 p i t i p i t i ln i (3.7) = ( ) 2 0. p i t i p i t i t Combintion of (3.7) with Lemm 3 yields the sttement of Theorem 4. Theorem 5. For monotonic sequence of positive numbers 0 < 1 2 nd positive weights p = (p 1, p 2,... ), if m < n, then (3.8) Equlity holds if 1 = 2 =. M m (p; ; r, s) M n (p; ; r, s). Proof. For r s, inequlity (3.8) reduces to m p i r i p i r i (3.9) m. p i s i p i s i Since 0 < 1 2, p i > 0, i 1, the sequences { } p i r i nd { } p i s i re positive nd monotonic. By mthemticl induction nd the elementry inequlities (2.8), we cn esily obtin the inequlity (3.9). The proof of Theorem 5 is completed. Lemm 4. If A = (A 1,..., A n ) nd B = (B 1,..., B n ) re two nondecresing (or nonincresing) sequences nd P = (P 1,..., P n ) is nonnegtive sequence, then n n (3.10) P i P i A i B i P i A i P i B i, with equlity if nd only if t lest one of the sequences A or B is constnt. If one of the sequences A or B is nonincresing nd the other nondecresing, then the inequlity in (3.10) is reversed. The inequlity (3.10) is known in the literture s Tchebycheff s (or Čebyšev s) inequlity in discrete form [7, p. 240].

GENERALIZED ABSTRACTED MEAN VALUES 7 Theorem 6. Let p = (p 1,..., p n ) nd q = (q 1,..., q n ) be positive weights, = ( 1,..., n ) sequence of positive numbers. If the sequences (p 1 /q 1,..., p n /q n ) nd re both nonincresing or both nondecresing, then (3.11) M n (p; ; r, s) M n (q; ; r, s). If one of the sequences of (p 1 /q 1,..., p n /q n ) or is nonincresing nd the other nondecresing, the inequlity (3.11) is reversed. Proof. Substitution of P = (q 1 s 1,..., q n s n), A = ( r s 1,..., r s n ) nd B = (p 1 /q 1,..., p n /q n ) into inequlity (3.10) nd the stndrd rguments produce inequlity (3.11). This completes the proof of Theorem 6. Theorem 7. Let p = (p 1,..., p n ) be positive weights, = ( 1,..., n ) nd b = (b 1,..., b n ) two sequences of positive numbers. If the sequences ( 1 /b 1,..., n /b n ) nd b re both incresing or both decresing, then (3.12) M n (p; ; r, s) M n (p; b; r, s) holds for i /b i 1, n i 1, nd r, s 0 or r 0 s. The inequlity (3.12) is reversed for i /b i 1, n i 1, nd r, s 0 or s 0 r. If one of the sequences of ( 1 /b 1,..., n /b n ) or b is nonincresing nd the other nondecresing, then inequlity (3.12) is vlid for i /b i 1, n i 1 nd r, s 0 or s 0 r; the inequlity (3.12) reverses for i /b i 1, n i 1, nd r, s 0 or r 0 s,. Proof. The inequlity (3.10) pplied to ( P i = p i b r i ) r, (3.13) i, A i = Bi = b s r i, 1 i n b i nd the stndrd rguments yield Theorem 7. Theorem 8. Suppose p nd g re defined on R. If f 1 g hs constnt sign nd if (f 1 /f 2 ) g is incresing (or decresing, respectively), then M(p; g; f 1, f 2 ;, y) hve the inverse (or sme) monotonicities s f 1 /f 2 with both nd y. Proof. Without loss of generlity, suppose (f 1 /f 2 ) g increses. By strightforwrd computtion nd using Lemm 1, we obtin ( d (3.14) p(t)f ) 1(g(t))dt dy p(t)f 2(g(t))dt = p(y)f 1(g(y)) p(t)f 1(g(t))dt ( p(t)f 2(g(t))dt ) 2 ( p(t)f ) 2(g(t))dt p(t)f 1(g(t))dt f 2(g(y)) 0. f 1 (g(y)) From Definition 4 nd its suitble bsic properties, Theorem 8 follows. Lemm 5. Let G, H : [, b] R be integrble functions, both incresing or both decresing. Furthermore, let Q : [, b] [0, + ) be n integrble function. Then (3.15) Q(u)G(u) du Q(u)H(u) du Q(u) du Q(u)G(u)H(u) du. If one of the functions of G or H is nonincresing nd the other nondecresing, then the inequlity (3.15) reverses. Inequlity (3.15) is clled Tchebycheff s integrl inequlity, plese refer to [1] nd [4] [7].

8 FENG QI Remrk 3. Using Tchebycheff s integrl inequlity, some inequlities of the complete elliptic integrls re estblished in [15], mny inequlities concerning the probbility function, the error function, nd so on, re improved in [12]. Theorem 9. Suppose f 2 g hs constnt sign on [, y]. When g(t) increses on [, y], if p 1 /p 2 is incresing, we hve (3.16) M(p 1 ; g; f 1, f 2 ;, y) M(p 2 ; g; f 1, f 2 ;, y); if p 1 /p 2 is decresing, inequlity (3.16) reverses. When g(t) decreses on [, y], if p 1 /p 2 is incresing, then inequlity (3.16) is reversed; if p 1 /p 2 is decresing, inequlity (3.16) holds. Proof. Substitution of Q(t) = f 2 (g(t))p 2 (t), G(t) = (f 1 /f 2 ) g(t) nd H(t) = p 1 (t)/p 2 (t) into Lemm 5 nd the stndrd rguments produce inequlity (3.16). The proof of Theorem 9 is completed. Theorem 10. Suppose f 2 g 2 does not chnge its sign on [, y]. (i) When f 2 (g 1 /g 2 ) nd (f 1 /f 2 ) g 2 re both incresing or both decresing, inequlity (3.17) M(p; g 1 ; f 1, f 2 ;, y) M(p; g 2 ; f 1, f 2 ;, y) holds for f 1 /f 2 being incresing, or reverses for f 1 /f 2 being decresing. (ii) When one of the functions f 2 (g 1 /g 2 ) or (f 1 /f 2 ) g 2 is decresing nd the other incresing, inequlity (3.17) holds for f 1 /f 2 being decresing, or reverses for f 1 /f 2 being incresing. Proof. The inequlity (3.15) pplied to Q(t) = p(t)(f 2 g 2 )(t), G(t) = f 2 ( ) f1 nd H(t) = g 2 (t), nd stndrd rguments yield Theorem 10. f 2 References ( g1 g 2 ) (t) [1] E. F. Beckenbch nd R. Bellmn, Inequlities, Springer, Berlin, 1983. [2] P. S. Bullen, D. S. Mitrinović nd P. M. Vsić, Mens nd Their Inequlities, D. Reidel Publ. Compny, Dordrecht/Boston/Lncster/Tokyo, 1988. [3] Yong Ding, Two clsses of mens nd their pplictions, Mthemtics in Prctice nd Theory 25 (1995), no. 2, 16 20. (Chinese) [4] G. H. Hrdy, J. E. Littlewood nd G. Póly, Inequlities, 2nd edition, Cmbridge University Press, Cmbridge, 1952. [5] Ji-Chng Kung, Applied Inequlities, 2nd edition, Hunn Eduction Press, Chngsh, Chin, 1993. (Chinese) [6] D. S. Mitrinović, Anlytic Inequlities, Springer-Verlg, Berlin, 1970. [7] D. S. Mitrinović, J. E. Pečrić nd A. M. Fink, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers, Dordrecht/Boston/London, 1993. [8] J. Pečrić, Feng Qi, V. Šimić nd Sen-Lin Xu, Refinements nd etensions of n inequlity, III, J. Mth. Anl. Appl. 227 (1998), no. 2, 439 448. [9] Feng Qi, Generlized weighted men vlues with two prmeters, Proc. Roy. Soc. London Ser. A 454 (1998), no. 1978, 2723 2732. [10] Feng Qi, Monotonicity results nd inequlities for the gmm nd incomplete gmm functions, submitted for publiction. [11] Feng Qi, Studies on Problems in Topology nd Geometry nd on Generlized Weighted Abstrcted Men Vlues, Thesis submitted for the degree of Doctor of Philosophy t University of Science nd Technology of Chin, Hefei City, Anhui Province, Chin, Winter 1998. (Chinese)

GENERALIZED ABSTRACTED MEAN VALUES 9 [12] Feng Qi, Li-Hong Cui nd Sen-Lin Xu, Some inequlities constructed by Tchebycheff s integrl inequlity, submitted for publiction. [13] Feng Qi nd Lokenth Debnth, Inequlities of power-eponentil functions, Intern. J. Mth. Mth. Sci. 23 (2000), in the press. [14] Feng Qi nd Bi-Ni Guo, Inequlities for generlized weighted men vlues of conve function, submitted for publiction. [15] Feng Qi nd Zheng Hung, Inequlities of the complete elliptic integrls, Tmkng Journl of Mthemtics 29 (1998), no. 3, 165 169. [16] Feng Qi, Sen-Lin Xu nd Lokenth Debnth, A new proof of monotonicity for etended men vlues, Intern. J. Mth. Mth. Sci. 22 (1999), no. 2, 415 420. [17] Feng Qi nd Shi-Qin Zhng, Note on monotonicity of generlized weighted men vlues, Proc. Roy. Soc. London Ser. A 455 (1999), no. 1989, 3259 3260. [18] Ming-Bo Sun, Inequlities for two-prmeter men of conve function, Mthemtics in Prctice nd Theory 27 (1997), no. 3, 193 197. (Chinese) [19] D-Feng Xi, Sen-Lin Xu, nd Feng Qi, A proof of the rithmetic men-geometric menhrmonic men inequlities, RGMIA Reserch Report Collection 2 (1999), no. 1, 85 88. Deprtment of Mthemtics, Jiozuo Institute of Technology, Jiozuo City, Henn 454000, The People s Republic of Chin E-mil ddress: qifeng@jzit.edu.cn