On the Generalized Weighted Quasi-Arithmetic Integral Mean 1

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1 Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, HIKARI Ltd, On the Generlized Weighted Qusi-Arithmetic Integrl Men 1 Hui Sun School of Mthemtics nd Computtion Science Hunn City University Yiyng, Hunn, , P. R. Chin Boyong Long School of Mthemtics Science Anhui University Heifei, Anhui, , P.R. Chin Yuming Chu College of Mthemtics nd Computtion Science Hunn City University Yiyng, Hunn, , P.R. Chin chuyuming@hutc.zj.cn Copyright c 2013 Hui Sun et l. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. Abstrct In this pper, we estblish severl inequlities for the generlized weighted qusi-rithmetic integrl men by use of the Chebyshev inequlity, Jensen inequlity nd convexity. Mthemtics Subject Clssifiction: 26E60 Keywords: generlized weighted qusi-rithmetic integrl men, Chebyshev inequlity, Jensen inequlity, convexity 1 This reserch ws supported by the Nturl Science Foundtion of Chin under Grnt

2 2040 Hui Sun, Boyong Long nd Yuming Chu 1. Introduction Let p be rel positive Lebesgue integrble function on [, b], f rel Lebesgue integrble function on [, b], nd g rel continuous nd strictly monotonic function defined on J, the rnge of f. In [6], the generlized weighted qusirithmetic integrl men of function f with respect to weight function p is defined by ( M g (p, f =M g (p, f;, b =g 1 p(xg(f(xdx p(xdx, (1.1 where g 1 denotes the inverse function to the function g. It is not difficult to verify tht α M g (p, f;, b β, where α = inf t [,b] f(t nd β = sup t [,b] f(t. Mny known mens in the integrl form of two vribles p nd f re specil cse of M g (p, f when tking the suitble functions p, f nd g. For instnce, for p(x C on [, b], where C is positive constnt, we get the generlized qusi-rithmetic integrl mens of function f ( M g (f =M g (C, f;, b =g 1 g(f(xdx ; b for f(x = x = id(x on[, b], we get the clssicl weighted qusirithmetic integrl mens ( M g (p, id =M g (p, id;, b =g 1 p(xg(xdx p(xdx ; for g(x =x = id(x onj, we get the weighted rithmetic integrl mens A(p, f =A(p, f;, b =M id (p, f;, b = p(xf(xdx p(xdx. Mens M g (p, f generlize lso other types of men, cf. [2], e.g. generlized weighted rithmetic, geometric nd hrmonic mens, logrithmic mens, intrinsic mens, power mens, one-prmeter mens, extended logrithmic mens, extended men vlues, generlized weighted men vlues, nd others. Hence, from M g (p, f we cn deduce most of the two vrible mens.

3 On the generlized weighted qusi-rithmetic integrl men 2041 Some bsic properties of mens M g (p, f relted to properties of input functions f nd g were studied in [2, 3] in connection with the weighted integrl Jensen inequlity for convex functions. In [4], some integrl inequlities nlogous to the well-known Hermite-Hdmrd inequlity for the generlized weighted qusi-rithmetic integrl men M g (p, f were estblished. And, in [1] the uthors discussed the Schur convexity of the generlized qusi-rithmetic integrl men M g (f nd the generlized weighted qusi-rithmetic integrl men M g (p,f, where p denote the derivtive of the weight p. The following Lemm 1.1 (Chebyshev inequlity cn be found in [5]. Lemm 1.1. Let p be positive Lebesgue integrble function, h nd k two Lebesgue integrble functions on [, b], both incresing or both decresing on [, b]. Then p(th(tk(tdt p(tdt p(th(tdt p(tdt p(tk(tdt b p(tdt. If one of the functions h nd k is incresing nd the other is decresing, then the bove inequlity is reversed. In generl mesure theoreticl nottion the Jensen inequlity theorem sounds s follows: let (Ω,A,μ be mesurble spce, such tht μ(ω = 1. If f is rel μ-integrble function nd φ is convex (concve function on the rnge of f, then ( φ fdμ ( φ fdμ. Ω Ω The purpose of this pper is to present some new nd interesting inequlities for the generlized weighted qusi-rithmetic integrl men. 2. Min Results Theorem 2.1. Let p be positive Lebesgue integrble function, f Lebesgue integrble function, g continuous nd strictly monotonic function, nd h positive monotonic function. Then the following sttements re true. (1 If h nd f hve the sme monotonicity, then M g (p, f M g (hp, f. (2.1 (2 If h nd f hve opposite monotonicity, then inequlity (2.1 is reversed. Proof. We only give the proof of Theorem 2.1(1 in detil. Similr rgument leds to the proof of Theorem 2.1(2. We divide the proof of inequlity (2.1 into two cses. Cse 1. h nd f re incresing. We divide the discussion into two subcses.

4 2042 Hui Sun, Boyong Long nd Yuming Chu Subcse 1.1. g is strictly incresing. Then h nd g f hve the sme monotonicity, nd the Chebyshev inequlity implies tht p(th(tg(f(tdt p(tdt p(th(tdt p(tdt p(tg(f(tdt p(tdt. Tht is p(th(tg(f(tdt p(th(tdt p(tg(f(tdt p(tdt. (2.2 From (2.2 nd the monotonicity of g we clerly see tht ( g 1 p(th(tg(f(tdt ( p(th(tdt g 1 p(tg(f(tdt p(tdt. (2.3 Therefore, inequlity (2.1 follows from (2.3. Subcse 1.2. g is strictly decresing. Then h nd g f hve opposite monotonicity, nd the Chebyshev inequlity leds to p(th(tg(f(tdt p(tdt p(th(tdt p(tdt p(tg(f(tdt p(tdt. Tht is p(th(tg(f(tdt p(th(tdt p(tg(f(tdt p(tdt. (2.4 Therefore, inequlity (2.3 follows from (2.4 nd the monotonicity of g. Cse 2. h nd f re decresing. We divide the proof into two subcses. Subcse 2.1. g is strictly incresing. Then h nd g f hve the sme monotonicity, nd the Chebyshev inequlity leds to the conclusion tht inequlity (2.2 holds gin. It follows from the monotonicity of g tht inequlity (2.1 holds. Subcse 2.2. g is strictly decresing. Then h nd g f hve opposite monotonicity, nd the Chebyshev inequlity implies tht inequlity (2.4 holds gin. Therefore, inequlity (2.1 follows from (2.4 nd the monotonicity of g. Corollry 2.2. Let p nd q be two positive monotonic functions, f monotonic function, nd g continuous nd strictly monotonic function. If p nd f hve the sme monotonicity, q nd f hve opposite monotonicity, then M g (p, f M g (f M g (q, f.

5 On the generlized weighted qusi-rithmetic integrl men 2043 Proof. Let h = 1. Then h nd f hve the sme monotonicity, nd Theorem q 2.1(1 leds to M g (f =M g (hq, f M g (q, f. The proof of the remining prt is similr. Theorem 2.3. Let g 1 nd g 2 be two continuous nd strictly monotonic functions, p positive Lebesgue integrble function, nd f Lebesgue integrble function. Then the following sttements re true. (1 If g 2 is strictly incresing nd g 2 g1 1 is convex, or g 2 is strictly decresing nd g 2 g1 1 is concve, then M g1 (p, f M g2 (p, f. (2.5 (2 If g 2 is strictly decresing nd g 2 g1 1 is convex, or g 2 is strictly incresing nd g 2 g1 1 is concve, then inequlity (2.5 is reversed. Proof. If g 2 g1 1 is convex, then the Jensen inequlity implies tht ( (g 2 g1 1 p(tg 1(f(tdt p(tdt p(tg 2(f(tdt p(tdt. If g 2 is strictly incresing, then the bove inequlity leds to ( ( g 1 1 p(tg 1(f(tdt p(tdt g 1 2 p(tg 2(f(tdt p(tdt nd inequlity (2.5 holds. The proofs of the remining prts re similr. Let g 1 = id or g 2 = id, where id is the identity trnsformtion, then Theorem 2.3 deduces the following Corollry 2.4. Corollry 2.4. Let g 1 nd g 2 be two continuous functions, p positive Lebesgue integrble function, nd f Lebesgue integrble function. If g 2 is strictly incresing nd convex or strictly decresing nd concve, nd g 1 is strictly decresing nd convex or strictly incresing nd concve, then M g1 (p, f A(p, f M g2 (p, f., Theorem 2.5. Let p be positive Lebesgue integrble function, f 1 nd f 2 two Lebesgue integrble function, nd g continuous nd strictly monotonic function. Then the following sttements re true. (1 If f 1 (t f 2 (t for ll t [, b], then M g (p, f 1 M g (p, f 2. (2.6 (2 If f 1 (t f 2 (t for ll t [, b], then inequlity (2.6 is reversed.

6 2044 Hui Sun, Boyong Long nd Yuming Chu Proof. (1 We divide the proof into two cses. Cse 1. g is strictly incresing. Then we clerly see tht for ll t [, b]. Therefore, p(tg(f 1(tdt p(tdt g(f 1 (t g(f 2 (t p(tg(f 2(tdt p(tdt. (2.7 From (2.7 nd the monotonicity of g we get inequlity (2.6. Cse 2. g is strictly decresing. Then we hve for ll t [, b], nd g(f 1 (t g(f 2 (t p(tg(f 1(tdt p(tdt p(tg(f 2(tdt p(tdt. (2.8 Therefore, inequlity (2.6 follows from (2.8 nd the monotonicity of g. The proof of prt (2 is similr. Corollry 2.6. Let p be positive Lebesgue integrble function, nd g continuous nd strictly monotonic function. If f 1 (t id(t f 2 (t for ll t [, b], then M g (p, f 1 M g (p, id M g (p, f 2. Theorem 2.7. Let g 1 nd g 2 be two continuous nd strictly monotonic functions, p positive Lebesgue integrble function, then we hve (1 If f 1 f 2, g 2 is incresing nd g 2 g1 1 is convex, or g 2 is decresing nd g 2 g1 1 is concve, then M g1 (p, f 1 M g2 (p, f 2. (2.9 (2 If f 1 f 2, g 2 is incresing nd g 2 g1 1 is concve, or g 2 is decresing nd g 2 g1 1 is convex, then inequlity (2.9 is reversed. Proof. We only give the proof of Theorem 2.7(1 in detil. If g 2 is incresing nd g 2 g1 1 is convex, or g 2 is decresing nd g 2 g1 1 is concve, then Theorem 2.3(1 implies tht M g1 (p, f 1 M g2 (p, f 1. (2.10

7 On the generlized weighted qusi-rithmetic integrl men 2045 If f 1 f 2, then Theorem 2.5(1 leds to M g2 (p, f 1 M g2 (p, f 2. (2.11 Therefore, inequlity (2.9 follows from inequlities (2.10 nd (2.11. Theorem 2.8. Let g 1 nd g 2 be two continuous nd strictly monotonic functions, p 1 nd p 2 two positive Lebesgue integrble function with p 2 = p 1 h. Then the following sttements re true. (1 If h nd f hve the sme monotonicity, g 2 is incresing nd g 2 g1 1 is convex or g 2 is decresing nd g 2 g1 1 is concve, then M g1 (p 1,f M g2 (p 2,f. (2.12 (2 If h nd f hve opposite monotonicity, g 2 is incresing nd g 2 g1 1 is concve or g 2 is decresing nd g 2 g1 1 is convex, then inequlity (2.12 is reversed. Proof. If g 2 is incresing nd g 2 g1 1 is convex, or g 2 is decresing nd g 2 g1 1 is concve, then Theorem 2.3(1 implies tht M g1 (p 1,f M g2 (p 1,f. (2.13 If h nd f hve the sme monotonicity, then Theorem 2.1(1 implies tht M g2 (p 1,f M g2 (p 2,f. (2.14 Therefore, inequlity (2.12 follows from inequlities (2.13 nd (2.14. The proof of prt (2 is similr. Theorem 2.9. Let g be continuous nd strictly monotonic function, p 1 nd p 2 two positive Lebesgue integrble functions with p 2 = p 1 h. Then the following sttements re true. (1 If f 1 f 2, nd f 1 or f 2 hve the sme monotonicity with h, then M g (p 1,f 1 M g (p 2,f 2. (2.15 (2 If f 1 f 2, nd f 1 or f 2 hve opposite monotonicity with h, then inequlity (2.15 is reversed. Proof. (1 Without loss of generlity, we ssume tht h nd f 1 hve the sme monotonicity. Then Theorem 2.1(1 leds to M g (p 1,f 1 M g (p 2,f 1. (2.16 If f 1 f 2, then Theorem 2.5(1 implies tht M g (p 2,f 1 M g (p 2,f 2. (2.17

8 2046 Hui Sun, Boyong Long nd Yuming Chu Therefore, inequlity (2.15 follows from inequlities (2.16 nd (2.17. The proof of prt (2 is similr. It is esy to see tht the following Theorem 2.10 follows from Theorems 2.1, 2.3 nd 2.5. Theorem Let g 1 nd g 2 be two strictly monotonic functions, p 1 nd p 2 two positive Lebesgue integrble functions with p 2 = p 1 h. Then the following sttements re true. (1 If f 1 f 2, f 1 or f 2 hve the sme monotonicity with h, g 2 is incresing such tht g 2 g1 1 is convex or g 2 is decresing such tht g 2 g1 1 is concve, then M g1 (p 1,f 1 M g2 (p 2,f 2. (2.18 (2 If f 1 f 2, f 1 or f 2 hve opposite monotonicity with h, g 2 is incresing such tht g 2 g1 1 is concve or g 2 is decresing such tht g 2 g1 1 is convex, then inequlity (2.18 is reversed. Theorem Let k be strictly monotonic nd differentible function on [, b]. Then the following sttements re true. (1 If f k 1 f k, (p k k nd f hve the sme monotonicity, g k is p incresing nd g k g 1 is convex, or g k is decresing nd g k g 1 is concve, then M g (p, f;, b k 1 (M g (p, f; k(,k(b. (2.19 (2 If f k 1 f k, (p k k nd f hve opposite monotonicity, g k is p decresing nd g k g 1 is convex, or g k is incresing nd g k g 1 is concve, then inequlity (2.19 is reversed. Proof. Eqution (1.1 leds to k 1 (M g (p, f; k(,k(b = (k 1 g 1 k(b k( p(xg(f(xdx k(b k( p(xdx. (2.20 Mking the chnge of the vrible x = k(t we obtin k(b (k 1 g 1 k( p(xg(f(xdx k(b p(xdx k( ( =(g k 1 p(k(tg(f(k(tk (tdt p(k(tk (tdt = M g k ((p k k,k 1 f k;, b. (2.21

9 On the generlized weighted qusi-rithmetic integrl men 2047 From equtions (2.20 nd (2.21 we hve k 1 (M g (p, f; k(,k(b = M g k ((p k k,k 1 f k;, b. (2.22 Therefore, Theorem 2.11 follows from Theorem 2.10 nd eqution (2.22. Let g = id. Then Theorem 2.11 leds to the following Corollry Corollry Let k be strictly monotonic nd differentible function on [, b]. Then the following sttements re true. (1 If f k 1 (p k k f k, nd f hve the sme monotonicity, k is p incresing nd convex, or k is decresing nd concve, then A(p, f;, b k 1 (A(p, f; k(,k(b. (2.23 (p k k p (2 If f k 1 f k, nd f hve opposite monotonicity, k is decresing nd convex, or k is incresing nd concve, then inequlity (2.23 is reversed. Let p = C, where C is positive constnt. Then Corollry 2.12 leds to the following Corollry Corollry Let k be strictly monotonic nd differentible function on [, b]. Then the following sttements re true. (1 If f k 1 f k, k nd f hve the sme monotonicity, k is incresing nd convex, or k is decresing nd concve, then f(tdt b k(b k 1 k( f(tdt. (2.24 k(b k( (2 If f k 1 f k, k nd f hve opposite monotonicity, k is decresing nd convex, or k is incresing nd concve, then inequlity (2.24 is reversed. References [1] V. Čuljk, I. Frnjić, G. Roqi nd J. Pečrić, Schur-convexity of verges of convex functions, J. Inequl. Appl. (2011, Article ID , 25 pges. [2] J. Hlušk nd O. Hutník, Some inequlities involving integrl mens, Ttr Mt. Mth. Publ., 35 (2007, [3] J. Hlušk nd O. Hutník, On generlized weighted qusi-rithmetic mens in integrl form, J. Electr. Eng., 56 (2005, 3-6.

10 2048 Hui Sun, Boyong Long nd Yuming Chu [4] O. Hutník, On Hdmrd type inequlities for generlized weighted qusirithmetic mens, JIPAM. J. Inequl. Pure Appl. Mth., 7(2006, no. 3, Article 96, 10 pges. [5] D. S. Mitrinović, J. E. Pečrić nd A. M. Fink, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers Group, Dordrecht, [6] F. Qi, Generlized bstrcted men vlues, JIPAM. J. Inequl. Pure Appl. Mth., 1(2000, no. 1, Article 4, 9 pges. Received: April 5, 2013