INEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION BAI-NI GUO AND FENG QI Abstrct. In the rticle, using the Tchebycheff s integrl inequlity, the suitble properties of double integrl nd the Cuchy s men vlue theorem in integrl form, the following result is proved: Suppose f(x is positive differentible function nd p(x 0 n integrble nonnegtive weight on the intervl [, b], if f (x nd f (x/p(x re integrble nd both incresing or both decresing, then for ll rel numbers r nd s, we hve ( M p,f (r, s;, b < E ( r + 1, s + 1; f(, f(b ; if one of the functions f (x or f (x/p(x is nondecresing nd the other nonincresing, then the inequlity ( reverses. Where M p,f (r, s;, b nd E(r, s;, b denote the generlized weighted men vlues of function f with two prmeters r, s nd weight p nd the extended men vlues, respectively. This inequlity ( generlizes the Hermite-Hdmrd s inequlity, nd the like. 1. Introduction Let f : [, b] R be convex function, then ( + b f 1 (1 f(tdt b f( + f(b. The inequlity (1 is clled Hermite-Hdmrd s inequlity in [4] nd [5, pp. 10 1]. The middle term of inequlity (1 is clled the rithmetic men of the function f(x on the intervl [, b], the right term in (1 is the rithmetic men of numbers f( nd f(b. Let f(x be positive integrble function on the intervl [, b], then the power men of f(x is defined s follows ( f α (xdx 1/α, α 0, ( M α (f = exp b ( ln f(xdx b, α = 0. The generlized logrithmic men (or Stolrsky s men on the intervl [, b] is defined for x > 0, y > 0 by 1991 Mthemtics Subject Clssifiction. Primry 6D15, Secondry 6B15. Key words nd phrses. inequlity, generlized weighted men vlues, convex function, power men, two-prmeter men, Hermite-Hdmrd s inequlity, Tchebycheff s integrl inequlity. The uthors were supported in prt by NSF of Henn Province, SF of the Eduction Committee of Henn Province (No. 1999110004, nd Doctor Fund of Jiozuo Institute of Technology, The People s Republic of Chin. This pper is typeset using AMS-LATEX. 1
( x α y α 1/(α 1, α 0, 1 x y 0; (3 α(x y y x, α = 0, x y 0; S α (x, y = ln y ln x ( 1 x x 1/(x y e y y, α = 1, x y 0; x, x y = 0. In [5, p. 1] nd [13], Zhen-Hng Yng hs given the following generliztions of the Hermite-Hdmrd s inequlity (1: If f(x > 0 hs derivtive of second order nd f (x > 0, for λ > 1, we hve (i f λ( + b < 1 f λ (xdx < f λ ( + f λ (b ; b ( + b (ii f < M λ (f < N λ (f(, f(b, where x λ + y λ, λ 0, N λ (x, y = xy, λ = 0; ( (iii For ll rel number α, M α (f < S α+1 f(, f(b ; ( + b ( (iv For α 1, f < M α (f < S α+1 f(, f(b. (v If f (x < 0 for x (, b, the bove inequlities re ll reversed. In [11], two-prmeter men is defined s ( f p (xdx 1/(p q M p,q (f = f, p q, q (xdx (4 ( exp f p (x ln f(xdx f, p = q. p (xdx When q = 0, M p,0 (f = M p (f; when f(x = x, the two-prmeter men is reduced to the extended men vlues E(r, s; x, y for positive x nd y: (5 (6 (7 [ r E(r, s; x, y = s ys x s ] 1/(s r y r x r, rs(r s(x y 0; [ 1 E(r, 0; x, y = r y r x r ] 1/r, r(x y 0; ln y ln x ( x E(r, r; x, y = e 1/r x r 1/(x r y r, r(x y 0; y yr E(0, 0; x, y = xy, x y; E(r, s; x, x = x, x = y. In 1997, Ming-Bo Sun [11] generlized Hermite-Hdmrd s inequlity (1 nd the results derived by Yng in [5, 13] to obtin tht, if the positive function f(x
hs derivtive of second order nd f (x > 0, then, for ll rel numbers p nd q, (8 M p,q (f < E ( p + 1, q + 1; f(, f(b. If f (x < 0, then inequlity (8 is reversed. Recently the first uthor estblished in [7, 8] the generlized weighted men vlues M p,f (r, s; x, y of positive function f defined on the intervl between x nd y with two prmeters r, s R nd nonnegtive weight p 0 by ( y x M p,f (r, s; x, y = p(uf s 1/(s r y (udu x p(uf, (r s(x y 0; r (9 ( y x M p,f (r, r; x, y = exp p(uf r (u ln f(udu y x p(uf, x y 0; r (udu M p,f (r, s; x, x = f(x, x = y. It is well-known tht the concepts of mens nd their inequlities not only re bsic nd importnt concepts in mthemtics (for exmple, some definitions of norms re often specil mens nd hve explicit geometric menings [10], but lso hve pplictions in electrosttics [6], het conduction nd chemistry [1]. Moreover, some pplictions to medicine re given in [1]. In this rticle, using the Tchebycheff s integrl inequlity, suitble properties of double integrl nd the Cuchy s men vlue theorem in integrl form, the following result is obtined: Min Theorem. Suppose f(x is positive differentible function nd p(x 0 n integrble nonnegtive weight on the intervl [, b], if f (x nd f (x/p(x re both incresing or both decresing nd integrble, then for ll rel numbers r nd s, we hve ( M p,f (r, s;, b < E ( r + 1, s + 1; f(, f(b ; if one of the functions f (x or f (x/p(x is nondecresing nd the other nonincresing, then the inequlity ( reverses.. Proof of Min Theorem In order to verify the Min Theorem, the following lemms re necessry. Lemm 1. Let G, H : [, b] R be integrble functions, both incresing or both decresing. Furthermore, let Q : [, b] [0, + be n integrble function. Then (10 Q(uG(udu Q(uH(udu Q(udu Q(uG(uH(udu, with equlity if nd only if one of the functions G nd H reduces to constnt. If one of the functions of G or H is nonincresing nd the other nondecresing, then the inequlity (10 reverses. Inequlity (10 is clled the Tchebycheff s integrl inequlity [, 5]. Lemm ([9]. Suppose tht f(t nd g(t 0 re integrble on [, b] nd the rtio f(t/g(t hs finitely mny removble discontinuity points. Then there exists t lest one point θ (, b such tht f(tdt (11 g(tdt = lim t θ 3 f(t g(t.
We cll Lemm 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 in stndrd textbook of mthemticl nlysis or clculus, there exists t lest one point θ (, b stisfying (1 Lemm follows. f(tdt = ( f(t g(tdt = f(θ g(t g(θ g(tdt. Proof of Min Theorem. It is sufficient to prove the Min Theorem only for s > r nd for f (x nd f (x/p(x both being incresing. The remining cses cn be done similrly. Cse 1. When s > r nd f( f(b, inequlity ( is equivlent to p(xf s (xdx f r (xf (xdx < b (13 p(xf r (xdx f s (xf (xdx. Tke G(x = f s r (x, H(x = f (x/p(x (being incresing nd Q(x = p(xf r (x 0 in inequlity (10. If f (x > 0, then f s r (x is incresing, inequlity (13 holds. If f (x < 0, then f s r (x decreses, inequlity (13 is still vlid. If f (x does not keep the sme sign on (, b, then there exists n unique point θ (, b such tht f (x > 0 on (θ, b nd f (x < 0 on (, θ. Further, if f( < f(b, then there exists n unique point (θ, b such tht f( = f(. Therefore, inequlity (13 is lso equivlent to (14 p(xf s (xdx f r (xf (xdx < Using inequlity (10 gin produces (15 p(xf s (xdx f r (xf (xdx < p(xf r (xdx p(xf r (xdx f s (xf (xdx. f s (xf (xdx. For x (,, y (, b, we hve f (y > 0, f(x < f( = f( < f(y nd f s r (x < f s r (y, therefore, suitble properties of double integrl leds to (16 = p(xf s (xdx f r (xf (xdx p(xf r (xdx f s (xf (xdx p(xf r (xf r (yf (y [ f s r (x f s r (y ] dxdy < 0. [,] [,b] From this, we conclude tht inequlity (14 is vlid, nmely, inequlity (13 holds. If f (x does not keep the sme sign on (, b nd f(b < f(, from the sme rguments s the cse of f(b > f(, inequlity (13 follows. Cse. When s > r nd f( = f(b, since f (x increses, we hve f(x < f( = f(b, x (, b. From the definition of E(r, s; x, y, inequlity ( is equivlent to (17 M p,f (r, s;, b < f( = f(b, 4
tht is (18 p(xf s (xdx p(xf r (xdx < f s r ( = f s r (b. This follows from Lemm. The proof of Min Theorem is completed. 3. Applictions It is well-known tht men S 0 is clled the logrithmic men denoted by L, nd S 1 the identric men or the exponentil men by I. The logrithmic men L(x, y cn be generlized to the one-prmeter mens: (19 J p (x, y = p(yp+1 x p+1 (p + 1(y p x p, x y, p 0, 1; J 0 (x, y = L(x, y, J p (x, x = x. J 1 (x, y = G L ; Here, J 1/ (x, y = h(x, y is clled the Heron s men nd J (x, y = c(x, y the centroidl men. Moreover, J (x, y = H(x, y, J 1 (x, y = A(x, y, J 1/ (x, y = G(x, y. The extended Heron s mens h n (x, y is defined by (0 h n (x, y = 1 n + 1 x1+1/n y 1+1/n x 1/n y 1/n. Let f nd p be defined nd integrble functions on the closed intervl [, b]. The weighted men M [r] (f; p; x, y of order r of the function f on [, b] with the weight p is defined [3, pp. 75 76] by (1 ( y x p(tf r (tdt 1/r y M [r] x (f; p; x, y = p(tdt, r 0; ( y p(t ln f(tdt x exp y, x p(tdt r = 0. It is cler tht M [r] (f; p; x, y = M p,f (r, 0; x, y, E(r, s; x, y = M 1,x (r 1, s 1; x, y, E(r, r + 1; x, y = J r (x, y. From these definitions of men vlues nd some reltionships between them, we cn esily get the following inequlities: Corollry. Let f(x be positive differentible function nd p(x 0 n integrble nonnegtive weight on the intervl [, b]. If f (x nd f (x/p(x re integrble nd 5
both incresing or both decresing, then for ll rel numbers r, s, we hve ( (3 M [r] (f; p;, b < S r+1 ( f(, f(b, Mp,f (0, 1;, b < L ( f(, f(b, M p,f (0, 0;, b < I ( f(, f(b, M p,f (0, 1;, b < A ( f(, f(b, (4 M p,f ( 1, 1;, b < G ( f(, f(b, M p,f ( 3, ;, b < H ( f(, f(b, (5 ( 1 M p,f, 1 ;, b < h ( f(, f(b ( 1, M p,f n, 1 n 1;, b ( < h n f(, f(b, M p,f (, 1;, b < c ( f(, f(b, M p,f ( 1, ;, b < G( f(, f(b (6 L ( f(, f(b, (7 ( M p,f (r, r + 1;, b < J r f(, f(b. If one of the functions f (x or f (x/p(x is nondecresing nd the other nonincresing, then ll of the inequlities from ( to (7 reverse. Remrk 1. If tke p(x 1 nd specil vlues of r nd s in Min Theorem or Corollry, we cn derive the Hermite-Hdmrd s inequlity (1 nd ll of the relted inequlities in [5, 11, 13], nd the like. Remrk. The men M p,f (0, 1;, b is clled the weighted rithmetic men, M p,f ( 1, 1;, b the weighted geometric men, M p,f ( 3, ;, b the weighted hrmonic men of the function f(x on the intervl [, b] with weight p(x, respectively. So, we cn seemingly cll M [r] (f; p;, b, M p,f (0, 1;, b, M p,f (0, 0;, b, M p,f ( 1 n, 1 n 1;, b nd M p,f (1, ;, b the weighted Stolrsky s (or generlized logrithmic men, the weighted logrithmic men, the weighted exponentil men, the weighted Heron s men nd the weighted centroidl men of the function f(x on the intervl [, b] with weight p(x, respectively. References 1. Yong Ding, Two clsses of mens nd their pplictions, Mthemtics in Prctice nd Theory 5 (1995, no., 16 0. (Chinese. Ji-Chng Kung, Applied Inequlities, nd edition, Hunn Eduction Press, Chngsh, Chin, 1993. (Chinese 3. D. S. Mitrinović, Anlytic Inequlities, Springer-Verlg, Berlin, 1970. 4. D. S. Mitrinović nd I. Lcković, Hermite nd convexity, Aequt. Mth. 8 (1985, 9 3. 5. D. S. Mitrinović, J. E. Pečric nd A. M. Fink, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers, Dordrecht/Boston/London, 1993. 6. G. Póly nd G. Szegö, Isoperimetric Inequlities in Mthemticl Physics, Princeton University Press, Princeton, 1951. 7. Feng Qi, Generlized weighted men vlues with two prmeters, Proceedings of the Royl Society of London Series A 454 (1998, no. 1978, 73 73. 8. Feng Qi, On two-prmeter fmily of nonhomogeneous men vlues, Tmkng Journl of Mthemtics 9 (1998, no., 155 163. 9. Feng Qi, Sen-Lin Xu, nd Lokenth Debnth, A new proof of monotonicity for extended men vlues, Intern. J. Mth. Mth. Sci. (1999, no., 415 40. 10. Feng Qi nd Qiu-Ming Luo, Refinements nd extensions of n inequlity, Mthemtics nd Informtics Qurterly 9 (1999, no. 1, 3 5. 11. Ming-Bo Sun, Inequlities for two-prmeter men of convex function, Mthemtics in Prctice nd Theory 7 (1997, no. 3, 193 197. (Chinese 1. K. Tettmnti, G. Sárkány, D. Krâlik nd R. Stomfi, Über die nnäherung logrithmischer funktionen durch lgebrische funktionen, Period. Polytech. Chem. Engrg. 14 (1970, 99 111. 6
13. Zhen-Hng Yng, Inequlities for power men of convex function, Mthemtics in Prctice nd Theory 0 (1990, no. 1, 93 96. (Chinese Deprtment of Mthemtics, Jiozuo Institute of Technology, Jiozuo City, Henn 454000, The People s Republic of Chin E-mil ddress: qifeng@jzit.edu.cn 7