SOME INEQUALITIES INVOLVING INTEGRAL MEANS. Introduction

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1 SOME INEQUALITIES INVOLVING INTEGRAL MEANS JÁN HALUŠKA nd ONDREJ HUTNÍK Abstrct. A clss of generlized weighted qusi-rithmetic mens in the integrl form M [,b,g p, f is studied using the weighted integrl form of Jensen s inequlity. In prticulr, vrious inequlities nd properties of the generlized weighted qusirithmetic mens re estblished with respect to the properties of input functions p, f nd g. Some well known inequlities s consequence of our results re derived. Introduction In the discrete cse, men of nonnegtive n-tuple of rel numbers =,..., n with respect to weight vector p = p,..., p n of positive rel numbers, where n i= p i =, n N the set of ll positive integers, is defined with the formul n Mp, = ϕ p i ϕ i, where ϕ is continuous nd strictly monotone function which hs its inverse function ϕ stisfying the sme conditions, [6. Here ϕ is clled the Kolmogoroff- Ngumo function ssocited with. The men of the form is lso referred to s the qusi-rithmetic men. This clss of mens comprises the commonly used lgebric mens nd lso other types of ggregtion opertors. A considerble mount of literture bout the concept of men or verge nd the properties of severl mens e.g. the rithmetic men, the geometric men, the power men, the hrmonic men hs been ppered in the 9th century. A. N. Kolmogoroff [ nd M. Ngumo [3 were the first who investigted the chrcteristic properties of mens in generl. They considered mostly the cse of equl weights. The generliztion to rbitrry weights nd the chrcteriztion of mens of the form re due to B. de Finetti [3, B. Jessen [8, T. Kitgw [9, J. Aczél [ nd mny others. Integrl nlogue of mens ws estblished in 93ties by A. N.Kolmogoroff, M. Ngumo nd B. de Finetti. They showed tht ll types of the so clled intrinsic i= 2 Mthemtics Subject Clssifiction. Primry 26D, 26D5. Key words nd phrses. Generlized weighted men, integrl men, Jensen s inequlity. This pper ws supported by grnt VEGA 2/565/5.

2 2 JÁN HALUŠKA nd ONDREJ HUTNÍK mens my be expressed vi M F ϕ = ϕ R ϕxdfx, 2 where Fx is distribution nd ϕx is continuous rel incresing function on R the set of ll rel numbers, cf. [6. The integrl 2 is ment in the sense of Lebesgue-Stieltjes. This form of mens coincides with clss of the so clled integrl ϕ-mens. Recently, it is well known tht mny types of mens my be rewritten ccording to the pttern given with the formul 2, e.g. the Stolrsky s mens [9. In this pper we study the qusi-rithmetic non-symmetricl weighted men 3 proposed by F. Qi in [8. In Section 2, we stte n nlogue of Jensen s inequlity for the weighted integrl mens s well s its conversion. This enbles us to derive vrious inequlities for mens M [,b,g p, f, < b, with respect to the convexity property of functions f nd g. Also comprison theorem between generlized weighted qusi-rithmetic mens is estblished. As consequence we derive the weighted integrl version of Wng-Wng s geometric-hrmonic men inequlity, Chebyshev s integrl inequlity nd the well known Póly-Knopp s inequlity.. Preliminries Let [, b R, < b, be n intervl. Denote by L [, b the vector spce of ll rel Lebesgue integrble functions defined on the intervl [, b with the clssicl Lebesgue mesure. Let us denote by L + [, b the positive cone of L [, b, i.e. the vector spce of ll rel positive Lebesgue integrble functions on [, b. In wht follows p [,b denotes the finite L -norm of function p L + [, b. Definition.. Let p, f L + [, b L+ [, b nd g : [, R be rel continuous nd strictly monotone function. The generlized weighted qusirithmetic men of function f with respect to weight function p is number M [,b,g p, f R, where M [,b,g p, f = g b pxgfx dx, 3 p [,b where g denotes the inverse function to the function g. In wht follows, g is lwys rel continuous nd strictly monotone function in ccordnce with Definition.. Mens M [,b,g p, f include mny commonly used two vrible integrl mens s prticulr cses when tking the suitble functions p, f nd g. For instnce, for gx = x = Ix the identity function we get the weighted rithmetic men M [,b,g p, f = A [,b p, f = p [,b pxfx dx;

3 SOME INEQUALITIES INVOLVING INTEGRAL MEANS 3 b for gx = x we obtin the weighted hrmonic men M [,b,g p, f = H [,b p, f = p [,b px fx dx ; c for gx = x r we hve the weighted power men of order r b M [,b,g p, f = M [r p f; p;, b = [,b pxfxr r dx, r b exp p [,b pxln fxdx, r =. The cse r = corresponds to the weighted geometric men nd r = 2 is clled the weighted Eucliden qudrtic men. d if we replce px by pxf r x nd gx by x s r in 3, then we get the generlized weighted men vlues M [,b,g p, f = M p,f r, s;, b = pxfs xdx pxfr xdx s r ; e if px is constnt function on [, 2π nd fx is of the form fx = { u n cos 2 x + v n sin 2 x n, n u cos2 x v sin2 x, n =, then we obtin the generliztion of rithmetic-geometric men of Guss M [,b,g p, f = M g,n u, v = g 2π 2π gfxdx for n. Choosing n = 2 nd gx = x we get originl Guss rithmetic-geometric men. For n = 2 nd gt = t 2 the men my be found in [6, the cse n = nd gt = lnt ws studied in [2. Note tht mens M [,b,g p, f generlize lso logrithmic mens L, b, identric mens I, b, one-prmeter mens J r, b, bstrcted mens M g, b, extended men vlues Er, s;, b, generlized logrithmic mens S r, b, nd mny others. Hence, from M [,b,g p, f we my deduce most of the two vrible mens. Further possible extension of mens M [,b,g p, f could be considered when f is of the form fθ = hre ıθ, where < r < nd h is n nlytic function in the open unit disk D = {z : z < } of the complex plne C. In tht cse choosing =, b = 2π, gx = x q for < q < nd px on [, 2π, we get the integrl men of order q, cf. [4, M [,b,g p, f = M q r, h = 2π 2π hre ıθ q q dθ.

4 4 JÁN HALUŠKA nd ONDREJ HUTNÍK 2. Jensen s inequlity nd its conversion Mny mthemticl investigtions del with problems bout opertor of mens depending of the behvior of the input functions p, f nd g, resp. how functions behve under the ction of mens. The most known cse is tht of Jensen s convex functions, which originlly del with the rithmetic men. In generl mesure theoreticl nottion the Jensen s inequlity theorem sounds s follows: let Ω, A, µ be mesurble spce, such tht µω =. If f is rel µ-integrble function nd ϕ is convex concve function on the rnge of f, then ϕ f dµ ϕ f dµ, Ω Ω ϕ f dµ Ω Ω ϕ f dµ, cf. [7. The following Lemm 2. is direct ppliction of this generl sttement to our prticulr sitution when µ[, b = nd dµ = px p [,b dx. Lemm 2. Jensen s Inequlity. Let p, f L + [, b L+ [, b such tht α < fx < β for ll x [, b, where < α < β <. i If g is convex function on α, β, then g A [,b p, f A [,b p, g f. ii If g is concve function on α, β, then g A [,b p, f A [,b p, g f, where A [,b p, f denotes the weighted rithmetic men of the function f on [, b. Corollry 2.2. Let p, f L + [, b L+ [, b such tht α < fx < β for ll x [, b, where < α < β <. i If g is convex incresing or concve decresing function on α, β, then A [,b p, f M [,b,g p, f. ii If g is convex decresing of concve incresing function on α, β, then A [,b p, f M [,b,g p, f. Proof. Let g be convex incresing function. According to Jensen s inequlity we get which is equivlent to g g A [,b p, f g A [,b p, g f, A [,b p, f M [,b,g p, f. Proofs of the remining prts re similr.

5 SOME INEQUALITIES INVOLVING INTEGRAL MEANS 5 Some elementry properties of M [,b,g p, f derived by the use of the weighted integrl nlogue of Jensen s inequlity my be found in [5. Also some well known inequlities mong weighted mens in integrl form my be obtined s its direct corollries. The following theorem corresponds to some conversions of the Jensen s inequlity for convex concve functions in the cse of M [,b,g p, f. Theorem 2.3. Let p, f L + [, b L+ [, b, such tht f : [, b [α, β, nd g : [α, β R, where < α < β <. i If g is convex on [α, β, then gα β A [,b p, f gβ A [,b p, f α g M [,b,g p, f +. ii If g is concve on [α, β, then gα β A [,b p, f g M [,b,g p, f + gβ A [,b p, f α Proof. We will prove only the item i. The item ii my be proved nlogously. Suppose tht g is convex function on the intervl [α, β. Let us consider the following integrl Putting pxgfx dx. λx = fx α, 4 we hve fx = λxα + λxβ, for ll x [, b. Since α fx β for ll x [, b nd g is convex on [α, β, we hve pxgfxdx px λxgα + λxgβ dx By 4 we get nd therefore = gα px λx dx + gβ pxλxdx = b pxfxdx α p [,b. pxλx dx. pxgfxdx gα b β p [,b pxfxdx + gβ b pxfxdx α p [,b.

6 6 JÁN HALUŠKA nd ONDREJ HUTNÍK Since p [,b is positive nd finite, we my write g M [,b,g p, f = p [,b gα β b p [,b pxfxdx = gα β A [,b p, f The proof is complete. pxgfx dx b gβ p [,b pxfxdx α + + gβ A [,b p, f α. 3. Some inequlities mong mens In this section we investigte some more properties of the clss of generlized weighted qusi-rithmetic mens expressed in the integrl form. Summrizing elementry properties of M [,b,g p, f relted to convexity concvity of functions f, g we obtin the following esy lemm. For the proof, cf. [5. Lemm 3.. Let p, k L + [, b L+ [, b nd let h i L + [, b be sequence of functions, i =,...,n; n N. Let δ R. If g : [, R is convex function nd f : [, R is concve function on [, b, then i M [,b,g p, k M [,b, g p, k nd M [,b,f p, k M [,b, f p, k; ii M [,b,f p, k M [,b,g p, k; iii M [,b,f p, δ δ M [,b,g p, δ, where δ = δx is constnt function for x [, b; iv n i= M [,b,fp, h i M [,b,g p, n i= h i; v if fx gx for ll x [, b, then M [,b,f p, g A [,b p, g A [,b p, f M [,b,g p, f. b If g : [, R is concve function nd f : [, R is convex function on [, b, then the bove inequlities i v re in the reversed order. Jensen s inequlity provides very importnt tool for kind of comprison of mens. Therefore, we stte the following comprison theorem between mens M [,b,gk p, f nd M [,b,gk p, f. Theorem 3.2. Let p, f L + [, b L+ [, b such tht f : [, b [α, β. Let g k, k =, 2,...,m be one to one functions defined on [α, β. Let us denote G = g, G 2 = g 2 g,..., G k+ = g k+ g k for k =, 2,...,m. i If either G k re concve incresing or convex decresing on the rnge of g k, for k =, 2,...,m, then M [,b,gk p, f M [,b,gk p, f, k =, 2,..., m.

7 SOME INEQUALITIES INVOLVING INTEGRAL MEANS 7 ii If either G k re concve decresing or convex incresing on the rnge of g k, for k =, 2,...,m, then M [,b,gk p, f M [,b,gk p, f, k =, 2,..., m. Proof. Let us suppose tht G k = g k g k re concve incresing functions for k =, 2,...,m. By Jensen s inequlity we hve b b pxg k g k fx dx G k pxg k fxdx, p [,b g k p [,b p [,b for k =, 2,...,m. Since the functions G k re incresing, it follows tht g k re incresing too, nd therefore for k =, 2,...,m we get b pxg k g k fx dx g k G k p [,b pxg k fxdx Using the fct g k G k = g k nd G k g k = g k, we finlly obtin the inequlity g b k pxg k fxdx g b k pxg k fxdx, p [,b p [,b for k =, 2,..., m, which corresponds to M [,b,gk p, f M [,b,gk p, f. Proofs of the remining prts re similr. Lemm 3.3. Let p L + [, b nd f : [, b R be continuous nd integrble function with the continuous first derivtive on, b. i If f is strictly monotone nd convex function on [, b, then A [,b p, f f A [,b pxf x, x. ii If f is strictly monotone nd concve function on [, b, then Proof. Let us define θ by A [,b p, f f A [,b pxf x, x. θ = pxf xxdx pxf xdx. 5 Since f is strictly monotone on [, b, it follows tht θ, b. The convexity of f ensures tht f is nondecresing on, b nd fx + f xθ x fθ, cf. [4. Multiplying both sides of the bove inequlity by. px p [,b, we hve pxfx + pxf xθ x pxfθ. 6 p [,b p [,b p [,b

8 8 JÁN HALUŠKA nd ONDREJ HUTNÍK Integrting 6 with respect to x we my write pxfxdx p [,b + θ Replcing θ by 5 we obtin pxfxdx p [,b pxf b xdx pxf xxdx p [,b p [,b f pxf xxdx pxf, xdx fθ. i.e. A [,b p, f f A [,b pxf x, x. Exmple 3.4. Let [, b =, /2 nd suppose p L +, /2. Let fx = ln x x on, /2. It is esy to verify tht function fx is strictly decresing nd convex on, /2 nd f x = xx, i.e. the ssumptions of Lemm 3.3 re stisfied. Thus, /2 pxln x /2 px dx ln xx dx /2 px x dx p,/2 x /2 px x dx /2 px x = ln dx. /2 px x dx The bove inequlity my be rewritten s follows /2 exp pxln x dx x p,/2 which is equivlent to /2 exp p,/2 pxln xdx /2 exp p,/2 pxln xdx Using the nottion G,/2 px, gx = exp for the weighted geometric men nd H,/2 px, gx = p,/2 p,/2 /2 /2 /2 px x /2 /2 px x /2 dx px x dx, dx 7 px x dx. pxln gxdx px gx dx, for the weighted hrmonic men, we my rewrite the inequlity 7 s follows G,/2 px, x G,/2 px, x H,/2px, x, H,/2 px, x,

9 SOME INEQUALITIES INVOLVING INTEGRAL MEANS 9 which is equivlent to the weighted integrl inequlity of Wng-Wng, cf. [, in the form H,/2 px, x H,/2 px, x G,/2px, x G,/2 px, x. The following theorem is n esy corollry of Lemm 3.3. Theorem 3.5. Let p L + [, b. Suppose f : [, b [α, β is continuous nd integrble function on [, b with the continuous first derivtive on, b, nd g : [α, β R. i If either g is convex incresing or concve decresing function on [α, β nd f is strictly monotone nd concve function on [, b, then f A [,b pxf x, x M [,b,g p, f. ii If either g is concve incresing or convex decresing function on [α, β nd f is strictly monotone nd convex function on [, b, then f A [,b pxf x, x M [,b,g p, f. Proof. Let g be concve decresing function on [α, β nd f be strictly monotone nd concve function on [, b. From Corollry 2.2 nd Lemm 3.3 we immeditely get the inequlities M [,b,g p, f A [,b px, fx f A [,b pxf x, x. Remining prts my be proved nlogously. Theorem 3.6. Let p, f L + [, b L+ [, b, where f : [, b [α, β is continuous function with the continuous first derivtive on, b. Let g : [α, β R. i If either g is convex incresing or concve decresing on [α, β, then x A [,b p, f f + M [,b,g px, f tdt. ii If either g is concve incresing or convex decresing on [α, β, then x A [,b p, f f + M [,b,g px, f tdt. Proof. Consider the cse when g is convex incresing on [α, β. The direct clcultion yields x x M [,b,g px, f tdt A [,b px, f tdt = p [,b pxfxdx f p [,b pxdx = A [,b p, f f.

10 JÁN HALUŠKA nd ONDREJ HUTNÍK Similr results we my obtin when considering integrls x f tdt. An esy corollry for the generlized weighted qusi-rithmetic men of product of two functions follows from the Chebyshev s integrl inequlity in the following form, cf. [2. Lemm 3.7 Chebyshev s Inequlity. Let p L + [, b nd let h, k : [, b R be two integrble functions, both incresing or both decresing on [, b. Then A [,b p, h A [,b p, k A [,b p, hk. 8 If one of the functions h or k is nonincresing nd the other nondecresing, then the inequlity in 8 is reversed. Theorem 3.8. Let p L + [, b, let h, k : [, b [α, β be two integrble functions nd f : [α, β R. Let g be rel continuous monotone function defined on the rnge of hk. i If g is convex incresing or concve decresing, f is concve incresing or convex decresing nd h, k re either both incresing or both decresing functions, then M [,b,f p, h M [,b,f p, k M [,b,g p, hk. ii If g is concve incresing or convex decresing, f is convex incresing or concve decresing nd one of the functions h, k is nonincresing nd the other one nondecresing, then M [,b,f p, h M [,b,f p, k M [,b,g p, hk. Proof. Let us prove the item i. Suppose tht h, k re both incresing functions, g is convex incresing nd f is concve incresing function. From Corollry 2.2 it follows tht M [,b,f p, h A [,b p, h nd M [,b,f p, k A [,b p, k. Since h, k re both incresing functions, then M [,b,f p, h M [,b,f p, k A [,b p, h A [,b p, k. Applying Lemm 3.7 nd Corollry 2.2 we get M [,b,f p, h M [,b,f p, k A [,b p, hk M [,b,g p, hk. Hence the result. Similrly we my prove the remining prts. 4. Some pplictions In this section we del with the geometric men opertor G : L + [, L + [, defined s follows: If f L+ [,, then x [Gfx = exp lnftdt, x,, 9 x i.e. we consider g = exp nd the weighted function pt.

11 SOME INEQUALITIES INVOLVING INTEGRAL MEANS There exist mny inequlities involving the geometric men opertor 9, cf. [5. We will prove the following generliztion of the inequlity considered in [7 nd give n esy corollry relted to Póly-Knopp s inequlity nd its weighted form. Theorem 4.. Let n be nturl nd r, s, q be rel numbers stisfying 2r n > q sr n. If then where x [r 2n x rn Ce C = t q srn ft rn dt <, t rn ln x s ft dt dx t q srn ft rn dt, r n 2r n + r n s q. Proof. Using the substitution t = xy, we my rewrite the left side of s follows [r 2n y rn ln x s fxy dy dx. [ Since e = exp r 2n lny dy, we hve tht hs the form yrn [r 2n y rn ln x s fxy dy dx C exp [ r 2n The direct clcultion yields C t q srn ft rn dt exp [r 2n = y rn lny dy y rn lny dy [r 2n [r 2n Therefore, we get the inequlity [r n y rn ln x s yfxy rn dy dx C t q srn ft rn dt. y rn ln x s fxy dy dx y rn ln x s yfxy dy dx. t q srn ft rn dt. By Jensen s inequlity, the left side is dominted by r n x q y rn x s yfxy rn dy dx.

12 2 JÁN HALUŠKA nd ONDREJ HUTNÍK Applying Fubini s theorem to nd using the substitution t = xy, we get r n y rn x q srn fxy rn dx dy = = r n y 2rn +sr n q 2 = Hence the result. r n 2r n + sr n q t q srn ft rn dt dy = t q srn ft rn dt. Note tht from the inequlity we my obtin some well known inequlities s direct consequences. For instnce, the following Póly Knopp s inequlity. Corollry 4.2. Let r =, q = s = n = nd ft. Then the inequlity reduces to [ x exp lnftdt dx e fxdx, 2 x or equivlently, [Gfxdx e fxdx. Some weighted versions of Póly Knopp s inequlity 2 my be lso directly derived from, for exmple: [ x x lnftdt dx e q x q fxdx, for every q <, which is more generl thn 2. Using Theorem 4. we my obtin some other interesting integrl inequlities. One of them is the Cochrn Lee s type inequlity. Corollry 4.3. Let s =, let r n = λ > nd q be rel numbers such tht 2λ > q. Then hs the form [ λ x q 2 x exp t λ ln ft dt dx Ce x q fx λ dx, where x λ C = λ 2λ q. References [ J. ACZÉL, On men vlues, Bull. Amer. Mth. Soc , [2 B.C. CARLSON, Invrince of n integrl verge of logrithm, Amer. Mth. Monthly , [3 B. DE FINETTI, Sul concetto di medi, Giornle di Istituto Itlino dei Atturii 2 93, [4 S. DUBORIJA, Two estimtes for integrl mens of nlytic functions in the unit disk, Mtemtički Vesnik ,

13 SOME INEQUALITIES INVOLVING INTEGRAL MEANS 3 [5 J. HALUŠKA nd O. HUTNÍK, On generlized weighted qusi-rithmetic mens in integrl form, Jour. Electricl Engineering 56, 2/s 25, 5. [6 G.H. HARDY nd J.E. LITTLEWOOD nd G. PÓLYA, Inequlities, Cmbridge University Press, Cmbridge, 934. [7 H.P. HEINIG, Some Extensions of Hrdy s Inequlity, SIAM J. Mth. Anl , [8 B. JESSEN, Über die Verllgemeinerung des rithmetischen Mittels, Act Sci. Mth. 4 93, 8-6. [9 T. KITAGAWA, On some clss of weighted mens, Proceedings Physico-Mthemticl Society of Jpn 6 934, [ A.N. KOLMOGOROFF, Sur l notion de l moyenne, Atti dell R. Accdemi Nzionle dei Lincei 2, 93, [ M.T. MCGREGOR, On some inequlities of Ky Fn nd Wng-Wng, J. Mth. Anl. Appl , [2 D. S. MITRINOVIĆ nd J. E. PEČARIĆ nd A. M. FINK, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers, Dordrecht/Boston/London, 993. [3 M. NAGUMO, Über eine Klsse von Mittelwerte, Jpnese Journl of Mthemtics 7 93, [4 C.P. NICULESCU nd L.E. PERSSON, Convex Functions nd their Applictions - A contemporry pproch, Cndin Mth. Series Books in Mthemtics, Springer, 26. [5 L.E. PERSSON, nd V.D. STEPANOV, Weighted Integrl Inequlities with the Geometric Men Opertor, J. of Inequl. nd Appl , [6 G. PÓLYA nd G. SZEGÖ, Isoperimetric Inequlities in Mthemticl Physics, Annls of Mthemtics Studies, no. 27, Princeton University Press, New Jersey, 95. [7 W. RUDIN, Rel nd Complex Anlysis, McGrw-Hill Book Co., New York 987, 3rd edition. [8 F. QI, Generlized Abstrcted Men Vlues, J. Ineq. Pure nd Appl. Mth. rticle 4 2. [ONLINE Avilble online t 99.html [9 K.B. STOLARSKY, The Power nd Generlized Logrithmic Mens, Amer. Mth. Monthly 8 974, Ján Hlušk, Mthemticl Institute of Slovk Acdemy of Science, Current ddress: Grešákov 6, 4 Košice, Slovki, E-mil ddress: jhlusk@sske.sk Ondrej Hutník, Deprtment of Mthemticl Anlysis nd Applied Mthemtics, University of Žilin, Current ddress: Hurbnov 5, 26 Žilin, Slovki, E-mil ddress: ondrej.hutnik@fpv.utc.sk