A generalization of the well-known Hilbert’s inequality is given. Several other results of this type in recent years follows as a special case of our result

Josip Pečarić

Ljubo Marangunić

English

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Rad HAZUVolume 503 (2009), 61–67ON SOME HILBERT’S TYPE INEQUALITIESLJ. MARANGUNIĆ AND J. PEČARIĆAbstract. A generalization of the well-known Hilbert’s inequality is given. Several other resultsof this type in recent years follows as a special case of our result.1. IntroductionFirst, let us repeat the well-known Hilbert’s integral inequality:THEOREM A. If f , g ∈ L2(0,∞) , then the following inequality holds:∞∫0∞∫0f (x)g(y)x+ ydxdy  π⎛⎝∞∫0f 2(x)dx∞∫0g2(x)dx⎞⎠12(1.1)where π is the best constant.In the recent years a lot of results with generalization of this type of inequalitywere obtained. Let us mention one of them which take our attention.Note that in all theorems we suppose that all integrals converge.THEOREM B. (Yang, Rassias, [1]): If f and g are real functions such that:∞∫0t1−λ f 2(t)dt < ∞ and∞∫0t1−λg2(t)dt < ∞then:i) for 0 < b < ∞ , we haveA)b∫0b∫0f (x)g(y)(x+ y)λdxdy < B(λ2,λ2)⎡⎣b∫0[1− 12( tb) λ2]t1−λ f 2(t)dt⎤⎦12××⎡⎣b∫0[1− 12( tb) λ2]t1−λg2(t)dt⎤⎦12; (1.2)Mathematics subject classification (2000): 25D15.Keywords and phrases: Hilbert’s inequality, Hölder’s inequality, Beta-function.(Accepted January 29, 2008)61Rad Hrvat. akad. znan. umjet. 503. Matematičke znanosti 16 (2009), str. 61–67Lj. Marangunić and J. Pečarić On some Hilbert’s type inequalitiesB)b∫0yλ−1⎡⎣b∫0f (x)(x+ y)λdx⎤⎦2dy <[B(λ2,λ2)]2 b∫0[1− 12( tb) λ2]t1−λ f 2(t)dt; (1.3)ii) for 0 < a < ∞ , we haveA)∞∫a∞∫af (x)g(y)(x+ y)λdxdy < B(λ2,λ2)⎡⎣∞∫a[1− 12(at) λ2]t1−λ f 2(t)dt⎤⎦12××⎡⎣∞∫a[1− 12(at) λ2]t1−λg2(t)dt⎤⎦12; (1.4)B)∞∫ayλ−1⎡⎣∞∫af (x)(x+ y)λdx⎤⎦2dy <[B(λ2,λ2)]2 ∞∫a[1− 12(at) λ2]t1−λ f 2(t)dt (1.5)where B is beta-function.In this paper we generalize inequalities (1.2)–(1.5).2. Main resultsTHEOREM 1. If f and g are real functions and p a real number, p > 1 , suchthat:∞∫0t p−1−pλ2 f p(t)dt < ∞ and∞∫0t p−1−pλ2 gp(t)dt < ∞then:i) for 0 < b < ∞ , 1p +1q = 1 , we haveA)b∫0b∫0f (x)g(y)(x+ y)λdxdy < B(λ2,λ2)⎡⎣b∫0[1− 12( tb) λ2]t p−1−pλ2 f p(t)dt⎤⎦1p××⎡⎣b∫0[1− 12( tb) λ2]tq−1−qλ2 gq(t)dt⎤⎦1q; (2.1)62Rad Hrvat. akad. znan. umjet. 503. Matematičke znanosti 16 (2009), str. 61–67Lj. Marangunić and J. Pečarić On some Hilbert’s type inequalitiesB)b∫0yλ p2 −1⎡⎣b∫0f (x)dx(x+ y)λ⎤⎦pdy <[B(λ2,λ2)]p b∫0[1− 12( tb) λ2]t p−1−pλ2 f p(t)dt;(2.2)ii) for 0 < a < ∞ , 1p +1q = 1 , we haveA)∞∫a∞∫af (x)g(y)(x+ y)λdxdy < B(λ2,λ2)⎡⎣∞∫a[1− 12(at) λ2]t p−1−pλ2 f p(t)dt⎤⎦1p××⎡⎣∞∫a[1− 12(at) λ2]tq−1−qλ2 gq(t)dt⎤⎦1q; (2.3)B)∞∫ayλ p2 −1⎡⎣∞∫af (x)dx(x+ y)λ⎤⎦pdy <[B(λ2,λ2)]p ∞∫a[1− 12(at) λ2]t p−1−pλ2 f q(t)dt.(2.4)Proof : i) A) We start with the following equality:b∫0b∫0f (x)g(y)(x+ y)λdxdy =b∫0f (x) x2−λ2qy2−λ2p(x+ y)λp·g(y) y2−λ2px2−λ2q(x+ y)λqdxdy. (2.5)By Hölder’s inequality and (2.5) we have:b∫0b∫0f (x)g(y)(x+ y)λdxdy⎡⎣b∫0b∫0f p(x)(x+ y)λ· xp(2−λ)2qy2−λ2dxdy⎤⎦1p⎡⎣b∫0b∫0gq(y)(x+ y)λ· yq(2−λ)2px2−λ2dxdy⎤⎦1q=⎡⎣b∫0b∫0f p(x)(x+ y)λ· xp(2−λ)2q x2−λ2x2−λ2 y2−λ2dxdy⎤⎦1p⎡⎣b∫0b∫0gq(y)(x+ y)λ· yq(2−λ)2p y2−λ2y2−λ2 x2−λ2dxdy⎤⎦1q=⎡⎢⎢⎣b∫0f p(x)x(2−λ)(p−q)2q⎛⎜⎜⎝b∫0(yx) λ−22(x+ y)λdy⎞⎟⎟⎠dx⎤⎥⎥⎦1p×63Rad Hrvat. akad. znan. umjet. 503. Matematičke znanosti 16 (2009), str. 61–67Lj. Marangunić and J. Pečarić On some Hilbert’s type inequalities×⎡⎢⎢⎣b∫0gq(y)y(2−λ)(q−p)2p⎛⎜⎜⎝b∫0(xy) λ−22(x+ y)λdx⎞⎟⎟⎠dy⎤⎥⎥⎦1q=⎡⎣b∫0f p(x)x(2−λ)(p−q)2q Ix dx⎤⎦1p⎡⎣b∫0gq(y)y(2−λ)(q−p)2p Iy dy⎤⎦1q, (2.6)where we denote:Ix =b∫0(yx) λ−22 (x+ y)−λ dy, Iy =b∫0(xy) λ−22 (x+ y)−λ dx.Using the change of variables y = xt , dy = xdt , we have for Ix :Ix =bx∫0tλ−22(x+ xt)λxdt = x1−λbx∫0tλ−22(1+ t)λdt = x1−λWλ(bx)and similarly:Iy = y1−λWλ(by)where we denote:Wλ (z) =z∫0tλ−22(1+ t)λdt, z > 1 (2.7)Using expressions for Ix and Iy , (2.6) can be rewritten as:b∫0b∫0f (x)g(y)(x+ y)λdxdy⎡⎣b∫0f p(x)x(2−λ)(p−q)2q +1−λWλ(bx)dx⎤⎦1p⎡⎣b∫0gq(y)y(2−λ)(q−p)2p +1−λWλ(by)dy⎤⎦1q=⎡⎣b∫0f p(x)xp−1−pλ2 Wλ(bx)dx⎤⎦1p⎡⎣b∫0gq(y)yq−1−qλ2 Wλ(by)dy⎤⎦1q. (2.8)In [2] Gavrea proved the inequality:∞∫αtλ−22(1+ t)λdt >12α−λ2 B(λ2,λ2), α > 1. (2.9)64Rad Hrvat. akad. znan. umjet. 503. Matematičke znanosti 16 (2009), str. 61–67Lj. Marangunić and J. Pečarić On some Hilbert’s type inequalitiesUsing (2.9) we have for Wλ(bx):Wλ(bx)=bx∫0tλ−22(1+ t)λdt =∞∫0tλ−22(1+ t)λdt−∞∫bxtλ−22(1+ t)λdt= B(λ2,λ2)−∞∫bxtλ−22(1+ t)λdt< B(λ2,λ2)− 12(bx)− λ2B(λ2,λ2)=[1− 12( xb) λ2]B(λ2,λ2)(2.10)and similarlyWλ(by)<[1− 12( yb) λ2]B(λ2,λ2). (2.11)Using inequalities (2.10) and (2.11) in (2.8), and taking into account 1p +1q = 1, weobtain (2.1).In the similar way we prove ii) A), using the equation:∞∫axtλ−22(1+ t)λdt =xa∫0tλ−22(1+ t)λdt = Wλ( xa). (2.12)REMARK 1. Inequalities (2.1) and (2.3) are generalizations of (1.2) and (1.4),respectively. We obtain (1.2) and (1.4) by putting p = q = 2 in (2.1) and (2.3).We begin our proof of i) B) by using the equation:J =b∫0yλ p2 −1[Wλ(by)]1−p⎛⎝b∫0f (x)dx(x+ y)λ⎞⎠pdy=b∫0yλ p2 −1[Wλ(by)]1−p⎛⎝b∫0f (x)dx(x+ y)λ⎞⎠p−1⎛⎝b∫0f (x)dx(x+ y)λ⎞⎠ dy=b∫0g(y)⎛⎝b∫0f (x)dx(x+ y)λ⎞⎠ dy =b∫0b∫0f (x)g(y)(x+ y)λdxdy (2.13)65Rad Hrvat. akad. znan. umjet. 503. Matematičke znanosti 16 (2009), str. 61–67Lj. Marangunić and J. Pečarić On some Hilbert’s type inequalitieswhere we denoteg(y) = yλ p2 −1[Wλ(by)]1−p⎛⎝b∫0f (x)dx(x+ y)λ⎞⎠p−1. (2.14)Using (2.8) and (2.14) we have:J ⎡⎣b∫0f p(x)xp−1−pλ2 Wλ(bx)dx⎤⎦1p⎡⎣b∫0gq(y)yq−1−qλ2 Wλ(by)dy⎤⎦1q=⎡⎣b∫0f p(x)xp−1−pλ2 Wλ(bx)dx⎤⎦1p××⎡⎢⎣b∫0yq(λ p2 −1) [Wλ(by)]q(1−p)⎛⎝b∫0f (x)dx(x+ y)λ⎞⎠q(p−1)yq−1−qλ2 Wλ(by)dy⎤⎥⎦1q=⎡⎣b∫0f p(x)xp−1−pλ2 Wλ(bx)dx⎤⎦1p⎡⎣b∫0yλ p2 −1[Wλ(by)]1−p⎛⎝b∫0f (x)dx(x+ y)λ⎞⎠pdy⎤⎦1q.Thus we obtain:J ⎡⎣b∫0f p(x)xp−1−pλ2 Wλ(bx)dx⎤⎦1p· J 1qwherefrom it follows:J =b∫0yλ p2 −1[Wλ(by)]1−p⎛⎝b∫0f (x)dx(x+ y)λ⎞⎠pdy b∫0f p(x)xp−1−pλ2 Wλ(bx)dx.(2.15)As 1− p < 0 and[1− 12(yb) λ2]1−p> 1 for y ∈ (0,b) , we obtain from (2.11):[Wλ(by)]1−p>{[1− 12( yb) λ2]B(λ2,λ2)}1−p>[B(λ2,λ2)]1−p.Using the last inequation in (2.15) we finally obtain:b∫0yλ p2 −1[B(λ2,λ2)]1−p⎛⎝b∫0f (x)dx(x+ y)λ⎞⎠pdy<b∫0f p(x)xp−1−pλ2[1− 12( xb) λ2]B(λ2,λ2)dx66Rad Hrvat. akad. znan. umjet. 503. Matematičke znanosti 16 (2009), str. 61–67Lj. Marangunić and J. Pečarić On some Hilbert’s type inequalitieswherefrom it follows (2.2).The proof of ii) B) is similar to the proof of i) B). Indeed, instead of the functiong(y) used in (2.14), here we define:g(y) = yλ p2 −1[Wλ( ya)]1−p⎛⎝∞∫af (x)dx(x+ y)λ⎞⎠p−1, y > a,and later we use the inequality[Wλ( ya)]1−p>{[1− 12(ay) λ2]B(λ2,λ2)}1−p>[B(λ2,λ2)]1−pfor y > a.REMARK 2. Inequalities (2.2) and (2.4) are generalization of (1.3) and (1.5), re-spectively. We obtain (1.3) and (1.5) by putting p = q = 2 in (2.2) and (2.4).RE F ER EN CE S[1] BICHENG YANG AND T. M. RASSIAS, On the Way of Weighted Coefficient and Research for theHilbert-type Inequalities, Math. Inequal. Appl., 6, 4 (2003), 625–658.[2] IOAN GAVREA, Some Remarks on Hilbert’s Integral Inequality, Math. Inequal. Appl., 5, 3 (2002),473–477.Lj. Marangunić, Faculty of Electrical Engineering and Computing, University of Zagreb, Unska 3, Zagreb,CroatiaJ. Pečarić, Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, Zagreb, Croatia6768

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