SOME NEW INEQUALITIES SIMILAR TO HILBERT TYPE INTEGRAL INEQUALITY WITH A HOMOGENEOUS KERNEL. 1. Introduction. sin(

Transcription

1 Journal of Mathematical Ineualities Volume 6 Number doi:.753/jmi-6-9 SOME NEW INEQUALITIES SIMILAR TO HILBERT TYPE INTEGRAL INEQUALITY WITH A HOMOGENEOUS KERNEL VANDANJAV ADIYASUREN AND TSERENDORJ BATBOLD Communicated b A. Čižmešija Abstract. In this aer we establish some new ineualities similar to Hilbert-te integral ineualit whose kernel is the homogeneous function the best constant factors are also derived.. Introduction If > + f g satisf then < f xdx < < g xdx < f xg dxd < x + π { } { sin π f xdx g xdx} { f xg } { dxd < f xdx g xdx} 2 max{x} where the constant factors π/sin π/ are the best ossible. Ineualities 2 are called Hard-Hilbert s ineualities [] are imortant in analsis their alications [2]. In the recent ears a lot of results with generalizations of these te of ineualities were obtained. In 25 Yang [3] has given a new Hilbert-te ineualit as follows: If > + < λ < f g satisf then < f xdx < < g xdx < { f xg } x λ dxd < k λ x λ f { xdx x λ g xdx} 3 Mathematics subject classification 2: 26D5. Kewords hrases: Hilbert s ineualit Hölder s ineualit Hard s ineualit Fubini s theorem. c D l Zagreb Paer JMI

2 84 V. ADIYASUREN AND TS. BATBOLD where the constant factor k λ B λ λ + B λ λ is the best ossible. In 28 W. Zhong [4] has given a new Hilbert-te integral ineualit with a homogeneous kernel of λ -degree as follows: Let > + s > r + s λ > f g ωxx λ /r ϖ λ /s. Further suose a Kx is a measurable homogeneous kernel function of λ -degree b the weight coefficient A λ s Kuuλ /s du is a ositive number deending onl on the arameters λ s. Then one has following ineualities. If f LωR + g L ϖr + f ω g ϖ > then Kx f xgdxd < A λ s f ω g ϖ 4 { / λ /s Kx f xdx d} < A λ s f ω 5 where the constant factor A λ s is the best ossible in both ineualities 45. For more general results lease refer to [5] [6] [7] where [5] rovides an unified treatment to Hilbert ineualities with general kernels while [6] [7] deals with the roblems of the best ossible constants in such ineualities homogeneous case. In the recent ears man new ineualities similar to 2 3 have been established [8] [3]. In 2 Das Sahoo [8] have given two new ineualities similar to Hard-Hilbert s ineualit as follows: Let > + λ sr > r + s λ f g Fx x f tdt Gx x gtdt.if< f xdx < < g xdx < then the following two ineualities hold: x r s { } { x + λ FxGdxd < Brs f xdx g xdx} 6 x r s x + λ Fxdx d < [Brs] f xdx 7 where the constant factors Brs [Brs] are the best ossible. In 2Das Sahoo [9] have also given two more new ineualities similar to Hard-Hilbert s ineualit 2 as follows: Let > + λ sr > r + s λ f g Fx x f tdt Gx x gtdt.if< f xdx < < g xdx < then the following two ineualities hold: x r s λ max{x λ λ FxGdxd < } rs { f xdx } { g xdx} 8

3 SOME NEW INEQUALITIES SIMILAR TO HILBERT-TYPE INEQUALITY 85 x r s max{x λ λ } Fxdx d < λ f xdx 9 rs where the constant factors λ rs λ rs are the best ossible. In 2 Sulaiman [ Theorem ] derived a new integral ineualit similar to 3 as follows: Let f g + < < α/2 < < β /2 αβ >. We define Fx Gx as: Then x Fx f tdt G gtdt. x β α F α + xg β + dxd x 2 α + 2 β { < K / α K / β } { f α + xdx g β xdx} + where K α 2 + α α + 2 B. α α Ver recentl Du Miao [3] obtained the following ineualit: Let f g x Fx f tdt G gtdt. Furthermore assume that > αβ st μν > hold + s > β + t> α + Then we have β + μ s + α + ν t +. x α β x + s+t F μ xg ν dxd μ < κ μ μ ν ν { } { f μ xdx g ν xdx} ν where κ B / β + s β + B / α + t α +.

4 86 V. ADIYASUREN AND TS. BATBOLD In [] [3] authors do not rove whether the constant factors are the best ossible or not. The main objective of this aer is to build some new ineualities similar to Hilbert-te integral ineualities 45 whose kernel is the homogeneousfunction with the best constant factors. As alications some articular results are given. 2. Preliminar lemmas In this section we shall rove lemmas which la crucial roles in roving our main results. LEMMA 2.. Let be conjugate arameters with > let λ sr > such that s + r λ. If k λ x : R 2 + R is non-negative homogeneous function of degree λ i.e. k λ uxuu λ k λ x then ω λ sxϖ λ r C λ s 2 where ω λ sx : k λ x s x r d ϖ λ r : k λ xx r s dx C λ s : k λ uu s du. Proof. Setting u x wefind ω λ sx k λ uu s du C λ s for > letting x u it is eas to find that ϖ λ r k λ u s r u r u 2 du k λ uu s du C λ s euation 2 is valid. This comletes the lemma. LEMMA 2.2. Hard s ineualit cf. [] If > f Fx x f tdt then Fx dx < f xdx 3 x unless f. The constant is the best ossible. LEMMA 2.3. Let > β < β n> β x β +/n β for x then β x β +/n. 4

5 SOME NEW INEQUALITIES SIMILAR TO HILBERT-TYPE INEQUALITY 87 Proof. For x we set Hxx β +/n β Simle comutations ield for x > H x β x β +/n +. β +/n x β +/n x +/n β β β >. f is increasing function on continuous on [. In articular we have f x f which gives the desired ineualit. 3. Main results THEOREM 3.. Let be conjugate arameters with > α > β < αβ let λ sr > such that s+r λ k λ x is non-negative homogeneous function of degree λ in R 2 +. Assume Fx : x f tdt G : gtdt. If < C λ s < < k λ uu s β du < < k λ uu r α du < f g satisf < f α xdx < < g β xdx < then the following two ineualities hold: k λ xx r α s β F α xg β dxd { } { < C λ αβ s f α xdx g β xdx} 5 k λ xx r α s F α xdx d < C α α λ s f α xdx α 6 α β where the constant factors C λ αβ s C λ s α β α β C α λ s α α are the best ossible.

6 88 V. ADIYASUREN AND TS. BATBOLD Proof. B Hölder s ineualit Lemma 2. wehave J : { { k λ xx r α s β F α xg β dxd k λ x s x r α F α xx r s β G β dxd Fx α } k λ x s x r dxd x G β k λ xx r dxd} s { Fx α } { } G β C λ s dx d. x Then b Hard s ineualit 5 is valid. Suosing there exists a ositive constant C < C λ αβ s such that 5 is still valid when C λ αβ s is relaced b C for n > β n N setting f x g as follows: { for x f x x +/n α for x [ then Denote φn { for g +/n β for [ { } C f α { xdx g β xdx} nc 7 { for x Fx α α for x [ G α +/n x α +/n { for β +/n β β +/n β for [. α α α +/n β β +/n as n for x b Lemma 2.3 wehave F α x G β φnx α +/n α φnx α +/n β α β.thenφn α β α β α β +/n β β β +/n > φnx α +/n β +/n x α +/n β +/n.

7 SOME NEW INEQUALITIES SIMILAR TO HILBERT-TYPE INEQUALITY 89 Then Jn > φn k λ xx r α s β F α x G β dxd k λ x φni I 2 I 3. x r n s n x r n s β x r α s n dxd Taking u x b Fubini s theorem we obtain I : n x n x n n n k λ xx r n s n dxd k λ x s n x r+ n d dx k λ uu s n du+ /x k λ uu s n du+ k λ uu s n du+ Again taking u x we obtain I 2 : < k λ uu s n du+ k λ xx r n s β dxd x β + n β + n Similarl we get k λ xx r n s β dxd I 3 : Hence b 7 we have < x n dx k λ uu s n du dx /x k λ uu s n du k λ uu s n du k λ uu s+ n du dx k λ uu s β du k λ uu s β du <. α + n k λ xx r α s n dxd k λ uu r α du <. x n dx /u. k λ xx r n s β dxd φnk λ uu s n du+ φnk λ uu s+ n du φn < C. n

8 9 V. ADIYASUREN AND TS. BATBOLD Then b Fatou lemma we have α α β β C λ αβ s k α β λ uu s du lim φnk λ uu s n du n + lim φnk λ uu s+ n φn du lim n n n lim φnk λ uu s n du n + φnk λ uu s+ n du φn n < C. Hence the constant factor C C λ αβ s is the best ossible. B Hölder s ineualit Lemma 2.weget L : { k λ xx r α s F α xdx k λ xx r α s F α xx r s dx } / { k λ xx r α s F α xdx { C λ s / k λ xx r α s F xdx} α. } / k λ xx r s dx Hence again aling Lemma 2. wehave L d C λ s k λ xx r s d x α F α xdx Fx α C λ s dx. x Then b Hard s ineualit 6 is valid. α If the constant factor C λ s α α in 6 is not the best ossible then there α exists a ositive constant K such that K < C λ s α α 6 still remains valid if C λ s α α α is relaced b K.ThenbHölder s ineualit 6 Hard s

9 SOME NEW INEQUALITIES SIMILAR TO HILBERT-TYPE INEQUALITY 9 ineualit we obtain J k λ xx r α s G β F α xdx d { } { / k λ xx r α s G β / F xdx α d d} β β { } < K f α { xdx g β xdx} β which results that the constant factor C λ αβ s in 5 is not the best ossible. α This contradiction shows that the constant factor C λ s α α in 6 is the best ossible. The theorem is roved. If k λ x/x + λ /max{x λ λ } or / x λ then we obtain the following corollaries corresondingl COROLLARY 3.2. Let be conjugate arameters with > α > β < αβ let λ sr > such that r+s λ f g Fx x f tdtg gtdt. If < f α xdx < < gβ xdx < then the following two ineualities hold: x r α s β x + λ F α xg β dxd α α β β { } < Brs f α { xdx g β xdx} α β x r α s α α x + λ F α xdx d < [Brs] f α xdx α α β α where the constant factors Brs α β α β [Brs] α α are the best ossible. COROLLARY 3.3. Let be conjugate arameters with > α > β < αβ let λ sr > such that r+s λ f g Fx x f tdtg gtdt. If < f α xdx < < gβ xdx < then the following two ineualities hold: x r α s β max{x λ λ } Fα xg β dxd < λ α rs α α β β { } f α { xdx g β xdx} β

10 92 V. ADIYASUREN AND TS. BATBOLD x r α s max{x λ λ } Fα xdx d < where the constant factors rs λ ossible. λ α α f α xdx rs α α β α α β α β λrs α α are the best COROLLARY 3.4. Let be conjugate arameters with > α > β < αβ let < λ < sr > such that r + s λ f g Fx x f tdtg gtdt. If < f α xdx < < gβ xdx < then the following two ineualities hold: x r α s β α α β < Bs λ +Br λ α β x λ F α xg β dxd β { } { f α xdx g xdx} β x r α s x λ F α xdx d < Bs λ +Br λ α α α f α xdx α β where the constant factors Bs λ +Br λ α β α β α Bs λ +Br λ α α are the best ossible. Acknowledgements The authors would like to exress their gratitude to the referee for his/her ver valuable comments suggestions. REFERENCES [] G. H. HARDY J. E. LITTLEWOOD AND G. POLYA Ineualities Cambridge Universit Press Cambridge 952. [2] D. S. MITRINOVIĆ J. E. PEČARIĆ AND A. M. FINK Ineualities Involving Function Their Integrals Derivatives Kluwer Acadamic Publishers Boston 99. [3] B. YANG A New Hilbert-Te Integral Ineualit Its Generalization Journal of Jilin Universit [4] W. ZHONG The Hilbert-Te Integral Ineualities With a Homogeneous Kernel of λ -Degree Journal of Ineualities Alications Vol. 28 Article ID ages 28. [5] M. KRNIĆ AND J. PEČARIĆ General Hilbert s Hard s Ineualities Math. Ineual. Al [6] M. KRNIĆG.MINGZHEJ.PEČARIĆ AND G. XUEMEI On The Best Constant in Hilbert s Ineualit Math. Ineual. Al

11 SOME NEW INEQUALITIES SIMILAR TO HILBERT-TYPE INEQUALITY 93 [7] I. PERIĆ AND P. VUKOVIĆ Hard-Hilbert s Ineualit With General Homogeneous Kernel Math. Ineual. Al [8] N. DAS AND S. SAHOO New Ineualities Similar to Hard-Hilbert s ineualit Turk. J. Math [9] N. DAS AND S. SAHOO On a Generalization of Hard-Hilbert s Integral Ineualit Bul. Acad. Sţiinţe Reub. Mold. Mat [] W. T. SULAIMAN On Two New Ineualities Similar to Hard-Hilbert s Integral Ineualit Int. J. Math. Anal [] W. T. SULAIMAN On Three Ineualities Similar to Hard-Hilbert s Integral Ineualit Acta Math. Univ. Comenianae LXXVI [2] W. T. SULAIMAN On Two New Ineualities Similar to Hard-Hilbert s Integral Ineualit Soochow J. Math [3] H. DU AND Y. MIAO Several New Hard-Hilbert s Ineualities Filomat Received Ma 2 Vanjav Adiasuren Deartment of Mathematical Analsis National Universit of Mongolia Ulaanbaatar Mongolia V Adiasuren@ahoo.com Tserendorj Batbold Institute of Mathematics National Universit of Mongolia Ulaanbaatar Mongolia tsbatbold@hotmail.com Journal of Mathematical Ineualities jmi@ele-math.com