A Hilbert-Type Inequality with Some Parameters and the Integral in Whole Plane

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1 Advances in Pe Mathematics,,, doi:.436/am..39 Pblished Online Ma (htt:// A Hilbet-Te Inealit with Some Paametes the Integal in Whole Plane Abstact Zitian Xie, Zheng Zeng Deatment of Mathematics, Zhaoing Univesit, Zhaoing, China Deatment of Mathematics, Shaogan Univesit, Shaogan, China gdzzt@63.com, zz@sg.ed.cn Received Feba 3, ; evised Mach 7, ; acceted Mach, In this ae, b intodcing some aametes estimating the weight coefficient, we give a new Hilbet s inealit with the integal in whole lane with a non-homogeneos the eivalent fom is given as well. The best constant facto is calclated b the wa of Comle Analsis. Kewods: Hilbet-Te Integal Inealit, Weight Fnction, Holde s Inealit. Intodction If, g ae non-negative fnctions sch that f d g d, f f g dd f d gd (.) whee the constant facto is the best ossible. Inealit (.) is well-known as Hilbet s integal inealit, which has been etended b Ha-Riesz as []: If,, f, g, sch that f d g d, we have the following Ha-Hilbet s integal inealit: f g dd (.) f g d sin d whee the constant facto also is the best sin ossible. In ecent eas, b intodcing some aametes estimating the wa of weight fnction, inealities (.) (.) have man genealizations vaiants (.) has been stenged b Yang othes. (inclding doble seies inealities) [3-5]. In 5 Yang gave a Hilbet-te Inealit [3] as follows: If,,,, min, d g f g dd f g, f d, d K f g d (.3) whee the constant facto K B, B, also is the best ossible. In 8 Xie gave a new Hilbet-te Inealit [4] as follows: If,, f, g f d g d, f g dd a a a d K f g d whee the constant facto K abbcca Coight SciRes.

2 Z. T. XIE ET AL. 85 also is the best ossible. In this ae, b sing the wa of weight fnction the technic of eal analsis b the wa of comle analsis, a new Hilbet-te inealit with the integal in whole lane is given. In the following, we alwas sose that:,,,,,.. Some Lemmas Lemma. If cos k : d, cos cos k : d, cos 4 sin sin k :, sin (.) 4 sin sin k : sin cos k : d cos k k 4 sin Poof We have setting f cos sin : cos d A z cos cos d cos : B (.) z z cos z zcos, i i z e, z e, We find i B Re s f, zre s f, z i e z z z z cos sin cos cos i i e zz z z cos[ ] A B. sin cos k d cos 4sin sin sin cos k d cos cos d cos 4 sin sin sin cos k d cos cos d cos cos d cos k k 4 sin cos sin The lemma is oved. Lemma. Define the weight fnctions as follow: cos w: d, cos cos w : d cos Coight SciRes.

3 86 Z. T. XIE ET AL. 4sin w w k cos sin w (.3) Poof We onl ove that k,. cos w cos Setting, w fo cos cos cos cos cos cos w w cos Similal, setting cos cos d k cos, cos w d k cos And w k k k, we have (.3). Lemma.3 Fo, ma,,, define both fnctions, f g as follow:, if,,, if,, f, if, ;, if,, g, if,,, if, ; I: f d g d, fo (.4) cos I : f g d cos k o Poof Easil, I : d d ; (.5) Let Y, sing f f, g g, cos f g cos Y cos Y f gy dy Y cos Y cos f g cos is an even fnction, we have that cos I f g d cos cos cos d d cos cos I I. Setting cos cos I d d d d cos cos cos cos d d cos d d cos Coight SciRes.

4 Z. T. XIE ET AL. cos cos d d d cos cos cos cos d d cos cos cos cos d ( ) d cos cos 4sin sin sin 87 Thee lim, we have I k (fo ). Similal I k (fo ). The lemma is oved. Lemma.4 If f d, we have cos J : f d cos k f d Poof B lemma., we find J cos f d cos cos cos cos cos cos cos (.6) f d f d d cos k f d, cos cos k f d cos k f d cos cos k f d (.7) 3. Main Reslts f Theoem If both fnctions, g ae nonnegative measable fnctions, satisf f d, g d, * cos I : f g cos d d k f g d (3.) cos J f d cos k f d (3.) Inealities (3.) (3.) ae eivalent, the constant factos in the two foms ae all the best ossible. Poof If (.7) takes the fom of ealit fo some,,, thee eist constants M N, sch that the ae not all zeo, M f a.e. in,,, Hence, thee eists a constant C, sch that a.e. in,, M f N C. We claim that M. In fact, if M, f C M a.e. in, which con- tadicts the fact that f d. In the same wa, we claim that N. This is too a contadiction hence b (.7), we have (3.). B Holde s inealit with weight (3.), we have, Coight SciRes.

5 88 Z. T. XIE ET AL. cos cos * I f d g J g (3.3) Using (3.), we have (3.). Setting cos g f d cos J g ( ) b (.7) we have J. if J (3.) is oved; if J, b (3.), we obtain * g ( ) J I d k f d g d, g J k f d Inealities (3.) (3.) ae eivalent. If the constant facto k in (3.) is not the best ossible, thee eists a ositive h (with h k ), sch that cos f g d d cos (3.4) h f d g d Fo, b (3.4), sing lemma.3, we have I ko d h f g d (3.5) Hence we find, k o h, fo, it follows that k h, which contadicts the fact that h k. Hence the constant h in (3.) is the best ossible. As (3.) (3.) ae eivalent, if the constant facto in (3.) is not the best ossible, b sing (3.), we can get a contadiction that the constant facto in (3.) is not the best ossible. Ths we comlete the ove of the theoem. Remak Fo, in (3.), we have the 4 3 following aticla eslt: f g d (3.6) k f d g d Whee the constant facto 5 4sin cos k 4 4 also is the best ossible. sin 4. Refeences [] G. H. Ha, J. E. Littlewood G. Pola, Inealities, Cambidge Univesit Pess, Cambidge, 95. [] G. H. Ha, Note on a Theoem of Hilbet Concening Seies of Positive Teems, Poceedings of London Mathematical Societ, Vol. 3, No., 95, [3] B. C. Yang, A New Hilbet-Te Integal Inealit with a Paametes, Jonal of Henan Univesit (Natal Science), Vol. 35, No. 4, 5, [4] Z. T. Xie Z. Zeng, A Hilbet-Te Integal Inealit whose Kenel is a Homogeneos Fom of Degee-3, Jonal of Mathematical Analsis Alication, Vol 339, No., 8, doi:.6/j.jmaa [5] B. C. Yang, A New Hilbet-Te Integal Inealit with Some Paametes, Jonal of Jilin Univesit (Science Edition), Vol. 46, No. 6, 8, [6] B. C. Yang, A Hilbet-Te Integal Inealit with the Homogeneos Kenel of Real Nmbe-Degee, Jonal of Jilin Univesit (Science Edition), Vol. 47, No. 5, 9, [7] Z. T. Xie X. D. Li, A New Hilbet-Te Integal Inealit its Revese, Jonal of Henan Univesit, (Science Edition), Vol. 39, No., 9,. -3. [8] Z. Zeng Z. T. Xie, On a New Hilbet-Te Integal Inealit with the Integal in Whole Plane, Jonal of Inealities Alications, Vol., Aticle ID 56796,. [9] Z. T. Xie, B. C. Yang Z. Zeng, A New Hilbet- te Integal Inealit with the Homogeneos Kenel of Real Nmbe-Degee, Jonal of Jilin Univesit (Science Edition), Vol. 48, No. 6,, [] Z. T. Xie Z. Zeng, On Genealit of Hilbet s Inealit with Best Constant Facto, Natal Science Jonal of Xiangtan Univesit, Vol. 3, No. 3,,. -4. [] Z. T. Xie B. L. F, A New Hilbet-Te Integal Inealit with a Best Constant Facto, Jonal of Whan Univesit (Natal Science Edition), Vol. 55, No. 6, 9, [] Z. T. Xie Z. Zeng, The Hilbet-Te Integal Inealit with the Sstem Kenel of Degee Homogeneos Fom, Kngook Mathematical Jonal, Vol. Coight SciRes.

6 Z. T. XIE ET AL. 89 5, No.,, [3] Z. T. Xie F. M. Zho, A Genealization of a Hilbet-Te Inealit with the Best Constant Facto, Jonal of Sichan Nomal Univesit (Natal Science), Vol. 3, No. 5, 9, [4] Z. T. Xie Z. Zeng, A Hilbet-Te Integal Inealit with a Non-Homogeneos Fom a Best Constant Facto, Advances Alications in Mathematical Sciens, Vol. 3, No.,, [5] Z. Zeng Z. T. Xie, A New Hilbet-Te Integal Inealit with a Best Constant Facto, Jonal of Soth China Nomal Univesit (Natal Science Edition), Vol. 3,, Coight SciRes.