A new half-discrete Mulholland-type inequality with multi-parameters

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1 Huang and Yang Journal of Ineualities and Alications 5) 5:6 DOI.86/s R E S E A R C H Oen Access A new half-discrete Mulholland-tye ineuality with multi-arameters Qiliang Huang and Bicheng Yang * * Corresondence: bcyang@gdei.edu.cn Deartment of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 5, P.R. China Abstract By means of weight functions and Hermite-Hadamard s ineuality, ew half-discrete Mulholland-tye ineuality with a best constant factor is given. A best etension with multi-arameters, some euivalent forms, the oerator eressions as well as some articular cases are considered. MSC: 6D5; 47A7 Keywords: Mulholland-tye ineuality; weight function; euivalent form Introduction Assuming that f, g L R + ), f f ) d} >, g >,wehavethefollowing Hilbert integral ineuality cf. [): gy) + y ddy < π f g,.) where the constant factor π is best ossible. If a a m } m, b b n} n l, a m a m } >, b >, then we have the following discrete Hilbert ineuality: m n a m b n < π a b,.) m + n with the same best constant factor π. Ineualities.) and.) areimortantinanalysis and its alications cf. [, ).Ontheotherhand,wehavethefollowingMulholland ineuality with the same best constant factor π cf. [, 4): m a m b n ln mn < π ma m m nbn}..) In 998, by introducing an indeendent arameter,, Yang [5gaveanetension of.). Generalizing the results from [5, Yang [ gave some etensions of.)and.)as follows: If >, +, + R, k, y) is on-negative homogeneous function of degree satisfying k ) k t,)t dt R +, 5 Huang and Yang. This article is distributed under the terms of the Creative Commons Attribution 4. International License htt://creativecommons.org/licenses/by/4./), which ermits unrestricted use, distribution, and reroduction in any medium, rovided you give aroriate credit to the original authors) and the source, rovide a link to the Creative Commons license, and indicate if changes were made.

2 Huangand YangJournal of Ineualities and Alications 5) 5:6 Page of 9 φ) ), ψ) ),, gy), f L,φ R + ) f f,φ : φ) } d < }, g L,ψ R + ), and f,φ, g,ψ >,then k, y)gy) ddy < k ) f,φ g,ψ,.4) where the constant factor k ) is best ossible. Moreover, if k, y) is finite and k, y) k, y)y )isdecreasingfor >y >),thenfora m, b n, } } a a m } m l,φ a a,φ : φm) a m <, m and b b n } n l,ψ, a,φ, b,ψ >,wehave k m, n)a m b n < k ) a,φ b,ψ,.5) m n where the constant factor k )isstillthebestossible.clearly,for,, k, y) +y and,.4)reducesto.), while.5)reducesto.). Some other results about Hilbert-tye ineualities can be found in [6. On halfdiscrete Hilbert-tye ineualities with the general non-homogeneous kernels, Hardy et al. rovided a few results in Theorem 5 of [.Buttheydidnotrovethattheconstant factors are best ossible. In 5, Yang [4 gave a result with the kernel by introducing a variable and roved that the constant factor is best ossible. Recently, Wang and +n) Yang [5 gave a more accurate reverse half-discrete Hilbert-tye ineuality, and Yang [6 rovided the following half-discrete Hilbert ineuality with best constant factor: n d < π f a..6) + n In this aer, by means of weight functions and Hermite-Hadamard s ineuality, ew half-discrete Mulholland-tye ineuality similar to.)and.6) with a best ossible constant factor is given as follows: a n +ln ln n d < π f ) d nan}..7) Moreover, a best etension of.7) with multi-arameters, some euivalent forms, the oerator eressions as well as some articular cases are considered.

3 Huangand YangJournal of Ineualities and Alications 5) 5:6 Page of 9 Some lemmas Lemma. If <σ < σ ), >,β, δ, }, the weight functions ωn) and ϖ ) are defined by ωn):ln βn) σ ln ) δσ + ln δ d, n N\},.) ln βn) ϖ ):ln ) δσ ln βn) σ ) n + ln δ ln βn),,,.) then we have ϖ )<ωn)bσ, σ )..) Proof Substituting t ln δ ln βn in.), and by a simle calculation, for δ, }, we have ωn) + t) tσ dt Bσ, σ ). For fied >, in view of the conditions, it is easy to find that h, y): ln βy) σ y + ln δ ln βy) y + ln δ ln βy) ln βy) σ is decreasing and strictly conve with h y, y)<andh y, y)>,fory, ). Hence by the Hermite-Hadamard ineuality cf. [7), we find ϖ )<ln ) δσ y + ln δ ln βy) ln βy) tln δ ln βy t σ dt Bσ, σ ), ln δ ln β) + t) σ dy and then.) follows. Lemma. Let the assumtions of Lemma. be fulfilled and, additionally, let >, +,, n N\}, is on-negative measurable function in, ). Then we have the following ineualities: J : [ } ln βn)σ n + ln δ ln βn) d [ Bσ, σ ) L : ϖ ) ln ) δσ) f ) d},.4) [ ln ) δ } [ϖ ) + ln δ d ln βn) Bσ, σ ) n ln βn) σ ) an}..5)

4 Huangand YangJournal of Ineualities and Alications 5) 5:6 Page 4 of 9 Proof By Hölder s ineuality cf. [7) and.), it followsthat [ d + ln δ ln βn) + ln δ ln βn) [ ln βn) σ )/ n ln ) δσ)/ [ ln ) δσ)/ ln βn) σ )/ n d} ln ) δσ) ) n ln βn) σ ) ) + ln δ d ln βn) ln ) δσ ωn)n ln βn) σ ) } [ Bσ, σ ) nln βn) σ f ) d + ln δ ln βn) nln βn) σ } ln ) δσ) ) + ln δ ln βn) f ) d nln βn) σ ln ) δσ) ) + ln δ ln βn) f ) d nln βn) σ. Then by Lebesgue term-by-term integration theorem cf. [8), we have J [ Bσ, σ ) [ Bσ, σ ) } ln ) δσ) ) f ) d + ln δ ln βn) nln βn) σ } ln ) δσ) ) f ) d + ln δ ln βn) nln βn) σ [ Bσ, σ ) ϖ ) ln ) δσ) f ) d}, hence,.4)follows. By Hölder s ineuality again, we have [ + ln δ ln βn) + ln δ ln βn) [ ln βn) σ )/ n ln ) δσ)/ [ ln ) δσ)/ ln βn) σ )/ } n n ln βn) σ ) ) + ln δ a ln βn) ln ) δσ n [ϖ ) ln ) δσ ln ) δσ) ) n + ln δ ln βn) ln βn) σ n + ln δ ln βn) ln )δσ ln βn) σ ) ) a n. }

5 Huangand YangJournal of Ineualities and Alications 5) 5:6 Page 5 of 9 By the Lebesgue term-by-term integration theorem, we have L } n + ln δ ln βn) ln )δσ ln βn) σ ) ) a n d [ln βn) σ ln ) δσ d + ln δ ln βn) ωn)n ln βn) σ ) an}, } n ln βn) σ ) a n and in view of.), ineuality.5) follows. Main results We introduce the functions δ ): ln ) δσ) > ), n):n ln βn) σ ) ) n N\}, wherefrom [ δ ) ln )δσ,and[ n) n ln βn)σ. Theorem. If <σ < σ ), >,β, δ, }, >, +,,, f L,, ), a } l,, f, δ >,and a, >,then we have the following euivalent ineualities: I : d + ln δ ln βn) d + ln δ ln βn) < Bσ, σ ) f, δ a,,.) [ [ } d J n) + ln δ < Bσ, σ ) f ln βn),.), δ [ L : δ ) [ } + ln δ d < Bσ, σ ) a,,.) ln βn) where the constant Bσ, σ ) isthebestossibleintheaboveineualities. Proof The two eressions for I in.) follow from Lebesgue s term-by-term integration theorem. By.4)and.), we have.). By Hölder s ineuality, we have I [ n) Then by.), we have.). d + ln δ ln βn) [ n) J a,..4)

6 Huangand YangJournal of Ineualities and Alications 5) 5:6 Page 6 of 9 On the other hand, assuming that.) is valid, we set : [ n) [ d + ln δ ln βn), n N\}. It follows that J a,.by.4), we find J <. IfJ,then.) is trivially valid; if J >,thenby.), we have a, J ) J I < Bσ, σ ) f, δ a,, namely, a, J < Bσ, σ ) f, δ.thatis,.)iseuivalentto.). By.) wehave[ϖ ) >[Bσ, σ ).Theninviewof.5), we have.). By Hölder s ineuality, we find I [ δ )[ ) δ + ln δ ln βn) Then by.), we have.). On the other hand, assume that.) is valid. Setting : [ δ ) [ ) + ln δ, ln βn),, d f, δ L..5) then L f, δ.by.5), we find L <. IfL,then.) is trivially valid; if L >, then by.), we have f, δ L ) L I < Bσ, σ ) f, δ a,, therefore f, δ L < Bσ, σ ) a,,thatis,.) iseuivalentto.). Hence, ineualities.),.), and.)areeuivalent. For < < σ ), setting E δ : ; >, lnδ, )}, ln )δσ + ), E δ ;, ; > } \E δ, and ã n n ln βn)σ, n N\}, if there eists a ositive number k Bσ, σ )), such that.) is valid when relacing Bσ, σ )withk,theninarticular,forδ ±, setting u ln δ, it follows that E δ Ĩ : d ln ) δ u δ du δ+ u +δ, k E δ + ln δ ln βn) ãn d < k f, δ ã, } d ln ) δ+ ln β) + + n } nln βn) +

7 Huangand YangJournal of Ineualities and Alications 5) 5:6 Page 7 of 9 ) } < k ln β) + d + ln β) + k } ln β) +,.6) + ln β) δσ + Ĩ ln βn)σ ln ) ) n + ln δ ln βn) d tln δ ln βn B σ +, σ > B σ +, σ We find E δ ln βn nln βn) + ) ln β) B σ +, σ A): <A) nln βn) + ln βn + t) tσ + dt A) nln βn) + ) dy A) yln βy) + ) A), nln βn) + ln βn nln βn) σ + <, + t +) tσ + dt..7) t tσ + dt and so A)O) + ). Hence by.6)and.7), it follows that ln β) B σ +, σ ) } O) < k ln β) +, + ln β) and Bσ, σ ) k + ). Hence k Bσ, σ )isthebestvalueof.). By the euivalence of the ineualities, the constant factor Bσ, σ )in.).)) is the best ossible. Otherwise, we would reach the contradiction by.4).5)) that the constant factor in.)isnotthebestossible. Remark. i) Define the first tye half-discrete Hilbert-tye oerator T : L, δ, ) l, as follows: For f L, δ, ), we define T f l, by T f n) + ln δ d, n N\}. ln βn) Then by.), T f, Bσ, σ ) f, δ and so T is a bounded oerator with T Bσ, σ ). Since by Theorem., theconstantfactorin.) isbestossible, we have T Bσ, σ ).

8 Huangand YangJournal of Ineualities and Alications 5) 5:6 Page 8 of 9 ii) Define the second tye half-discrete Hilbert-tye oerator T : l, L,, ) δ as follows: For a l,,wedefinet a L,, )by δ T a) ) + ln δ ln βn),,. Then by.), T a, Bσ, σ ) a, and so T is a bounded oerator with δ T Bσ, σ ). Since by Theorem., theconstantfactorin.) is best ossible, we have T Bσ, σ ). Remark. For,,σ, δ in.),.), and.), i) if β,then we have.7) and the following euivalent ineualities: n ) +ln ln n d < π f ) d,.8) +ln ln n ) d < π na n ;.9) ii) if β, then we have the following euivalent ineualities: n d +ln ln n < π f ) d ) +ln ln n d < π +ln ln n nan},.) f ) d,.) ) d < π na n..) Remark.4 For δ in.),.), and.), setting F)ln ), μ σ > ), and ): ln ) μ), we have the following new euivalent ineualities with the same best ossible constant factor Bσ, μ): F) d ln βn) [ [ n) [ [ ) F) ln βn) d < Bσ, μ) F, a,,.) F) d ln βn) } < Bσ, μ) F,,.4) } ln d < Bσ, μ) a,..5) βn) Cometing interests The authors declare that they have no cometing interests.

9 Huangand YangJournal of Ineualities and Alications 5) 5:6 Page 9 of 9 Authors contributions BY carried out the mathematical studies, articiated in the seuence alignment and drafted the manuscrit. QH articiated in the design of the study and erformed the numerical analysis. All authors read and aroved the final manuscrit. Acknowledgements The authors wish to eress their thanks to the referees for their careful reading of the manuscrit and for their valuable suggestions. This work is suorted by the National Natural Science Foundation No. 6786), and Knowledge Construction Secial Foundation Item of Guangdong Institution of Higher Learning College and University No. KJCX4). Received: 8 March 5 Acceted: July 5 References. Hardy, GH, Littlewood, JE, Pólya, G: Ineualities. Cambridge University Press, Cambridge 94). Mitrinović, DS, Pečarić, JE, Fink, AM: Ineualities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Boston 99). Yang, BC: The Norm of Oerator and Hilbert-Tye Ineualities. Science Press, Beijing 9) 4. Yang, BC: An etension of Mulholand s ineuality. Jordan J. Math. Stat. ), 5-57 ) 5. Yang, BC: On Hilbert s integral ineuality. J. Math. Anal. Al., ) 6. Yang, BC,Brnetić, I, Krnić, M, Pečarić, J: Generalization of Hilbert and Hardy-Hilbert integral ineualities. Math. Ineual. Al. 8),59-7 5) 7. Krnić, M, Pečarić, J: Hilbert s ineualities and their reverses. Publ. Math. Debr.) 67-4),5-5) 8. Jin, JJ, Debnath, L: On a Hilbert-tye linear series oerator and its alications. J. Math. Anal. Al. 7, ) 9. Azar, L: On some etensions of Hardy-Hilbert s ineuality and alications. J. Ineual. Al. 9, ). Arad, B, Choonghong, O: Best constant for certain multi-linear integral oerator. J. Ineual. Al. 6, 858 6). Kuang, JC, Debnath, L: On Hilbert s tye ineualities on the weighted Orlicz saces. Pac. J. Al. Math. ),95-7). Zhong, WY: The Hilbert-tye integral ineuality with a homogeneous kernel of Lambda-degree. J. Ineual. Al. 8, 979 8). Li,YJ,He,B:OnineualitiesofHilbert stye.bull.aust.math.soc.76), - 7) 4. Yang, BC: A mied Hilbert-tye ineuality with a best constant factor. Int. J. Pure Al. Math. ), 9-8 5) 5. Wang, AZ, Yang, BC: A more accurate reverse half-discrete Hilbert-tye ineuality. J. Ineual. Al. 5, 85 5) 6. Yang, BC: A half-discrete Hilbert s ineuality. J. Guangdong Univ. Educ. ), -7 ) 7. Kuang, JC: Alied Ineualities. Shangdong Science Technic Press, Jinan 4) 8. Kuang, JC: Introduction to Real Analysis. Hunan Education Press, Chansha 996)