袁 琴， 俞芳婷， 王淼坤

(湖州師范學(xué)院 理學(xué)院， 浙江 湖州 313000)

第二類完全橢圓積分的平均值不等式

袁 琴， 俞芳婷， 王淼坤

(湖州師范學(xué)院 理學(xué)院， 浙江 湖州 313000)

研究一個(gè)與第二類完全橢圓積分相關(guān)的平均值,證得它關(guān)于調(diào)和平均、反調(diào)和平均的算術(shù)凸組合與幾何凸組合的兩個(gè)最佳雙邊不等式,進(jìn)而得到第二類完全橢圓積分在某種形式下的兩個(gè)最優(yōu)不等式.

調(diào)和平均； 反調(diào)和平均； 完全橢圓積分； 不等式

MSC 2010:33E05; 26E60

0 引 言

對t∈[0,1],第一類完全橢圓積分K(t)和第二類完全橢圓積分E(t)分別定義如下[1]：

(1)

(2)

自上世紀(jì)90年代以來,完全橢圓積分被廣泛研究,并應(yīng)用于擬共形映射偏差定理的估計(jì).關(guān)于完全橢圓積分的基本性質(zhì)及其應(yīng)用見文獻(xiàn)[1-6].

(3)

定理1 當(dāng)a,b>0且a≠b時(shí),不等式

(4)

成立當(dāng)且僅當(dāng)α≤α0=4/π2及β≥β0=7/16.

定理2 當(dāng)a,b>0且a≠b時(shí),不等式

(5)

若在定理1和定理2中令a=1,b=t′2,便得第二類完全橢圓積分的不等式.這里及下面均記：

(0,1)).

成立當(dāng)且僅當(dāng)α≤α0=4/π2及β≥β0=7/16.

1 主要結(jié)果的證明

定理1的證明 通過不等式變形,定理1可改寫為:當(dāng)a,b>0且a≠b時(shí),

(6)

當(dāng)且僅當(dāng)α≤4/π2及β≥7/16.

則b∈(0,1)且

(7)

令p∈(0,1),則

(8)

記

(9)

計(jì)算得:

(10)

(11)

再令

(12)

則

(13)

(14)

令

(15)

則

(16)

下面分兩種情況進(jìn)行討論：

最后證明α0和β0是使得雙邊不等式(4)成立的最佳參數(shù).

定理2的證明 通過對不等式兩邊同時(shí)取對數(shù)變形后，定理2可改寫為：當(dāng)a,b>0且a≠b時(shí),

(17)

(18)

令

(19)

則

(20)

(21)

其中：

λ(4t2+12)(2E-t′2K)(1-t4)+4(1-λ)(2E-t′2K)(1+t2)(t4+6t2+1)；

(22)

(23)

下面分兩種情形進(jìn)行討論：

(24)

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[2]ABRAMOWITZ M,STEGUN I A.Handbook of Mathematical Functions,with Formulas,Graphs,and Mathematical Tables[M].New York:Dover Publications,1965.

[3]ANDERSON G D,VAMANAMURTHY M K,VUORINEN M.Functional inequalities for complete elliptic integrals and their ratios[J].SIAM J Math Anal,1990,21(2):536-549.

[4]ALZER H,QIU S L.Monotonicity theorems and inequalities for the complete elliptic integrals[J].J Comput Appl Math,2004,172(2):289-312.

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[6]ALZER H,RICHARDS K C.A note on a function involving complete elliptic integrals:monotonicity,convexity,inequalities[J].Anal Math,2015,41(3):133-139.

[7]WANG M K,QIU S L,CHU Y M,et al.Generalized Hersch-Pfluger distortion function and complete elliptic integrals[J].J Math Anal Appl,2012,385(1):221-229.

[8]TOADER G H.Some mean values related to the arithmetic-geometric mean[J].J Math Anal Appl,1998,218(2):358-368.

[9]BULLEN P S,MITRINOVIC D S,VASIC P M.Means and Their Inequalities[M].Dordrecht:D Reidel Publishing Co,1988.

[10]NEUMAN E.On some means derived from the Schwab-Borchardt mean[J].J Math Inequal,2014,8(1):171-183.

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MSC 2010:33E05; 26E60

[責(zé)任編輯 高俊娥]

Inequalities for the Complete Elliptic Integrals of the Second Kind in Terms of Means

YUAN Qin, YU Fangting, WANG Miaokun

(School of Science, Huzhou University, Huzhou 313000, China)

A mean value related to the complete elliptic integrals of the second kind is investigated, and some optimal double inequalities in terms of harmonic mean and contra-harmonic mean are proved. These results lead to two best possible inequalities for the complete elliptic integrals of the second kind in some form.

harmonic mean; contra-harmonic mean; complete elliptic integrals; inequality

2016-12-23

浙江省教育廳科研計(jì)劃項(xiàng)目(Y201635325);湖州師范學(xué)院“大學(xué)生創(chuàng)新訓(xùn)練計(jì)劃”項(xiàng)目(2016-100).

王淼坤，博士，講師，研究方向：特殊函數(shù)、擬共形映射.E-mail:wangmiaokun@zjhu.edu.cn

O172

A

1009-1734(2017)02-0012-05