关于推广的Shapiro不等式及其应用

导读:?nxir???i?1???xti??rn????n??s?r?xr?s???x,i?1nxirn2?rmr??.(12)nm?mi?1?m?xi?n显然(12),本文定理及其推论在分式不等式的证明中有着广泛的应用,是建立新分式不等式和推广已有不等式的有力工具.,1999(11):13-15.[3]曾小平.一道分式不等式的推广与运用[J].数,[5]文开庭.Shapiro不等式及其变形的新推广

关于推广的Shapiro不等式及其应用

?r?n?ts?r??xi???xi?i?1故

???s?r???ss?r???n???n???xi??????xit?. ?i?1??i?1??r??sr?nxi ???i?1???xti???s?r?n??. ?n??s?rs?ss?r??rnn?s???t???ts?rs?r????xi??????xi???xi??i?1??i?1? 由(a)(b)(c)(d)可知定理3.1中的(8)式成立.其等号成立的条件有引理2.1,2.2,2.3中不等式取等号条件可得到.

s??(e) 若若r?0,s?0,r?s??,??0,n?1,则?1.分别运用引理2.1和引理2.2,得

r????n????xi??i?1?r?????????sn??xir???xit? ??n??rt?si?1?x???x??i?i??i?1???s????????t????xi??nt????x??i?i?1??????????ss?????????s??r

???????s?n?xir???xit? ???n???i?1??xir???xit?s???i?1???s????????t????xi?n?t????x??i?i?1??????????ss?????????s??r

???n??? ??????i?1?s????????n??? ??????i?1?s?????所以

xir???xitn???srtx???x?iii?1???ss?s??????x??????x?t?ini?1t?i??????????????????????????s??r

s??rx???xnrii?1ri?t?sit?si??x????x??ss??????x??????x?t?iini?1rt?s???????s??s????s??r?1.

??n????xi????i?1? ?sn????nr?tt???xx???x?????iii?i?1??i?1??????????1. ???s??????1r 6

r?nxi故 ???i?1???xti??????s??n??????xit?i?1??n????xi??i?1?r??????s.

(f) 若r?0,s?0,r?s??,??0,n?1,则?r?0,?s?0,?s?(?r)??,??0,n?1.利用(e)中已证结论,有

nn??t?????????xx?????ii?rt?s?n?n??xi???i?1??????????xi???i?1?rs?r?i?1???xt??i?1?xi?n???ni??????xi??????xit?i?1??i?1由(e),(f)可知定理3.1中的(9)式成立.

r?s(g) 若r?0,s?0,r?s??,则0??1.利用引理2.1,得

?sr????????????s.

?r?nxi???i?1???xti???n??r????????xir?s???xits???i?1????????sr?s???? ?????r?s??r?sn?r?1?????xir?s???xit ??n??i?1??????sr?s???????rn???s?r?r??xi?s???xit??n???i?1??????sr?s????r?s.

利用引理2.3,得

r???n???nt???xi??????xi???i?1??i?1????????????s1r?s?n??r??rs???????xi???i?1??r?sr?????rr?s?n??t??????xii?1???????sr?s???rr?s?s??????sr?s??xi?1n?rr?si????x???str?si.

所以

?nxir???i?1???xti??rn????n??s?r?xr?s???xtis???ii?1???????????xit??i?1? 故定理3.1中的(10)式成立.其等号成立的条件由引理2.1中取等号的条件可得. 故由(a)(b)(c)(d)(e)(f)(g)可知定理3.1成立. 4 定理的应用

在定理3.1中令??1,可得如下结论

??????sr?s????r?s?n??s?r?n????xi??i?1?n??s.

推论4.1 设xi?0,??R,??R,t?R,???x?0(i?1,2,?,n),?xi?m,m?0,则

tini?1(Ⅰ)当rs?0,r?s?1(或r?0,s?0)时,有

7

?????x?i?1nxirtsi?n1?s?rmr???????xit??i?1?n??s,

(Ⅱ)当rs?0,r?s?1,n?1时,有

?????x?i?1nxirtsi?mr???????xit??i?1?n??s,

(Ⅲ)当r?0,s?0,r?s?1时,有

?????x?i?1nxirtsi?n1?s?rmr???????xit??i?1?n??s.

在定理3.1的不等式(8)中令???xi??xi??xit,可得:

i?1n推论4.2 设xi?0,??0,??R,t?R,rs?0,r?s??(或r?0,s?0),?xi?m,m?0,

i?1n则有

?n?xir??s?r? ?. ?n?s?i?1?m?x?s?n?i???????m?xi???i?1?在推论4.2中令??1,可得:

??n????xi??i?1?r推论4.3 设xi?0,??R,t?R,rs?0,r?s?1(或r?0,s?0)?xi?m,m?0,则有

i?1nxirn1?s?rmr? ?. (11) ss?nm?m?i?1?m?xi?n特别低在推论4.3的(11)中令s?1,可得

推论4.4 设xi?0,??R,t?R,r?2或r?0,?xi?m,m?0,则有

i?1nxirn2?rmr? ?. (12) nm?mi?1?m?xi?n 显然(12)式是文[10]结论的推广.

本文定理及其推论在分式不等式的证明中有着广泛的应用,是建立新分式不等式和推广已有不等式的有力工具.

参考文献:

8

[1]匡继昌.常用不等式(第三版)[M].济南:山东科学技术出版社,2004.180-182. [2]张光华.用“取等匹配”技巧证明非严格不等式[J].中学数学,1999(11):13-15. [3]曾小平.一道分式不等式的推广与运用[J].数学通讯,1999(9):33-34

[4]沈艳.Shapiro不等式的改进[J].湖南科技学院学报,2005,,27(5):28-30.

[5]文开庭.Shapiro不等式及其变形的新推广与应用[J].贵州教育学院学报(自然科学),17(2):4-6. [6]周昱,高明哲.Shapiro不等式的一个改进[J].湘潭师范学院学报(自然科学版),2006,28(4):13-14. [7]隆建军.Shapiro不等式的指数推广[J].佳木斯大学学报(自然科学版),2012,30(1):121-123,131. [8]刘玉琏,傅沛仁.数学分析讲义:上册[M].北京:高等教育出版社,1992.249-250.

[9]王向东,苏化明,王方汉.不等式·理论·方法[M].郑州:河南教育出版社,1994.434-35. [10]徐丹,杨露.一个不等式的再推广[J].数学通报,2001,39(10):43-44.

On the Generalized Shapiro Inequality and Its Application

LONG Jian-jun

(DaHe Middle School of Panzhihua,Sichuan panzhihua 617061,China)

Abstract:This paper gives an improvement of the Hardy-Hilbert's inequality:Letxi?0,??0,

??R,??R,t?R,???xit?0(i?1,2,?,n),Then we have: (Ⅰ)if rs?0,r?s??(或r?0,s?0),we

?r?nxi???i?1???xti??????n??s?rs??n??????xit?i?1?n????xi??i?1?r??????s,

(Ⅱ)if rs?0,r?s??,n?1,we

?r?nxi???i?1???xti??????s??n??????xit?i?1?n????xi??i?1?r??????s,

(Ⅲ)if r?0,s?0,r?s??,we

????n??s?r. s?sn?????????xit??i?1?The results presented in this paper improve and unify some recent results in this field. Key words: Shapiro inequality;fractional inequality;Holder inequality

r?nxi???i?1???xti???????xi???i?1?nr?? 9

五星文库wxphp.com包含总结汇报、人文社科、IT计算机、计划方案、资格考试、考试资料、文档下载、党团工作以及关于推广的Shapiro不等式及其应用等内容。

本文共2页12