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浙江大学学报（理学版）
浙江大学学报（理学版）  2019, Vol. 46 Issue (3): 295-301    DOI: 10.3785/j.issn.1008-9497.2019.03.005
数学与计算机科学     
关于Neuman-Sándor平均的一些特殊组合不等式
徐仁旭1, 徐会作2, 钱伟茂3
1.浙江建设职业技术学院 人文与信息系，浙江 杭州 311231
2.温州广播电视大学 教师教学发展中心，浙江;温州 325000
3.湖州广播电视大学 远程教育学院，浙江 湖州 313000
On some special combination inequalities for Neuman-Sándor mean
Renxu XU1, Huizuo XU2, Weimao QIAN3
1.Department of Humanity and Information,Zhejiang College of Construction,Hangzhou 311231,China
2.Teachers’ Teaching Development Center, Wenzhou Broadcast and TV University, Wenzhou 325000, Zhejiang Province,China
3.School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, Zhejiang Province,China
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摘要： 研究了Neuman-Sándor平均NS(a,b)关于调和平均H(a,b)、算术平均A(a,b)、二次平均Q(a,b)若干特殊组合的序关系，给出最佳参数α1,α2,α3,α14,β1,β2,β3,β4∈(0,1),使得下列双向不等式：$\sqrt{a_{1}Q^{2}(a,b)+(1-a_{1})A^{2}(a,b)}< NS(a,b)<\sqrt{\beta_{1}Q^{2}(a,b)+(1-\beta_{1})A^{2}(a,b),}\\ \sqrt{[a_{2}Q(a,b)+(1-a_{2})A(a,b)]A(a,b)}< NS(a,b)<\sqrt{[\beta_{2}Q(a,b)+(1-\beta_{2})A(a,b)]A(a,b),}\\ \sqrt{a_{e}Q^{2}(a,b)+(1-a_{3})H^{2}(a,b)}< NS(a,b)<\sqrt{\beta_{3}Q^{2}(a,b)+(1-\beta_{3})H^{2}(a,b),}\\ \sqrt{[a_{4}Q(a,b)+(1-a_{4})H(a,b)]A(a,b)}< NS(a,b)<\sqrt{[\beta_{4}Q(a,b)+(1-\beta_{4})H(a,b)]A(a,b),}$对所有不同的正实数a和b均成立。
关键词： Neuman-Sándor平均;  调和平均;  算术平均;  二次平均    
Abstract: In the article, we deal with the order relation on the Neuman-Sándor mean NS(a,b) with respect to the special combinations of the harmonic mean H(a,b),arithmetic mean A(a,b) and quadratic mean Q(a,b) ，and present the best possible parameters α1,α2,α3,α14,β1,β2,β3,β4∈(0,1) such that the double inequalities $\sqrt{a_{1}Q^{2}(a,b)+(1-a_{1})A^{2}(a,b)}< NS(a,b)<\sqrt{\beta_{1}Q^{2}(a,b)+(1-\beta_{1})A^{2}(a,b),}\\ \sqrt{[a_{2}Q(a,b)+(1-a_{2})A(a,b)]A(a,b)}< NS(a,b)<\sqrt{[\beta_{2}Q(a,b)+(1-\beta_{2})A(a,b)]A(a,b),}\\ \sqrt{a_{e}Q^{2}(a,b)+(1-a_{3})H^{2}(a,b)}< NS(a,b)<\sqrt{\beta_{3}Q^{2}(a,b)+(1-\beta_{3})H^{2}(a,b),}\\ \sqrt{[a_{4}Q(a,b)+(1-a_{4})H(a,b)]A(a,b)}< NS(a,b)<\sqrt{[\beta_{4}Q(a,b)+(1-\beta_{4})H(a,b)]A(a,b),}$ hold for all a,b>0 with a≠b, where H(a,b), A(a,b), Q(a,b) and NS(a,b) are the harmonic, arithmetic, quadratic and Neuman-Sándor means of a and b, respectively.
Key words: Neuman-Sàndor mean    harmonic mean    arithmetic mean    quadratic mean
收稿日期: 2018-08-27 出版日期: 2019-05-25
CLC:  O174.6  
基金资助: 国家自然科学基金资助项目（61673169, 61374086, 11371125, 11401191）；浙江广播电视大学科学研究课题（XKT-17Z04，XKT-17G26）；浙江省教育厅2017年度高校访问学者“教师专业发展项目”（FX2017108，FX2017084）.
作者简介: 徐仁旭（1980—），https://orcid.org/0000-0003-0383-7917，男，硕士，讲师，主要从事解析不等式研究，E-mail：55444765@qq.com.
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徐仁旭, 徐会作, 钱伟茂. 关于Neuman-Sándor平均的一些特殊组合不等式[J]. 浙江大学学报（理学版）, 2019, 46(3): 295-301.

Renxu XU, Huizuo XU, Weimao QIAN. On some special combination inequalities for Neuman-Sándor mean. Journal of Zhejiang University (Science Edition), 2019, 46(3): 295-301.

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https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2019.03.005    或    https://www.zjujournals.com/sci/CN/Y2019/V46/I3/295

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