单参数平均的最优不等式-基础数学专业论文.docx

摘 摘 要 I I 摘 要 解析不等式在数学、物理学和工程技术等各相关领域有广泛的应用。均值不等式作 为基本不等式的重要组成部分,在学术研究中起着极其重要的作用,比如可以用其解决 最值问题,数列收敛的证明和设计优化问题等等。本文研究以下问题：对于 a, b 0 , a ≠ b , 0 α 1,求使双边不等式 J p1  q(a, b) Aα (a, b)G1?α (a, b) J q 1  (a, b), J p (a, b) αA(a, b)+ (1-α )G(a, b) J q (a, b), 2 J p (a, b) αC(a, b)+ (1? α )A(a, b) Jq (a, b) 3 和 J p (a, b) αC(a, b)+ (1? α )H (a, b) Jq (a, b) 4 成立的最优的 p1 (α ), p2 (α ), p3 (α ), p4 (α ), q1 (α ), q2 (α ), q3 (α ), q4 (α ) 的值。这里对于 p ∈ R , a, b 0 ,单参数平均 J p (a, b),算术平均 A(a, b) 和几何平均 G(a, b)定义如下： ? p(a p+1 ? b p+1 ) ??(p +1)(a p ? b p ), a ≠ b, p ≠ 0,?1, ? J (a, b) = ? a ? b ,  a ≠ b, p = 0, ?p ?log a ? log b ? ? ab(log a ?log b),  a ≠ b, p = ?1, ?? ? ?a, A(a, b) = a + b , G(a, b) = 2 a ? b ab .  a = b, 关键词 最优不等式 单参数平均 算术平均 几何平均 解析不等式 Abstract Abstract II II Abstract Analytic inequalities are widely used in mathematics, physics and engineering technology and other related fields. As an important part of the basic inequalities, the mean inequality plays a very important role in academic research, for example, it can be used in solving maximum value problems,the proof of convergence of series and design optimization problems. In the present paper, we answer the question: for 0 α 1 fixed, what are the greatest value p1 (α ),  p2 (α ), p3 (α ), p4 (α )  and the least values  q1 (α ), q2 (α ), q3 (α ), q4 (α ),  such that the inequalities J p1 (a, b) Aα (a, b)G1?α (a, b) J q1 q (a, b), J p (a, b) αA(a, b)+ (1-α )G(a, b) J q (a, b), 2 2 J p (a, b) αC(a, b)+ (1-α )A(a, b) Jq (a, b), 3 3 J p (a, b) αC(a, b)+ (1-α )H (a, b) Jq (a, b) 4 hold for all a, b 0 with  a ≠ b ? where for 4 p ∈ R , the one-parameter mean J p (a, b),arithmetic mean A(a, b) and geometric mean G(a, b)  of two positive real numbers a and b are defined by ? p(a p+1 ? b p+1 ) ??(p +1)(a p ? b p ), a ≠ b, p ≠ 0,?1, ? J (a, b) = ? a ? b ,  a ≠ b, p = 0, ?p ?log a ? log b ? ? ab(log a ?log b),  a ≠ b, p = ?1, ?? ? ?a, a ? b  a = b, A(