摘要
基于二次型的正定性,本文利用Fibonacci数列和Lucas数列的性质找到了2个数列在特定条件下成立的新不等式.该结论将《The Fibonacci Quarterly》上提出的B-1338问题,即对∀n≥0,1/F_(n+1)+1/F_(n+2)>16/9L_(n+1)-16F_(n),1/L_(n+1)+1/L_(n+2)>16/45F_(n+1)-16L_(n),推广到了更一般的结果:对∀n∈Ζ^(+)和∀a,b,c∈Ζ^(+,)当a,b,c满足a^(2)+b^(2)+c^(2)<2ab+2ac+2bc时,都有1/F_(n+1)+1/F_(n+2)>c/(a+b)L_(n+1)-(2a+b)F_(n),1/L_(n+1)+1/L_(n+2)>c/5(a+b)F_(n+1)-(a+2b)L_(n)成立.同时,利用函数的凹凸性得到了2个数列在双曲余弦函数和反正切函数上的新不等式.
In this paper,based on the positive definiteness of quadratic forms,by using the properties of Fi-bonacci and Lucas sequences,we derive new inequalities that hold for two sequences under specific conditions.This extends the problem B-1338 in The Fibonacci Quarterly:for∀n≥0,1/F_(n+1)+1/F_(n+2)>16/9L_(n+1)-16F_(n),1/L_(n+1)+1/L_(n+2)>16/45F_(n+1)-16L_(n),to a more general results:for∀n∈Ζ^(+)and ∀a,b,c∈Ζ^(+),when a,b,c satisfy a^(2)+b^(2)+c^(2)<2ab+2ac+2bc,the inequalities 1/F_(n+1)+1/F_(n+2)>c/(a+b)L_(n+1)-(2a+b)F_(n),1/L_(n+1)+1/L_(n+2)>c/5(a+b)F_(n+1)-(a+2b)L_(n)hold true.Meanwhile,by using the concavity and convexity of functions,we obtain newinequalities related to the two sequences on hyperbolic cosine function and arctangent function.
作者
柯翠菊
王霞
杨苗苗
KE Cuiju;WANG Xia;YANG Miaomiao
出处
《辽宁师专学报(自然科学版)》
2026年第1期6-9,共4页
Journal of Liaoning Normal Colleges(Natural Science Edition)
基金
2025年贵州省基础研究计划(自然科学)面上项目(黔科合基础MS[2025]283)。
关键词
FIBONACCI数列
LUCAS数列
正定二次型
函数凹凸性
不等式
Fibonacci sequence
Lucas sequence
positive definite quadratic forms
concavity and convexity of functions
inequalities
