与Fibonacci数列和Lucas数列有关的不等式探究

An exploration of inequalities related to Fibonacci sequences and Lucas sequences

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∈N⁺和∀a,b,c∈N⁺,当a,b,c满足a²+b²+c²<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.

Abstract

In this paper, based on the positive definiteness of quadratic forms, by using the properties of Fibonacci 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∈N⁺ and ∀a, b, c∈N⁺, when a, b, c satisfy a²+b²+c² < 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 new inequalities related to the two sequences on hyperbolic cosine function and arctangent function.

关键词

Fibonacci数列 / Lucas数列 / 正定二次型 / 函数凹凸性 / 不等式

Key words

Fibonacci sequence / Lucas sequence / positive definite quadratic forms / concavity and convexity of functions / inequalities

参考文献

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基金资助

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)