Inequalities and monotonicity of the ratio for the geometric means of a positive arithmetic sequence with arbitrary difference

Authors

  • Bai-Ni Guo
  • Feng Qi

DOI:

https://doi.org/10.5556/j.tkjm.34.2003.319

Abstract

For all nonnegative integers $ k $ and natural numbers $ n $ and $ m $, we have

$$ \frac{a(n+k+1)+b}{a(n+m+k+1)+b} < \frac{[\prod_{i=k+1}^{n+k} (ai+b)]^{\frac{1}{n}} }{ [\prod_{i=k+1}^{n+m+k} (ai+b)]^{\frac{1}{n+m}}} \le \sqrt{\frac{a(n+k)+b}{a(n+m+k)+b} } , $$

where $ a $ and $ b $ are positive constants. The equality above is valid for $ n = 1 $ and $ m = 1 $. Moreover, some monotonicity results for the sequences involving $ \sqrt[n]{\prod_{i=k+1}^{n+k} (ai+b)} $ are obtained.

Author Biographies

Bai-Ni Guo

Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, Jiaozuo City, henan 454000, CHINA.

Feng Qi

Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, Jiaozuo City, henan 454000, CHINA.

Published

2003-09-30

How to Cite

Guo, B.-N., & Qi, F. (2003). Inequalities and monotonicity of the ratio for the geometric means of a positive arithmetic sequence with arbitrary difference. Tamkang Journal of Mathematics, 34(3), 261–270. https://doi.org/10.5556/j.tkjm.34.2003.319

Issue

Section

Papers