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

Main Article Content

Bai-Ni Guo
Feng Qi

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.

Article Details

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
Section
Papers
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.