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

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.