Characterization of Monotone Sequences of Positive Numbers Prescribed by Means

The aim of this article is to investigate the relations between the exponent of the convergence of sequences and other characteristics defined for monotone sequences of positive numbers. Another main goal is to characterize such monotone sequences (an) of positive numbers that, for each n≥2, satisfy the equality an=K(an−1,an+1), where the function K:R+×R+→R+ is the mean, i.e., each value of K(x,y) lies between min{x,y} and max{x,y}. Well-known examples of such sequences are, for example, arithmetic (geometric) progression, because starting from the second term, each of its terms is equal to the arithmetic (geometric) mean of its neighboring terms. Furthermore, this accomplishment generalized and extended previous results, where the properties of the logarithmic sequence (an) are referred to, i.e., in such a sequence that every n≥2 satisfies an=L(an−1,an+1), where L(x,y) is the logarithmic mean of positive numbers x,y defined as follows: L(x,y):=y−xlny−lnxifx≠y,xifx=y.