Abstract

We call a sequence of real numbers, $\{a_n\}_{n\geq1}$, an asymptotically arithmetic sequence, if its increment $a_{n+1}-a_{n}$ approaches a real number $d$, as $n\toınfty$. For each $pın[-ınfty,ınfty]$, we compute the limit of the increment $H_p(a_1,\dots,a_n,a_{n+1})-H_p(a_1,\dots,a_n)$, of the $p$-Hölder mean sequence, $\{H_p(a_1,\dots,a_n)\}_{n\geq1}$, of an asymptotically arithmetic sequence $\{a_n\}_{n\geq1}$, with positive terms. Moreover, for $p\leq-1$, we not only show that this limit is $0$, but we also compute the rate with which the increment approaches zero.

Keywords: Hölder means; Stolz-Ces\`{a}ro theorem; D'Alembert theorem; Lagrange Mean Value theorem; Lalescu sequence.

DOI: 10.57016/TM-PVJD7224

Pages:  1$-$14     

Volume  XXVII ,  Issue  1 ,  2024