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Distribution of terms of a logarithmic sequence
Abstract
The number L(a, b) = a b ln a ln b for a 6= b and L(a, a) = a, is said to be the logarithmic mean of the positive numbers a, b. We shall say that a sequence (an) 1 n=1 with positive terms is a logarithmic sequence if an = L(an 1, an+1). In the present paper some basic estimations of the terms of logarithmic sequences are investigated.
... In this part, we recall some basic definitions. The following definitions are from papers [1,2,7,[9][10][11]. ...
... ( [1], Corollary 2.3., and Theorem 2.1.(i)) (ii) The inequality a n+1 − a n > ( √ a 2 − √ a 1 ) 2 (n + 1) holds for every n ≥ 2. ...
... ( [1], Theorem 2.6.) (iii) We have lim n→∞ a n+1 a n = 1 and lim n→∞ a n q n = 0 for every real q > 1. ...
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.
... This area has been studied by many mathematicians. For this paper we were inspired by [2,3,4,5,7]. ...
The paper deals with the generalized Gauss composition of arbitrary means. We give sufficient conditions for the existence of this generalized Gauss composition. Finally, we show that these conditions cannot be improved or changed.
Means of positive numbers and certain types of series
- J Bukor
- T Šalát
- J Tóth
- L Zsilinszky
Bukor, J., Šalát, T., Tóth, J. and Zsilinszky, L., Means of positive numbers and certain types of series, Acta Mathematica et Informatica Nitra, Vol. 1 (1992) 49–57.
Problems and Theorems in Analysis, I Springer–Verlag rDistribution of terms of a logarithmic sequence 45 Peter Csiba
- G Pólya
- G Szegő
Pólya, G. and Szegő, G., Problems and Theorems in Analysis, I Springer–Verlag, Berlin, Heidelberg (1962). rDistribution of terms of a logarithmic sequence 45 Peter Csiba, Ferdinánd Filip, János T. Tóth Department of Mathematics, J. Selye University, P.O.Box 54, 945 01 Komárno, Slovakia János T. Tóth Department of Mathematics, University of Ostrava, 30. dubna 22, 701 03 Ostrava, Czech Republic