a b s t r a c tThe palindrome complexity function pal w of a word w attaches to each n ∈ N the number of palindromes (factors equal to their mirror images) of length n contained in w. The number of all the nonempty palindromes in a finite word is called the total palindrome complexity of that word. We present exact bounds for the total palindrome complexity and construct words which have any palindrome complexity between these bounds, for binary alphabets as well as for alphabets with the cardinal greater than 2. Denoting by M q (n) the average number of palindromes in all words of length n over an alphabet with q letters, we present an upper bound for M q (n) and prove that the limit of M q (n)/n is 0. A more elaborate estimation leads to M q (n) = O( √ n).

L'accès aux archives de la revue « Annales de la faculté des sciences de Toulouse » (http://picard.ups-tlse.fr/~annales/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

The concept of strictly super-stabilizability for bivariate means has been defined recently by Raïsoulli and Sándor (J. Inequal. Appl. 2014:28, 2014). We answer into affirmative to an open question posed in that paper, namely: Prove or disprove that the first Seiffert mean P is strictly (G, A)-super-stabilizable. We use series expansions of the functions involved and reduce the main inequality to three auxiliary ones. The computations are performed with the aid of the computer algebra systems Maple and Maxima. The method is general and can be adapted to other problems related to sub-or super-stabilizability. MSC: 26E60

No abstract

We consider the problem of finding the optimal values \(\alpha,\ \beta\in\mathbb{R}\) for which the inequality\[\alpha G(a,b)+(1-\alpha)C(a,b)<L(a,b)<\beta G(a,b)+(1-\beta)C(a,b)\]holds for all \(a,b>0\), \(a\neq b\), where \(G(a,b),L(a,b)\) and \(C(a,b)\) are respectively the geometric, logarithmic and anti-harmonic means of \(a\) and \(b\).