منابع مشابه
Bell Numbers, Log-concavity, and Log-convexity
Let fb k (n)g 1 n=0 be the Bell numbers of order k. It is proved that the sequence fb k (n)=n!g 1 n=0 is log-concave and the sequence fb k (n)g 1 n=0 is log-convex, or equivalently, the following inequalities hold for all n 0, 1 b k (n + 2)b k (n) b k (n + 1) 2 n + 2 n + 1 : Let f(n)g 1 n=0 be a sequence of positive numbers with (0) = 1. We show that if f(n)g 1 n=0 is log-convex, then (n)(m) (n...
متن کاملLog-convexity and log-concavity of hypergeometric-like functions
We find sufficient conditions for log-convexity and log-concavity for the functions of the forms a 7→ ∑ fk(a)kx , a 7→ ∑ fkΓ(a + k)x k and a 7→ ∑ fkx k/(a)k. The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Turán inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexi...
متن کاملConvexity and Log Convexity for the Spectral Radius
The starting point of this paper is a theorem by J. F. C. Kingman which asserts that if the entries of a nonnegative matrix are log convex functions of a variable then so is the spectral radius of the matrix. A related result of J. Cohen asserts that the spectral radius of a nonnegative matrix is a convex function of the diagonal elements. The first section of this paper gives a new, unified pr...
متن کاملLog–convexity of Combinatorial Sequences from Their Convexity
A sequence (xn)n 0 of positive real numbers is log-convex if the inequality xn xn−1xn+1 is valid for all n 1 . We show here how the problem of establishing the log-convexity of a given combinatorial sequence can be reduced to examining the ordinary convexity of related sequences. The new method is then used to prove that the sequence of Motzkin numbers is log-convex.
متن کاملOn the log-convexity of combinatorial sequences
Here presented is a survey for the log-convexity of some famous combinatorial sequences. We develop techniques for dealing with the log-convexity of sequences satisfying a three-term recurrence. We also introduce the concept of q-log-convexity and establish the link with linear transformations preserving the log-convexity. MSC: 05A20; 11B73; 11B83; 11B37
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2007
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-07-08991-5