Recently, Došlić, and Liu and Wang developed techniques for dealing with the log-convexity of sequences. In this paper, we present a criterion for the log-convexity of some combinatorial sequences. In order to prove the log-convexity of a sequence satisfying a three-term recurrence, by our method, it suffices to compute a constant number of terms at the beginning of the sequence. For example, i...

In this paper, we discuss the properties of the hyperfibonacci numbers F [r] n and hyperlucas numbers L [r] n . We investigate the log-concavity (log-convexity) of hyperfibonacci numbers and hyperlucas numbers. For example, we prove that {F [r] n }n≥1 is log-concave. In addition, we also study the log-concavity (log-convexity) of generalized hyperfibonacci numbers and hyperlucas numbers.

This note provides another proof for the convexity (strict convexity) of log det(I + KX) over the positive definite cone for any given positive semidefinite matrix K 0 (positive definite matrix K ≻ 0) and the strictly convexity of log det(K + X) over the positive definite cone for any given K 0. Equivalent optimization representation with linear matrix inequalities (LMIs) for the functions log ...

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.

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...