We establish here an integral inequality for real log-concave functions, which can be viewed as an average monotone likelihood property. This inequality is then applied to examine the monotonicity of failure rates.

We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for integer moments (after neglecting the first $p^2$ terms of the sequence).

Given a sequence of polynomials $\{x_k(q)\}_{k \ges 0}$, define the transformation

$$y_n(q) = a^n\sum\limits_{i = 0}^n {\left( \matrix{n \cr i} \right)} b^{n-i}x_i(q)$$

$n\ges 0$

$[x_{i+j}(q)]_{i,j \ges 0}$

$\{y_n(q)\}_{n\ges 0}$

Recently, Sun posed a series of conjectures on the log-concavity of the sequence , where is a familiar combinatorial sequence of positive integers. Luca and Stănică, Hou et al. and Chen et al. proved some of Sun's conjectures. In this paper, we present a criterion on the log-concavity of the sequence . The criterion is based on the existence of a function f(n) that satisfies some inequalities involving terms related to the sequence . Furthermore, we present a heuristic approach to compute f(n). As applications, we prove that, for the Zagier numbers , the sequences are strictly log-concave, which confirms a conjecture of Sun. We also prove the log-concavity of the sequence of Cohen–Rhin numbers.

Let [An,k]n,k⩾0 be an infinite lower triangular array satisfying the recurrence

for n ⩾ 1 and k ⩾ 0, where A0,0 = 1, A0,k = Ak,–1 = 0 for k > 0. We present some criteria for the log-concavity of rows and strong q-log-convexity of generating functions of rows. Our results can be applied to many well-known triangular arrays, such as the Pascal triangle, the Stirling triangle of the second kind, the Bell triangle, the large Schröder triangle, the Motzkin triangle, and the Catalan triangles of Aigner and Shapiro, in a unified approach. In addition, we prove that the binomial transformation not only preserves the strong q-log-convexity property, but also preserves the strong q-log-concavity property. Finally, we demonstrate that the strong q-log-convexity property is preserved by the Stirling transformation and Whitney transformation of the second kind, which extends some known results for the strong q-log-convexity property.

The aim of this paper is to develop analytic techniques to deal with the monotonicity of certain combinatorial sequences. On the one hand, a criterion for the monotonicity of the function is given, which is a continuous analogue of a result of Wang and Zhu. On the other hand, the log-behaviour of the functions

is considered, where ζ(x) and Γ(x) are the Riemann zeta function and the Euler Gamma function, respectively. Consequently, the strict log-concavities of the function θ(x) (a conjecture of Chen et al.) and for some combinatorial sequences (including the Bernoulli numbers, the tangent numbers, the Catalan numbers, the Fuss–Catalan numbers, and the binomial coefficients are demonstrated. In particular, this contains some results of Chen et al., and Luca and Stănică. Finally, by researching the logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequence

is proved. This generalizes two results of Chen et al. that both the Catalan numbers and the central binomial coefficients are infinitely log-monotonic, and strengthens one result of Su and Wang that is log-convex in n for positive integers d > δ. In addition, the asymptotically infinite log-monotonicity of derangement numbers is showed. In order to research the stronger properties of the above functions θ(x) and F(x), the logarithmically complete monotonicity of functions

is also obtained, which generalizes the results of Lee and Tepedelenlioǧlu, and Qi and Li.

In this paper, we use the Riemann zeta function ζ(x) and the Bessel zeta function ζμ(x) to study the log behaviour of combinatorial sequences. We prove that ζ(x) is log-convex for x > 1. As a consequence, we deduce that the sequence {|B2n|/(2n)!}n ≥ 1 is log-convex, where Bn is the nth Bernoulli number. We introduce the function θ(x) = (2ζ(x)Γ(x + 1)) 1/x, where Γ(x) is the gamma function, and we show that logθ(x) is strictly increasing for x ≥ 6. This confirms a conjecture of Sun stating that the sequence is strictly increasing. Amdeberhan et al. defined the numbers an(μ) = 2 2n+1 (n + 1)!(μ+ 1)nζμ(2n) and conjectured that the sequence {an(μ)}n≥1 is log-convex for μ = 0 and μ = 1. By proving that ζμ(x) is log-convex for x > 1 and μ > -1, we show that the sequence {an(≥)}n>1 is log-convex for any μ > - 1. We introduce another function θμ,(x) involving ζμ(x) and the gamma function Γ(x) and we show that logθμ(x) is strictly increasing for x > 8e(μ + 2)2. This implies that

Based on Dobinski’s formula, we prove that

where Bn is the nth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of and Holder’s inequality in probability theory.

Brown (1980), (1981) proved that the renewal function is concave if the interrenewal distribution is DFR (decreasing failure rate), and conjectured the converse. This note settles Brown's conjecture with a class of counterexamples. We also give a short proof of Shanthikumar's (1988) result that the DFR property is closed under geometric compounding.

This note introduces shape orderings for stationary time series autocorrelation and partial autocorrelation functions and explores some of their convergence rate ramifications. The shapes explored include decreasing hazard rate and new better than used, orderings that are familiar from stochastic processes settings. Time series models where these shapes arise are presented. The shapes are used to obtain explicit geometric convergence rates for mean squared errors of one-step-ahead forecasts.

This paper considers the expression of expectations and density function of the extinction time for a continuous-time Markov branching process.