English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The aim of this paper is to develop analytic techniques to deal with certainmonotonicity of combinatorial sequences. (1) A criterion for the monotonicityof the function $\sqrt[x]{f(x)}$ is given, which is a continuous analog for oneresult of Wang and Zhu. (2) The log-behavior of the functions$\theta(x)=\sqrt[x]{2 \zeta(x)\Gamma(x+1)}$ and$F(x)=\sqrt[x]{\frac{\Gamma(ax+b+1)}{\Gamma(c x+d+1)\Gamma(e x+f+1)}}$ isconsidered, where $\zeta(x)$ and $\Gamma(x)$ are the Riemann zeta function andthe Euler Gamma function, respectively. As consequences, the strictlog-concavities of the function $\theta(x)$ (a conjecture of Chen {\it et al.})and $\{\sqrt[n]{z_n}\}$ for some combinatorial sequences (including theBernoulli numbers, the Tangent numbers, the Catalan numbers, the Fuss-Catalannumbers and some Binomial coefficients) are demonstrated. In particular, thiscontains some results of Chen {\it et al.}, Luca and St\u{a}nic\u{a}. (3). Byresearching logarithmically complete monotonicity of some functions, theinfinite log-monotonicity of the sequence$\{\frac{(n_{0}+ia)!}{(k_0+ib)!(\overline{k_0}+i\overline{b})!}\}_{i\geq0}$ isproved. This generalizes two results of Chen {\it et al.} that both the Catalannumbers $\frac{1}{n+1}\binom{2n}{n}$ and central binomial coefficients$\binom{2n}{n}$ are infinitely log-monotonic and strengths one result of Su andWang that $\binom{dn}{\delta n}$ is log-convex in $n$. (4) The asymptoticallyinfinite log-monotonicity of derangement numbers is showed. (5)Thelogarithmically complete monotonicity of functions $1/\sqrt[x]{a\zeta(x+b)\Gamma(x+c)}$ and$\sqrt[x]{\rho\prod_{i=1}^n\frac{\Gamma(x+a_i)}{\Gamma(x+b_i)}}$ is alsoobtained, which generalizes the results of Lee and Tepedelenlio\v{g}lu, Qi andLi.

Monotonicity and log-behavior of some functions related to the Euler Gamma function