Advanced Search

Let We prove that $F$ is completely monotonic on $(0,\mathrm{\infty })$. This complements a result of Miller and Moskowitz (2006), who proved that $F$ is positive and strictly decreasing on $(0,\mathrm{\infty })$. The sequence $\{S(k)\}$ $(k=1,2,\mathrm{\dots })$ plays a role in information theory.

In this note we prove a new extension and a converse of an inequality due to Gauss.

Let $p\in (0,1)$ be a real number and let $n\ge 2$ be an even integer. We determine the largest value $c_n\left(p\right)$ such that the inequality $$\sum _{i=1}^n{\left|a_i\right|}^p\ge c_n\left(p\right)$$ holds for all real numbers $a_1,...,a_n$ which are pairwise distinct and satisfy ${min}_{i\ne j}|a_i-a_j|=1$. Our theorem completes results of Ozeki, Mitrinović-Kalajdžić, and Russell, who found the optimal value $c_n\left(p\right)$ in the case $p>0$ and $n$ odd, and in the case $p\ge 1$ and $n$ even.

We prove: If $$\frac{1}{2}+\sum _{k=1}^n{a}_k\left(n\right)cos\left(kx\right)\ge 0\text{for}\text{all}x\in [0,2\pi ),$$ then $$1-a_k\left(n\right)\ge \frac{1}{2}\frac{k^2}{n^2}\text{for}k=1,\cdots ,n.$$ The constant $1/2$ is the best possible.

We prove: If $A\left(n\right)$ and $G\left(n\right)$ denote the arithmetic and geometric means of the first $n$ positive integers, then the sequence $n\mapsto nA\left(n\right)/G\left(n\right)-(n-1)A(n-1)/G(n-1)$ $(n\ge 2)$ is strictly increasing and converges to $e/2$, as $n$ tends to $\infty$.

In this paper we refine an inequality for infinite series due to Astala, Gehring and Hayman, and sharpen and extend a Holder-type inequality due to Daykin and Eliezer.

We prove: (I) For all integers n ≥ 2 and real numbers x ∈ (0,π) we have $\alpha \le {\sum }_{j=1}^{n-1}1/(n²-j²)sin\left(jx\right)\le \beta$, with the best possible constant bounds α = (15-√2073)/10240 √(1998-10√2073) = -0.1171..., β = 1/3. (II) The inequality $0<{\sum }_{j=1}^{n-1}(n²-j²)sin\left(jx\right)$ holds for all even integers n ≥ 2 and x ∈ (0,π), and also for all odd integers n ≥ 3 and x ∈ (0,π - π/n].

We prove that $|{\sum }_{k=1}^{n}sin\left((2k-1)x\right)/k|<Si\left(\pi \right)=1.8519...$ for all integers n ≥ 1 and real numbers x. The upper bound Si(π) is best possible. This result refines inequalities due to Fejér (1910) and Lenz (1951).

We study the Diophantine equations ${(k!)}^n-k^n={(n!)}^k-n^k$ and ${(k!)}^n+k^n={(n!)}^k+n^k,$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k=n$ or $(k,n)=(1,2),(2,1)$ and that the second one holds if and only if $k=n$.