Zobrazeno 1 - 10
of 446
pro vyhledávání: '"Alzer, Horst"'
Autor:
Alzer, Horst, Pedersen, Henrik L.
We prove that the function $g(x)= 1 / \bigl( 1 - \cos(x) \bigr)$ is completely monotonic on $(0,\pi]$ and absolutely monotonic on $[\pi, 2\pi)$, and we determine the best possible bounds $\lambda_n$ and $\mu_n$ such that the inequalities $$ \lambda_n
Externí odkaz:
http://arxiv.org/abs/2406.08932
Autor:
Alzer, Horst, Kwong, Man Kam
We present several new inequalities for trigonometric sums. Among others, we show that the inequality $$ \sum_{k=1}^n (n-k+1)(n-k+2)k\sin(kx) > \frac{2}{9} \sin(x) \bigl( 1+2\cos(x) \bigr)^2 $$ holds for all $n\geq 1$ and $x\in (0, 2\pi/3)$. The cons
Externí odkaz:
http://arxiv.org/abs/2307.04464
Autor:
Alzer, Horst, Yakubovich, Semyon
Let $$ A_{m,n}(a)=\sum_{j=0}^m (-4)^j {m+j\choose 2j}\sum_{k=0}^{n-1} \sin(a+2k\pi/n) \cos^{2j}(a+2k\pi/n) $$ and $$ B_{m,n}(a)=\sum_{j=0}^m (-4)^j {m+j+1\choose 2j+1}\sum_{k=0}^{n-1} \sin(a+2k\pi/n) \cos^{2j+1}(a+2k\pi/n), $$ where $m\geq 0$ and $n\
Externí odkaz:
http://arxiv.org/abs/2212.14841
Autor:
Alzer, Horst, Kouba, Omran
We give two new proofs of the Chaundy-Bullard formula $$ (1-x)^{n+1} \sum_{k=0}^m {n+k\choose k} x^k +x^{m+1}\sum_{k=0}^n {m+k\choose k} (1-x)^k=1 $$ and we prove the "twin formula" $$ \frac{ (1-x)^{(n+1)}}{(n+1)!} \sum_{k=0}^m \frac{n+1}{n+k+1} \fra
Externí odkaz:
http://arxiv.org/abs/2205.00480
Autor:
Alzer, Horst, Choi, Junesang
Publikováno v:
Applicable Analysis and Discrete Mathematics, 2022 Oct 01. 16(2), 524-533.
Externí odkaz:
https://www.jstor.org/stable/27174772
Autor:
Alzer, Horst, Kwong, Man Kam
Let $$ T(q)=\sum_{k=1}^\infty d(k) q^k, \quad |q|<1, $$ where $d(k)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we proved that $$ H(q)
Externí odkaz:
http://arxiv.org/abs/2010.05018
Autor:
Alzer, Horst, Kwong, Man Kam
The arc lemniscate sine function is given by $$ \mbox{arcsl}(x)=\int_0^x \frac{1}{\sqrt{1-t^4}}dt. $$ In 2017, Mahmoud and Agarwal presented bounds for $\mbox{arcsl}$ in terms of the Lerch zeta function $$ \Phi(z,s,a)=\sum_{k=0}^\infty \frac {z^k}{(k
Externí odkaz:
http://arxiv.org/abs/1903.03897