All Questions
14 questions
14
votes
3
answers
664
views
(Sharp) inequality for Beta function
I am trying to prove the following inequality concerning the Beta Function:
$$
\alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0,
$$
where as usual $B(a,b) = \...
- 315
13
votes
1
answer
638
views
A question on the sine function
The Fejer-Jackson-Gronwall inequality involving the sine function is as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
Here I ask the ...
- 15.6k
7
votes
3
answers
515
views
Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$
The question is to prove:
$$
\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1.
$$
Numerically it seems to hold true. So I have made some attempts to ...
- 419
5
votes
3
answers
383
views
The exact constant in a bound on ratios of Gamma functions
The answer to another question (Upper bound of the fraction of Gamma functions) gave an asymptotic upper bound for an expression with Gamma functions:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\...
- 343
4
votes
1
answer
387
views
$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$
The Fejer-Jackson inequality as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
I conjecture that the inequality as follows holds:
$$\sum_{...
- 1,776
3
votes
2
answers
731
views
Non-asymptotic upper bound of right tail of Gamma function
I'm wondering if there is any non-asymptotic upper bound for the following Gamma function:
$$f_a(x)=\int_{x}^{\infty}t^a\exp(-t)dt$$
for $x>0,a>0$? Something like $x^a\exp(-x)$?
- 1,495
3
votes
2
answers
698
views
Estimate of a ratio of two incomplete gamma functions
I would like to bound from above the expression
$$
\frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}
$$
for $x>y>0$. By plotting the above expression I have found that ...
- 263
3
votes
1
answer
223
views
Ratio of Selberg integral
I'm considering a ratio of incomplete Selberg integral:
$$f_n(a,b)=\frac{\int_{\Delta_a}\prod_{i=1}^nx_i^{\alpha-\frac{n+1}{2}}\prod_{i=1}^n(1-x_i)^{-1/2}\prod_{i<j}|x_i-x_j|}{\int_{\Delta_b}\prod_{...
- 1,495
2
votes
1
answer
214
views
Ratio of hypergeometric function
Given $a>b>0$, is there any upper bound of the following ratio of hypergeometric function?
$$\frac{_2F_1(a,1-b;a+1;x)}{_2F_1(a,1-b;a+1;y)}$$
for $1>x>y>0$ ideally in the form like some ...
- 1,495
1
vote
1
answer
444
views
Upper bound of a ratio of integrals
I'm wondering how to upper bound the following ratio of integrals:
$$\frac{\int_{\Delta_a}(\prod_{i=1}^n\lambda_i)^{p-1}\prod_{i<j}|\lambda_i-\lambda_j|}{\int_{\Delta_b}(\prod_{i=1}^n\lambda_i)^{p-...
- 1,495
1
vote
1
answer
507
views
Upper bound of the fraction of Gamma functions
Is there a simple upper bound of the following fraction of Gamma functions for any $a,b\geq1/2$:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}$$
An upper bound in the following form is ...
- 1,495
1
vote
0
answers
112
views
How to prove a simple Struve inequality
It appears from the figure 11.3.4 of dlmf that the Struve function $M_1(x)$ is monotonically decreasing for positive $x$. The asymptotic expansion (11.6.2) shows that the limit is $-2/\pi$. So it ...
- 141
0
votes
1
answer
229
views
Inequality with even powers of trigonometric functions
For $m>0$,
$0 < n\leqslant m+1$ ($m,n\in \mathbb{Z} $) , and $0 < a < 1$ , prove that
$$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi ...
0
votes
0
answers
43
views
The reciprocal of the normalized tail of the Maclaurin power series expansion of the hyperbolic sinc function is a convex function
The classical Bernoulli numbers $B_j$ are generated by
\begin{equation}\label{Bernoulli-No-Generating}
\frac{x}{\operatorname{e}^x-1}=\sum_{j=0}^\infty B_j\frac{x^j}{j!}=1-\frac{x}2+\sum_{j=1}^\infty ...
- 1,091