# Tagged Questions

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### Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
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### lower bound of modified Besselfunction

i'm looking for an lower bound for the modified Bessel function of the first kind $I_\nu(x)$ of a +ive real argument. There should be one of the form $$ce^{x^\alpha} \le I_\nu(x)$$.
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### How prove this $p(x)>0$ if $p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$

let the polynomials $$p(x)=\sum_{i=0}^{n}\binom{n}{i}a_{i}x^i(1-x)^{n-i}$$ and such $$a_{0}+\sum_{a_{i}<0}(1-\dfrac{i}{n})\binom{n}{i}a_{i}>0$$ and ...
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### Upper bound on the modified Bessel function of the first kind

I am looking for an upper bound tighter than $e^x$ for the modified Bessel function of the first kind. Does there exist $\epsilon\in (0,1)$ such that $I_0(x)\le e^{x^{1-\epsilon}}$ for all $x\ge 0$ ? ...
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### Proving Inequality with the Greatest Integer Function

Show that $$[(m+n)x]+[(m+n)y] \ge [mx+(n-1)y]+[my+(n-1)x]$$ where $m,~n \in \Bbb{N}$ and $0\le x,~y < 1$. I've tried everything for about half a day and still couldn't figure it out. ...
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### Inequality $\Gamma(\alpha+x)\Gamma(\beta+y)\leq C(\alpha; \beta)\Gamma(x+y-1)$

I need to prove $$\Gamma(\alpha+x)\Gamma(\beta+y)\leq C(\alpha; \beta)\Gamma(x+y),$$ for $x>\alpha$ and $y>\beta$ with $0<\alpha,\beta \leq \frac{1}{2}$ constants, and $C(\alpha, \beta)$ is a ...
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### Flipping inequality sign with $\operatorname{Erfc}^{-1}$

Say I have inequality $\operatorname{Erfc}(x)\geq y$ where $\operatorname{Erfc}(x)=\frac{2}{\sqrt{\pi}}\int_{x}^\infty e^{-t^2}dt$ is the complimentary error function. I can use the inverse ...
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### Calculation of integral with Bessel function

I have a trouble with to calculating (or bounding from above) the following integral: $$\int_{-\infty}^{\infty}\left(\frac{J_2(x)}{x^2}\right)^p\, dx, \quad p\geq 1,$$ where $J_2(x)$ is a Bessel ...
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### Inequality between incomplete beta and gamma functions

Let the regularized incomplete beta and gamma functions be defined as usual: $$I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt,$$ ...
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### Cardinality of Solutions to an Inequality [duplicate]

Show that the number of solutions in nonneg. int. of the ineq. $$x_1+x_2+\cdots +x_n\leq M,$$ where $M$ is a nonneg. int., is $C(M+n,n)$.
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### Upper bound for the quotient of gamma functions?

I am looking for an upper bound for $$\frac{\Gamma(x+\beta)}{\Gamma(x)},\,\,\,\beta>0.$$ In this question it was shown that $$\frac{\Gamma(x+\beta)}{\Gamma(x)} \approx x^\beta.$$ Then, I ...
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### Bounding the modified Bessel function of the first kind

i'm looking for an upper bound for the modified Bessel function of the first kind of a +ive real argument. It seems that it satisfies the inequality : $$I_{n}(x)\leqslant \frac{x^{n}}{2^{n}n!}e^{x}$$ ...
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### Inequalities involving regularized incomplete Gamma functions

I am new to the world of the Gamma functions and am wondering if there exist positive functions $f_1(x)>0$ and $g_1(x)>0$, and non-negative functions $f_2(x)\geq0$ and $g_2(x)\geq0$ such that ...
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### looking for upper bound on quantity with “erf”

Math people: I am looking for an upper bound on $$g(x,t) = \frac{-\operatorname{erf}(t)+2\operatorname{erf}(\frac{1}{2}t(x+1))-\operatorname{erf}(xt)}{(x-1)^2},$$ where $x > 1$ and $t > 0$. ...
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### How to solve $x \geq \frac{y}{z-\ln{x}}$ for positive variables?

How can you solve $x \geq \frac{y}{z-\ln{x}}$ for $x$ when the variables are real positive values? I am only really interested in the case where the values are large and $z > \ln x$. How ...
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### The cdf of a beta variable, evaluated at the mean

Consider a Beta random variable $X$ with shape parameters $k/2$ and $(d-k)/2$, where the parameters $k, d$ are integers that satisfy $0 < k < d$. What is the best possible upper bound for the ...
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### An inequality from the handbook of mathematical functions (by Abramowitz and Stegun)

Prove that $$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$
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Let $k, n ,m \in N$ and such that $0\leq k \leq n \leq m$. When the following ineuality is true? $$... 1answer 59 views ### reference needed for Gamma function Please help me to find a reference (book) for the following upper bound of Gamma function For x \geq 1$$ \Gamma(x)\leq x^{x-1}. $$Thank you. 1answer 286 views ### Upper bound for a gamma function Let n \in N. How to find a non-asymptotic upper bound for \Gamma(n) and \Gamma(\frac n2+1)? Thank you 0answers 64 views ### inequality with gamma function Help me please to prove the following inequality For x,y>1, x \neq y.$$ \frac{1}{\Gamma(x)\Gamma(y)}\leq 2\sqrt{2\pi}\frac{\sqrt{x+y}}{\Gamma(x+y)}. $$Thank you. 1answer 171 views ### Expressing solution to an inequality with Lambert W function I'm new to Lambert functions, any ideas on how to solve this are welcome:$$ \theta \rho^{\theta}+r \theta>v $$where \theta \in \mathbb{R}^{+}, -1<r,v<1, \ 0<\rho<1. I've tried ... 1answer 128 views ### Upper bound for \Gamma(x+y) Let x, y \geq 1 be two real numbers. I am wondering if one can get an upper bound for \Gamma(x+y) in terms of \Gamma(x)\Gamma(y)? Any references or ideas are very appreciated. Thank you. 3answers 677 views ### Derivatives of the Riemann zeta function at s=0 It's a curious fact that for n>0, \zeta^{(n)}(0)\approx -n!. Apostol gave a table for \frac{\zeta^{(n)}(0)}{n!}, among other results on \zeta^{(n)}(0) . the sequence :$$\delta_{n}=\left | ...
Let $\psi$ be the digamma function. I have a conjecture that $$\frac ax > \log(x) - \psi(x)$$ holds for all $x > 0$ if (and only if) $a \ge 1$. I do not know how to prove it. Please help.