2
votes
2answers
53 views

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 ...
2
votes
1answer
16 views

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 $$ce^{x^\alpha} \le I_\nu(x)$$.
6
votes
2answers
171 views

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 ...
2
votes
1answer
30 views

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 $x\ge 0$ ? ...
5
votes
2answers
289 views

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. ...
4
votes
1answer
123 views

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 ...
0
votes
0answers
23 views

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 ...
1
vote
0answers
75 views

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 ...
5
votes
0answers
206 views

Inequality between incomplete beta and gamma functions

Let the regularized incomplete beta and gamma functions be defined as usual: \begin{equation} I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt, \end{equation} \begin{equation} ...
1
vote
1answer
34 views

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 $C(M+n,n)$.
3
votes
2answers
72 views

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 ...
1
vote
1answer
347 views

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 : $$I_{n}(x)\leqslant \frac{x^{n}}{2^{n}n!}e^{x}$$ ...
3
votes
0answers
74 views

An inequality with Gamma Function

Consider the following function for any $a, b > 0$ $$ \ g\left( a,b\right) = \frac{% 3\Gamma \left( 3b+1\right) }{\Gamma \left( \frac{1}{a}+3b+1\right) }-\frac{% 5\Gamma \left( 2b+1\right) ...
3
votes
1answer
109 views

Understanding the Gamma Function

Is this valid? If $x_1 > x_2 > x_3 > 0$ and $\Delta{t_1} = \Delta{t_2} + \Delta{t_3}$, Does it follow that: $$\frac{\Gamma(x_1 + \Delta{t_1})}{\Gamma(x_1)} \ge \frac{\Gamma(x_2 + ...
1
vote
0answers
89 views

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 ...
2
votes
1answer
174 views

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 $t > 0$. ...
2
votes
2answers
108 views

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 ...
1
vote
0answers
45 views

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 ...
7
votes
2answers
453 views

An inequality from the handbook of mathematical functions (by Abramowitz and Stegun)

Prove that $$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$
2
votes
1answer
157 views

Inequality for Gamma functions

Let $k, n ,m \in N$ and such that $0\leq k \leq n \leq m$. When the following ineuality is true? $$ ...
0
votes
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.
1
vote
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
0
votes
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.
0
votes
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 ...
2
votes
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.
5
votes
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 | ...
3
votes
2answers
115 views

Bounding an expression involving digamma function

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.
3
votes
1answer
119 views

Hypergeometric functions inequality

Let $_2F_1(a,b;c,z)$ be the (Gauss) hypergeometric function, and $m$ and $n$ positive integers. From a simple plot it looks like $_2F_1(m+n,1,m+1,\frac{m}{m+n})>\frac{m}{n} ...
1
vote
2answers
209 views

Relating Gamma and factorial function for non-integer values.

We have $$\Gamma(n+1)=n!,\ \ \ \ \ \Gamma(n+2)=(n+1)!$$ for integers, so if $\Delta$ is some real value with $$0<\Delta<1,$$ then $$n!\ <\ \Gamma(n+1+\Delta)\ <\ (n+1)!,$$ because ...
2
votes
2answers
182 views

a proof for the following Gamma function inequality

Could you please provide or point me to a proof of inequality 5.6.8 found at this site? That is, $\left|\frac{\Gamma(z+a)}{\Gamma(z+b)}\right| \leq \frac{1}{|z|^{b-a}}$ for $z\in \mathbb{C}$, ...
7
votes
2answers
722 views

Are there well known lower bounds for the upper incomplete gamma function?

Let $a >0, b >0$, and $r \in \mathbb{R}$. I am trying to find a lower bound for the integral $$\int_a^\infty y^{-r} \exp\left( - b(y-a)^2\right) \,\mathrm dy.$$ After consulting the Wikipedia ...
6
votes
2answers
332 views

Inequality involving the regularized gamma function

Prove that $$Q(x,\ln 2) := \frac{\int_{\ln 2}^{\infty} t^{x-1} e^{-t} dt}{\int_{0}^{\infty} t^{x-1} e^{-t} dt} \geqslant 1 - 2^{-x}$$ for all $x\geqslant 1$. ($Q$ is the regularized gamma function.) ...
2
votes
2answers
248 views

How big is the integral $\int_0^\infty \frac{x\exp(-x^2/4)\cosh(x)}{\sqrt{\cosh(x)-1}} dx$

I can't seem to get Maple to approximate the integral $$\int_0^\infty \frac{x\exp(-x^2/4)\cosh(x)}{\sqrt{\cosh(x)-1}} dx.$$ Could somebody tell me why? This integral "should be" well-defined. (My ...
0
votes
1answer
79 views

Is the hypergeometric function $F(5/4,3/4; 2, z)$ bounded on $(0,1]$

Consider the classical hypergeometric function $F(5/4,3/4; 2, z)$ for $z\in (0,1]$. Is this bounded by some real number (independent of $z$)? I'm aware of Euler's formula: $$F(5/4,3/4; 2, z) = ...
4
votes
2answers
227 views

Bound for the Legendre function of the second kind of degree $1/2$

Let $Q_{1/2}(u)$ be the Legendre function of the second kind of degree $1/2$. One can show that $Q_{1/2}(u) = O(u^{-3/2})$ as $u\to \infty$; see Equation 21 in Section 3.9.2 of Higher transcendental ...
3
votes
1answer
301 views

Concave functions on discrete domain

We are given a positive, non-decreasing function $f$ defined on natural numbers with $f(0) = 0$. $f$ has a submodularity-like property: $f(x+y) \leq f(x) + f(y) $ for all natural numbers $x$ and ...
1
vote
3answers
102 views

finding bound for the integral

I am trying to get bound for the following integral $$ \int_0^{\infty}\frac{1}{|x|^r}dx, \mbox{for } 1\leq r< \infty $$ In particular, the bound of the form $\frac{constant}{r}$. Sorry, we can ...
16
votes
1answer
568 views

How do you prove Gautschi's inequality for the gamma function?

A few answers here on math.SE have used as an intermediate step the following inequality that is due to Walter Gautschi: $$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > ...
3
votes
1answer
191 views

Proof of an inequality under Equivalence of Weierstrass and Euler's Definitions of Gamma Function

I am using my lecturer's notes on Special Functions. When dealing with Gamma Functions under the title Equivalence of Wierstrass and Euler's Definitions, he has used a lemma (with no proof). I tried ...
1
vote
0answers
195 views

Bound on Bessel function of the first order

Let $I_1(z)$ be the Bessel function of the first order with purely imaginary argument. Can we explicitly bound $I_1$ on $[0,x]$, where $x>0$ is a real number in terms of $x$?
2
votes
0answers
160 views

Help with an integral inequality involving an incomplete beta function

I would like to determine if the following inequality is true: ...
5
votes
1answer
190 views

Lower bounding a ratio of gamma functions

I am trying to show that the following function has a lower bound of $\ \frac{1}{2}$ for all $c\geq 2$. Or, alternatively, that that function increases with $\ c$: ...
6
votes
2answers
277 views

Beta Function — finding a lower bound based on parameters

I would like to show that $$ 1-\frac{1}{c}Beta\left(c+1,\frac{1}{c}\right) \geq \frac{1}{c+1}.$$ for all $c \geq 2$. I have plotted it out for $c$ up through 200, and it seems to hold. Does anyone ...
9
votes
5answers
899 views

Proof that $x \Phi(x) + \Phi'(x) \geq 0$ $\forall x$, where $\Phi$ is the normal CDF

As title. Can anyone supply a simple proof that $$x \Phi(x) + \Phi'(x) \geq 0 \quad \forall x\in\mathbb{R}$$ where $\Phi$ is the standard normal CDF, i.e. $$\Phi(x) = \int_{-\infty}^x ...
5
votes
0answers
147 views

Bounding function involving Beta functions

Given $\frac{a}{x-1} \leq \frac{b}{y-1} \leq \frac{c}{z-1}$ with $a,b,c > 0$ and $x,y,z > 1$, I want to show that $$\frac{(\frac{a}{a+b})^{x-1}(\frac{b}{a+b})^{y-1}}{B(x,y)\cdot (x+y-1)} + ...
1
vote
1answer
70 views

Can one prove $\text{erf}\left(\frac{c}{t}\right) \ge \delta \, \min(1,\frac{c}{t})$?

Let $c>1/2$ be an arbitrary big fixed constant. Can one prove that for all $t\geq 1$: $$\text{erf}\left(\frac{c}{t}\right) \ge \delta \, \min\left(1,\frac{c}{t}\right)$$ for some small constant ...
9
votes
5answers
658 views

A gamma function inequality

I would like to prove $$\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n+1)} \le \frac{1}{\sqrt{n}}$$ for all natural $n \ge 1$. The inequality does seem to be true numerically, but the proof eludes me.
1
vote
1answer
155 views

Inequality concerning the Gamma function

For each $n>0$, how do we prove that $$\Gamma'(n+1)> \log{n} \cdot \Gamma(n+1)$$ I had spent about half an hour on this question, but just could find any way of proceeding for the solution. ...
12
votes
1answer
480 views

Proving $\sqrt{1-x^2}\ge \operatorname{erf}(\sqrt{-\log x})$

Can anyone see a nice way to prove the following for $0\le x \le 1$? $$\sqrt{1-x^2}\ge \operatorname{erf}(\sqrt{-\log x})$$ $\operatorname{erf}$ is defined as $$\operatorname{erf}(z) = ...