Skip to main content
MathOverflow

Return to Question

Bumped by Community user
Bumped by Community user
Bumped by Community user
added 1 character in body
Source Link
user44143
user44143

The exact constant in the simplea bound of the fractionon ratios of Gamma functions

In theThe answer to another question : (Upper bound of the fraction of Gamma functions the) gave an asymptotic upper bound for the fraction ofan expression with Gamma functions have been established: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\leq C\cdot f(a,b), \forall a,b\geq1/2$$.$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\!\leq \,C\,\frac{a+b}a, \forall a,b\geq\frac12$$

What is a non-asymptotic boundthe best possible value for this ratio, i.e. what is the constant $C$ in that statement?

The exact constant in the simple bound of the fraction of Gamma functions

In the question : Upper bound of the fraction of Gamma functions the asymptotic upper bound for the fraction of Gamma functions have been established: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\leq C\cdot f(a,b), \forall a,b\geq1/2$$.

What is a non-asymptotic bound for this ratio, i.e. what is the constant $C$?

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: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\!\leq \,C\,\frac{a+b}a, \forall a,b\geq\frac12$$

What is the best possible value for the constant $C$ in that statement?

Bumped by Community user
Bumped by Community user
Bumped by Community user
fixed capital, added tag
Source Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 285

The exact constant in the simple bound of the fraction of Gamma Functionsfunctions

In the Questionquestion : Upper bound of the fraction of Gamma functions the asymptotic upper bound for the fraction of Gamma functions have been established: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\leq C\cdot f(a,b), \forall a,b\geq1/2$$.

What is a non-asymptotic bound for this ratio, i.e. what is the constant C$C$?

The exact constant in the simple bound of the fraction of Gamma Functions

In the Question : Upper bound of the fraction of Gamma functions the asymptotic upper bound for the fraction of Gamma functions have been established: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\leq C\cdot f(a,b), \forall a,b\geq1/2$$.

What is a non-asymptotic bound for this ratio, i.e. what is the constant C?

The exact constant in the simple bound of the fraction of Gamma functions

In the question : Upper bound of the fraction of Gamma functions the asymptotic upper bound for the fraction of Gamma functions have been established: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\leq C\cdot f(a,b), \forall a,b\geq1/2$$.

What is a non-asymptotic bound for this ratio, i.e. what is the constant $C$?

Bumped by Community user
Bumped by Community user
Bumped by Community user
Bumped by Community user
Bumped by Community user
Bumped by Community user
Bumped by Community user
Source Link
user124297
user124297
  • 343
  • 2
  • 8

The exact constant in the simple bound of the fraction of Gamma Functions

In the Question : Upper bound of the fraction of Gamma functions the asymptotic upper bound for the fraction of Gamma functions have been established: $$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\leq C\cdot f(a,b), \forall a,b\geq1/2$$.

What is a non-asymptotic bound for this ratio, i.e. what is the constant C?