Return to Question

I am working with a certain Gauss hypergeometric function, $_{2}F_{1}(a,a-b;2a;1-z)$, where $a, b \in {\mathbb R}$ with $a, a-b > 0$ and $0<b<1$.

It appears that $\left| _{2}F_{1}(a,a-b;2a;1-z) \right| \geq 1$ for all $|z| \leq 1$.

More generally, it seems that $$ \left| _{2}F_{1}(a,b;c;1-z) \right| \geq 1 $$ for all $|z| \leq 1$ when $a,b,c \in {\mathbb R}$ with $a,b>0$ and $c \geq \max( a,b)$.

For example, if $-1 \leq z \leq 1$, then this is true since all the coefficients in the expansion of $_{2}F_{1}(a,b;c;z)$ are positive.

But I have been unable to prove the more general result for all $z$ with $|z| \leq 1$ myself or find a proof in the literature.

Would anyone have any ideas on this, or references I may have missed?

I am working with a certain Gauss hypergeometric function, $_{2}F_{1}(a,a-b;2a;1-z)$, where $a, b \in {\mathbb R}$ with $a, a-b > 0$ and $0<b<1$.

It appears that $\left| _{2}F_{1}(a,a-b;2a;1-z) \right| \geq 1$ for all $|z| \leq 1$.

More generally, it seems that $$ \left| _{2}F_{1}(a,b;c;1-z) \right| \geq 1 $$ for all $|z| \leq 1$ when $a,b,c \in {\mathbb R}$ with $a,b>0$ and $c \geq \max( a,b)$.

For example, if $0 \leq z \leq 1$, then this is true since all the coefficients in the expansion of $_{2}F_{1}(a,b;c;z)$ are positive.

But I have been unable to prove the more general result for all $z$ with $|z| \leq 1$ myself or find a proof in the literature.

Would anyone have any ideas on this, or references I may have missed?

I am working with a certain Gauss hypergeometric function, $_{2}F_{1}(a,a-b;2a;1-z)$, where $a, b \in {\mathbb R}$ with $a, a-b > 0$ and $0<b<1$.

It appears that $\left| _{2}F_{1}(a,a-b;2a;1-z) \right| \geq 1$ for all $|z| \leq 1$.

For example, if $0 \leq z \leq 1$, then this is true since all the coefficients in the expansion of $_{2}F_{1}(a,a-b;2a;z)$ are positive.

But I have been unable to prove the more general result for all $z$ with $|z| \leq 1$ myself or find a proof in the literature.

Would anyone have any ideas on this, or references I may have missed?

I am working with a certain Gauss hypergeometric function, $_{2}F_{1}(a,a-b;2a;1-z)$, where $a, b \in {\mathbb R}$ with $a, a-b > 0$ and $0<b<1$.

It appears that $\left| _{2}F_{1}(a,a-b;2a;1-z) \right| \geq 1$ for all $|z| \leq 1$.

More generally, it seems that $$ \left| _{2}F_{1}(a,b;c;1-z) \right| \geq 1 $$ for all $|z| \leq 1$ when $a,b,c \in {\mathbb R}$ with $a,b>0$ and $c \geq \max( a,b)$.

For example, if $-1 \leq z \leq 1$, then this is true since all the coefficients in the expansion of $_{2}F_{1}(a,b;c;z)$ are positive.

But I have been unable to prove the more general result for all $z$ with $|z| \leq 1$ myself or find a proof in the literature.

Would anyone have any ideas on this, or references I may have missed?

I am working with a certain Gauss hypergeometric function, $_{2}F_{1}(a,a-b;2a;1-z)$, where $a, b \in {\mathbb R}$ with $a, a-b > 0$. and $0<b<1$.

It appears that $\left| _{2}F_{1}(a,a-b;2a;1-z) \right| \geq 1$ for all $|z| \leq 1$.

For example, if $0 \leq z \leq 1$, then this is true since all the coefficients in the expansion of $_{2}F_{1}(a,a-b;2a;z)$ are positive.

But I have been unable to prove the more general result for all $z$ with $|z| \leq 1$ myself or find a proof in the literature.

Would anyone have any ideas on this, or references I may have missed?

I am working with a certain Gauss hypergeometric function, $_{2}F_{1}(a,a-b;2a;1-z)$, where $a, b \in {\mathbb R}$ with $a, a-b > 0$ and $0<b<1$.

It appears that $\left| _{2}F_{1}(a,a-b;2a;1-z) \right| \geq 1$ for all $|z| \leq 1$.

For example, if $0 \leq z \leq 1$, then this is true since all the coefficients in the expansion of $_{2}F_{1}(a,a-b;2a;z)$ are positive.

But I have been unable to prove the more general result for all $z$ with $|z| \leq 1$ myself or find a proof in the literature.

Would anyone have any ideas on this, or references I may have missed?