Looking for bounds on $|I_n(z)|$ for $z in mathbbC$ where $I_n(z)$ is modified Bessel function

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I am looking for some bounds on the magnitude of the modified Bessel function in the complex setting.



That is let $I_v(z), z in mathbbC$ be modified Bessel function of order $v$.



Are there any known anlytic (closed form) bounds on
beginalign
|I_v(z)|,
endalign
and
beginalign
left| fracI_v(x)I_v-1(x) right| ?
endalign



I found a number of bounds on $I_v(x)$ (in the real-setting) and ratios of Bessel function such as $fracI_v(x)I_v-1(x)$. However, I was not able to find any bounds on $|I_v(z)|$ or $fracI_v(z)I_v-1(z)$ in the complex setting. I am esspecially interested in the case of $v=1$



Any reference on how to approach this problem would be greatly appreciated!!!



Below is the plot of $|I_0(z)|$ and $frac$ vs. the real and imaginary part of $z$.



For the ratio, there seem to be singularities. I am not sure now if it is correct.



dd



ddd




reference-request bessel-functions




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  • 1




    What do you mean with bounds? Even for real $x$ the function $I_nu(x)$ is unbounded.
    – gammatester
    Aug 21 at 12:47











  • @gammatester I am looking for closed-form bounds. For example, $fracI_1(x)I_0(x) le fracx1/2+sqrt1/2+x^2, x>0$
    – Boby
    Aug 21 at 13:00















up vote
5
down vote

favorite
2












I am looking for some bounds on the magnitude of the modified Bessel function in the complex setting.



That is let $I_v(z), z in mathbbC$ be modified Bessel function of order $v$.



Are there any known anlytic (closed form) bounds on
beginalign
|I_v(z)|,
endalign
and
beginalign
left| fracI_v(x)I_v-1(x) right| ?
endalign



I found a number of bounds on $I_v(x)$ (in the real-setting) and ratios of Bessel function such as $fracI_v(x)I_v-1(x)$. However, I was not able to find any bounds on $|I_v(z)|$ or $fracI_v(z)I_v-1(z)$ in the complex setting. I am esspecially interested in the case of $v=1$



Any reference on how to approach this problem would be greatly appreciated!!!



Below is the plot of $|I_0(z)|$ and $frac$ vs. the real and imaginary part of $z$.



For the ratio, there seem to be singularities. I am not sure now if it is correct.



dd



ddd




reference-request bessel-functions




share|cite|improve this question


















  • 1




    What do you mean with bounds? Even for real $x$ the function $I_nu(x)$ is unbounded.
    – gammatester
    Aug 21 at 12:47











  • @gammatester I am looking for closed-form bounds. For example, $fracI_1(x)I_0(x) le fracx1/2+sqrt1/2+x^2, x>0$
    – Boby
    Aug 21 at 13:00













up vote
5
down vote

favorite
2









up vote
5
down vote

favorite
2






2





I am looking for some bounds on the magnitude of the modified Bessel function in the complex setting.



That is let $I_v(z), z in mathbbC$ be modified Bessel function of order $v$.



Are there any known anlytic (closed form) bounds on
beginalign
|I_v(z)|,
endalign
and
beginalign
left| fracI_v(x)I_v-1(x) right| ?
endalign



I found a number of bounds on $I_v(x)$ (in the real-setting) and ratios of Bessel function such as $fracI_v(x)I_v-1(x)$. However, I was not able to find any bounds on $|I_v(z)|$ or $fracI_v(z)I_v-1(z)$ in the complex setting. I am esspecially interested in the case of $v=1$



Any reference on how to approach this problem would be greatly appreciated!!!



Below is the plot of $|I_0(z)|$ and $frac$ vs. the real and imaginary part of $z$.



For the ratio, there seem to be singularities. I am not sure now if it is correct.



dd



ddd




reference-request bessel-functions




share|cite|improve this question














I am looking for some bounds on the magnitude of the modified Bessel function in the complex setting.



That is let $I_v(z), z in mathbbC$ be modified Bessel function of order $v$.



Are there any known anlytic (closed form) bounds on
beginalign
|I_v(z)|,
endalign
and
beginalign
left| fracI_v(x)I_v-1(x) right| ?
endalign



I found a number of bounds on $I_v(x)$ (in the real-setting) and ratios of Bessel function such as $fracI_v(x)I_v-1(x)$. However, I was not able to find any bounds on $|I_v(z)|$ or $fracI_v(z)I_v-1(z)$ in the complex setting. I am esspecially interested in the case of $v=1$



Any reference on how to approach this problem would be greatly appreciated!!!



Below is the plot of $|I_0(z)|$ and $frac$ vs. the real and imaginary part of $z$.



For the ratio, there seem to be singularities. I am not sure now if it is correct.



dd



ddd





reference-request bessel-functions





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 27 at 11:47

























asked Aug 21 at 12:33









Boby

9451828




9451828







  • 1




    What do you mean with bounds? Even for real $x$ the function $I_nu(x)$ is unbounded.
    – gammatester
    Aug 21 at 12:47











  • @gammatester I am looking for closed-form bounds. For example, $fracI_1(x)I_0(x) le fracx1/2+sqrt1/2+x^2, x>0$
    – Boby
    Aug 21 at 13:00













  • 1




    What do you mean with bounds? Even for real $x$ the function $I_nu(x)$ is unbounded.
    – gammatester
    Aug 21 at 12:47











  • @gammatester I am looking for closed-form bounds. For example, $fracI_1(x)I_0(x) le fracx1/2+sqrt1/2+x^2, x>0$
    – Boby
    Aug 21 at 13:00








1




1




What do you mean with bounds? Even for real $x$ the function $I_nu(x)$ is unbounded.
– gammatester
Aug 21 at 12:47





What do you mean with bounds? Even for real $x$ the function $I_nu(x)$ is unbounded.
– gammatester
Aug 21 at 12:47













@gammatester I am looking for closed-form bounds. For example, $fracI_1(x)I_0(x) le fracx1/2+sqrt1/2+x^2, x>0$
– Boby
Aug 21 at 13:00





@gammatester I am looking for closed-form bounds. For example, $fracI_1(x)I_0(x) le fracx1/2+sqrt1/2+x^2, x>0$
– Boby
Aug 21 at 13:00
















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