Reconstruction of proof in complex analysis paper
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I am reading through this paper of Musin:
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=2326&option_lang=eng
On the second page (visible in the preview of the english version) he twice uses a Phragmén-Lindelöf type theorem. However, he cites from sources that are not accessible online, so I want to reconstruct what he does. The first use of (a different) Phragmén-Lindelöf I could understand, however not the second one.
The thing I do not understand is the following:
Let $varphi$ be a holomorphic function on the upper half plane $ Pi = { operatorname{Im } z > 0 } = { 0 < arg z < pi }$. Let $C$ be a convex cone, $T_C = mathbb R^n + iC$ and $a$ be a non-negative, convex function continuous on $T_{overline C}$ and homegeneous of degree $1$. He has the estimate $$lvert varphi(u_1 + is) rvert leq M exp(sigma s), quad sigma := max_{substack{zeta in T_{overline C} \ lVert zeta rVert = 1}} a(z)$$
and $$limsup_{s to infty}frac{ln lvertvarphi(is)rvert}{s} leq a(iy^0)$$
and concludes by the (not accessible) Phragmén-Lindelöf theorem that $$ln lvert varphi(u_1 + is) rvert leq ln M + a(iy^0)s.$$
Now the closest theorem to this I could find is in the book of Ronkin, Introduction to the theory of entire functions of several variables (in fact the same book he cited the other theorem from):
Let $G_alpha = { zin mathbb C: 0 < arg z < alpha pi }, 0 < alpha < 2 $, and let $f(z)$ be analytic in $G_alpha$ satisfying the following conditions:
(a) $$limsup_{substack{ z to zeta \ z in G_alpha}} lvert f(z) rvert leq C quad forall zeta in partial G_alpha$$
(b) $$liminf_{rto infty} r^{-1/alpha} ln M_f(r) leq 0,$$
where $$M_f(r) := sup_{substack{lvert z rvert = r \ 0 < arg z < alpha pi}} lvert f(z) rvert.$$ Then $lvert f(z) rvert leq C$ everywhere in the angle $G_alpha$, and the equality $lvert f(z_0) rvert = C$ can hold if and only if $f(z) equiv C$.
Now setting $alpha = 1$ in this theorem comes quite close to what we have. However, it doesn't directly give me what I want. I wouldn't see why $M_varphi(r) = lvert varphi(ir) rvert$ and also why the inequality follows even if the theorem could be applied.
Can anyone help maybe? The complete setup, i.e. what $a(iy)$ is etc. can be found on the first page of the paper, but I don't think it is relevant. Thanks in advance!
complex-analysis analysis inequality holomorphic-functions maximum-principle
asked Nov 17 at 13:37
Staki42
1,187518
add a comment |
up vote
2
down vote
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I am reading through this paper of Musin:
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=2326&option_lang=eng
On the second page (visible in the preview of the english version) he twice uses a Phragmén-Lindelöf type theorem. However, he cites from sources that are not accessible online, so I want to reconstruct what he does. The first use of (a different) Phragmén-Lindelöf I could understand, however not the second one.
The thing I do not understand is the following:
Let $varphi$ be a holomorphic function on the upper half plane $ Pi = { operatorname{Im } z > 0 } = { 0 < arg z < pi }$. Let $C$ be a convex cone, $T_C = mathbb R^n + iC$ and $a$ be a non-negative, convex function continuous on $T_{overline C}$ and homegeneous of degree $1$. He has the estimate $$lvert varphi(u_1 + is) rvert leq M exp(sigma s), quad sigma := max_{substack{zeta in T_{overline C} \ lVert zeta rVert = 1}} a(z)$$
and $$limsup_{s to infty}frac{ln lvertvarphi(is)rvert}{s} leq a(iy^0)$$
and concludes by the (not accessible) Phragmén-Lindelöf theorem that $$ln lvert varphi(u_1 + is) rvert leq ln M + a(iy^0)s.$$
Now the closest theorem to this I could find is in the book of Ronkin, Introduction to the theory of entire functions of several variables (in fact the same book he cited the other theorem from):
Let $G_alpha = { zin mathbb C: 0 < arg z < alpha pi }, 0 < alpha < 2 $, and let $f(z)$ be analytic in $G_alpha$ satisfying the following conditions:
(a) $$limsup_{substack{ z to zeta \ z in G_alpha}} lvert f(z) rvert leq C quad forall zeta in partial G_alpha$$
(b) $$liminf_{rto infty} r^{-1/alpha} ln M_f(r) leq 0,$$
where $$M_f(r) := sup_{substack{lvert z rvert = r \ 0 < arg z < alpha pi}} lvert f(z) rvert.$$ Then $lvert f(z) rvert leq C$ everywhere in the angle $G_alpha$, and the equality $lvert f(z_0) rvert = C$ can hold if and only if $f(z) equiv C$.
Now setting $alpha = 1$ in this theorem comes quite close to what we have. However, it doesn't directly give me what I want. I wouldn't see why $M_varphi(r) = lvert varphi(ir) rvert$ and also why the inequality follows even if the theorem could be applied.
Can anyone help maybe? The complete setup, i.e. what $a(iy)$ is etc. can be found on the first page of the paper, but I don't think it is relevant. Thanks in advance!
complex-analysis analysis inequality holomorphic-functions maximum-principle
asked Nov 17 at 13:37
Staki42
1,187518
What is $sigma$? If there is some way to show that $$sigma < lim_{s to infty}frac{ln lvertvarphi(is)rvert}{s}$$, then the proof is trivial. But without even knowing what the definition of $sigma$ is, we can't help you.
– Paul Sinclair
Nov 17 at 21:21
Hi, thanks for trying to help me! I've added some details on what $sigma$ is. For the complete setup, you can check out the english version of the paper I linked (without institutional login just a preview, but the pages relevant are shown there). Also fixed a typo (should have been $limsup$, not $lim$).
– Staki42
Nov 17 at 22:02
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am reading through this paper of Musin:
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=2326&option_lang=eng
On the second page (visible in the preview of the english version) he twice uses a Phragmén-Lindelöf type theorem. However, he cites from sources that are not accessible online, so I want to reconstruct what he does. The first use of (a different) Phragmén-Lindelöf I could understand, however not the second one.
The thing I do not understand is the following:
Let $varphi$ be a holomorphic function on the upper half plane $ Pi = { operatorname{Im } z > 0 } = { 0 < arg z < pi }$. Let $C$ be a convex cone, $T_C = mathbb R^n + iC$ and $a$ be a non-negative, convex function continuous on $T_{overline C}$ and homegeneous of degree $1$. He has the estimate $$lvert varphi(u_1 + is) rvert leq M exp(sigma s), quad sigma := max_{substack{zeta in T_{overline C} \ lVert zeta rVert = 1}} a(z)$$
and $$limsup_{s to infty}frac{ln lvertvarphi(is)rvert}{s} leq a(iy^0)$$
and concludes by the (not accessible) Phragmén-Lindelöf theorem that $$ln lvert varphi(u_1 + is) rvert leq ln M + a(iy^0)s.$$
Now the closest theorem to this I could find is in the book of Ronkin, Introduction to the theory of entire functions of several variables (in fact the same book he cited the other theorem from):
Let $G_alpha = { zin mathbb C: 0 < arg z < alpha pi }, 0 < alpha < 2 $, and let $f(z)$ be analytic in $G_alpha$ satisfying the following conditions:
(a) $$limsup_{substack{ z to zeta \ z in G_alpha}} lvert f(z) rvert leq C quad forall zeta in partial G_alpha$$
(b) $$liminf_{rto infty} r^{-1/alpha} ln M_f(r) leq 0,$$
where $$M_f(r) := sup_{substack{lvert z rvert = r \ 0 < arg z < alpha pi}} lvert f(z) rvert.$$ Then $lvert f(z) rvert leq C$ everywhere in the angle $G_alpha$, and the equality $lvert f(z_0) rvert = C$ can hold if and only if $f(z) equiv C$.
Now setting $alpha = 1$ in this theorem comes quite close to what we have. However, it doesn't directly give me what I want. I wouldn't see why $M_varphi(r) = lvert varphi(ir) rvert$ and also why the inequality follows even if the theorem could be applied.
Can anyone help maybe? The complete setup, i.e. what $a(iy)$ is etc. can be found on the first page of the paper, but I don't think it is relevant. Thanks in advance!
complex-analysis analysis inequality holomorphic-functions maximum-principle
asked Nov 17 at 13:37
Staki42
1,187518
I am reading through this paper of Musin:
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mzm&paperid=2326&option_lang=eng
On the second page (visible in the preview of the english version) he twice uses a Phragmén-Lindelöf type theorem. However, he cites from sources that are not accessible online, so I want to reconstruct what he does. The first use of (a different) Phragmén-Lindelöf I could understand, however not the second one.
The thing I do not understand is the following:
Let $varphi$ be a holomorphic function on the upper half plane $ Pi = { operatorname{Im } z > 0 } = { 0 < arg z < pi }$. Let $C$ be a convex cone, $T_C = mathbb R^n + iC$ and $a$ be a non-negative, convex function continuous on $T_{overline C}$ and homegeneous of degree $1$. He has the estimate $$lvert varphi(u_1 + is) rvert leq M exp(sigma s), quad sigma := max_{substack{zeta in T_{overline C} \ lVert zeta rVert = 1}} a(z)$$
and $$limsup_{s to infty}frac{ln lvertvarphi(is)rvert}{s} leq a(iy^0)$$
and concludes by the (not accessible) Phragmén-Lindelöf theorem that $$ln lvert varphi(u_1 + is) rvert leq ln M + a(iy^0)s.$$
Now the closest theorem to this I could find is in the book of Ronkin, Introduction to the theory of entire functions of several variables (in fact the same book he cited the other theorem from):
Let $G_alpha = { zin mathbb C: 0 < arg z < alpha pi }, 0 < alpha < 2 $, and let $f(z)$ be analytic in $G_alpha$ satisfying the following conditions:
(a) $$limsup_{substack{ z to zeta \ z in G_alpha}} lvert f(z) rvert leq C quad forall zeta in partial G_alpha$$
(b) $$liminf_{rto infty} r^{-1/alpha} ln M_f(r) leq 0,$$
where $$M_f(r) := sup_{substack{lvert z rvert = r \ 0 < arg z < alpha pi}} lvert f(z) rvert.$$ Then $lvert f(z) rvert leq C$ everywhere in the angle $G_alpha$, and the equality $lvert f(z_0) rvert = C$ can hold if and only if $f(z) equiv C$.
Now setting $alpha = 1$ in this theorem comes quite close to what we have. However, it doesn't directly give me what I want. I wouldn't see why $M_varphi(r) = lvert varphi(ir) rvert$ and also why the inequality follows even if the theorem could be applied.
Can anyone help maybe? The complete setup, i.e. what $a(iy)$ is etc. can be found on the first page of the paper, but I don't think it is relevant. Thanks in advance!
complex-analysis analysis inequality holomorphic-functions maximum-principle
complex-analysis analysis inequality holomorphic-functions maximum-principle
asked Nov 17 at 13:37
Staki42
1,187518
asked Nov 17 at 13:37
Staki42
1,187518
edited Nov 17 at 21:59
asked Nov 17 at 13:37
Staki42
1,187518
asked Nov 17 at 13:37
Staki42
1,187518
asked Nov 17 at 13:37
Staki42
1,187518
1,187518
What is $sigma$? If there is some way to show that $$sigma < lim_{s to infty}frac{ln lvertvarphi(is)rvert}{s}$$, then the proof is trivial. But without even knowing what the definition of $sigma$ is, we can't help you.
– Paul Sinclair
Nov 17 at 21:21
Hi, thanks for trying to help me! I've added some details on what $sigma$ is. For the complete setup, you can check out the english version of the paper I linked (without institutional login just a preview, but the pages relevant are shown there). Also fixed a typo (should have been $limsup$, not $lim$).
– Staki42
Nov 17 at 22:02
add a comment |
What is $sigma$? If there is some way to show that $$sigma < lim_{s to infty}frac{ln lvertvarphi(is)rvert}{s}$$, then the proof is trivial. But without even knowing what the definition of $sigma$ is, we can't help you.
– Paul Sinclair
Nov 17 at 21:21
Hi, thanks for trying to help me! I've added some details on what $sigma$ is. For the complete setup, you can check out the english version of the paper I linked (without institutional login just a preview, but the pages relevant are shown there). Also fixed a typo (should have been $limsup$, not $lim$).
– Staki42
Nov 17 at 22:02
What is $sigma$? If there is some way to show that $$sigma < lim_{s to infty}frac{ln lvertvarphi(is)rvert}{s}$$, then the proof is trivial. But without even knowing what the definition of $sigma$ is, we can't help you.
– Paul Sinclair
Nov 17 at 21:21
What is $sigma$? If there is some way to show that $$sigma < lim_{s to infty}frac{ln lvertvarphi(is)rvert}{s}$$, then the proof is trivial. But without even knowing what the definition of $sigma$ is, we can't help you.
– Paul Sinclair
Nov 17 at 21:21
Hi, thanks for trying to help me! I've added some details on what $sigma$ is. For the complete setup, you can check out the english version of the paper I linked (without institutional login just a preview, but the pages relevant are shown there). Also fixed a typo (should have been $limsup$, not $lim$).
– Staki42
Nov 17 at 22:02
Hi, thanks for trying to help me! I've added some details on what $sigma$ is. For the complete setup, you can check out the english version of the paper I linked (without institutional login just a preview, but the pages relevant are shown there). Also fixed a typo (should have been $limsup$, not $lim$).
– Staki42
Nov 17 at 22:02
add a comment |
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What is $sigma$? If there is some way to show that $$sigma < lim_{s to infty}frac{ln lvertvarphi(is)rvert}{s}$$, then the proof is trivial. But without even knowing what the definition of $sigma$ is, we can't help you.
– Paul Sinclair
Nov 17 at 21:21
Hi, thanks for trying to help me! I've added some details on what $sigma$ is. For the complete setup, you can check out the english version of the paper I linked (without institutional login just a preview, but the pages relevant are shown there). Also fixed a typo (should have been $limsup$, not $lim$).
– Staki42
Nov 17 at 22:02