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Is there exist an open connected domain $U$ on which $f$ is never zero but $|f_u|$ attains its minimum at some points of U
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up vote
1
down vote
favorite
Is the following statement is True/false ?
suppose that $f$ is non constant analytics function defined over $mathbbC$ then
there exist an open connected domain $U$ on which $f$ is never zero but $|f_|$ attains its minimum at some points of U
my attempts : i thinks This statement is False take $f(z) = e^z$
complex-analysis
asked Jul 14 at 16:10
stupid
58419
add a comment |Â
up vote
1
down vote
favorite
Is the following statement is True/false ?
suppose that $f$ is non constant analytics function defined over $mathbbC$ then
there exist an open connected domain $U$ on which $f$ is never zero but $|f_|$ attains its minimum at some points of U
my attempts : i thinks This statement is False take $f(z) = e^z$
complex-analysis
asked Jul 14 at 16:10
stupid
58419
i edits its ...
– stupid
Jul 14 at 16:22
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Is the following statement is True/false ?
suppose that $f$ is non constant analytics function defined over $mathbbC$ then
there exist an open connected domain $U$ on which $f$ is never zero but $|f_|$ attains its minimum at some points of U
my attempts : i thinks This statement is False take $f(z) = e^z$
complex-analysis
asked Jul 14 at 16:10
stupid
58419
Is the following statement is True/false ?
suppose that $f$ is non constant analytics function defined over $mathbbC$ then
there exist an open connected domain $U$ on which $f$ is never zero but $|f_|$ attains its minimum at some points of U
my attempts : i thinks This statement is False take $f(z) = e^z$
complex-analysis
asked Jul 14 at 16:10
stupid
58419
edited Jul 14 at 16:21
asked Jul 14 at 16:10
stupid
58419
asked Jul 14 at 16:10
stupid
58419
asked Jul 14 at 16:10
stupid
58419
58419
i edits its ...
– stupid
Jul 14 at 16:22
add a comment |Â
i edits its ...
– stupid
Jul 14 at 16:22
i edits its ...
– stupid
Jul 14 at 16:22
i edits its ...
– stupid
Jul 14 at 16:22
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
Hint: Use Liouville theorem which says That is, every holomorphic/Analytic function f for which there exists a positive number $M$ such that $f(z)$ $leq$ $M$
for all $z$ in $mathbbC$ is constant.
Use function $g(z)$ $=$ 1/$f(z)$ since f is never zero it is analytic what can you say if minimum of f exist at open domain U.
answered Jul 14 at 16:19
sscool
198116
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Hint: Use Liouville theorem which says That is, every holomorphic/Analytic function f for which there exists a positive number $M$ such that $f(z)$ $leq$ $M$
for all $z$ in $mathbbC$ is constant.
Use function $g(z)$ $=$ 1/$f(z)$ since f is never zero it is analytic what can you say if minimum of f exist at open domain U.
answered Jul 14 at 16:19
sscool
198116
add a comment |Â
up vote
1
down vote
accepted
Hint: Use Liouville theorem which says That is, every holomorphic/Analytic function f for which there exists a positive number $M$ such that $f(z)$ $leq$ $M$
for all $z$ in $mathbbC$ is constant.
Use function $g(z)$ $=$ 1/$f(z)$ since f is never zero it is analytic what can you say if minimum of f exist at open domain U.
answered Jul 14 at 16:19
sscool
198116
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Hint: Use Liouville theorem which says That is, every holomorphic/Analytic function f for which there exists a positive number $M$ such that $f(z)$ $leq$ $M$
for all $z$ in $mathbbC$ is constant.
Use function $g(z)$ $=$ 1/$f(z)$ since f is never zero it is analytic what can you say if minimum of f exist at open domain U.
answered Jul 14 at 16:19
sscool
198116
Hint: Use Liouville theorem which says That is, every holomorphic/Analytic function f for which there exists a positive number $M$ such that $f(z)$ $leq$ $M$
for all $z$ in $mathbbC$ is constant.
Use function $g(z)$ $=$ 1/$f(z)$ since f is never zero it is analytic what can you say if minimum of f exist at open domain U.
answered Jul 14 at 16:19
sscool
198116
answered Jul 14 at 16:19
sscool
198116
answered Jul 14 at 16:19
sscool
198116
answered Jul 14 at 16:19
sscool
198116
198116
add a comment |Â
add a comment |Â
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i edits its ...
– stupid
Jul 14 at 16:22