Question

$f,g$ entire such that $f(0)=g(0)\neq 0$ and $|f(z)|\leq |g(z)|$ for all $z\in\mathbb{C}$, then $f=g$.

score 203 · Answer 1 · 2021-08-10 20:29:52

203

Views

$f,g$ entire such that $f(0)=g(0)\neq 0$ and $|f(z)|\leq |g(z)|$ for all $z\in\mathbb{C}$, then $f=g$.

Published on 10 Aug 2021 - 20:29

score 106 · Answer 2 · 2021-08-15 21:32:40

106

Views

Does there exist $z_0 \notin \{a_n\}_{n \geq 1}$ such that $f(z_0) = 0\ $?

Published on 15 Aug 2021 - 21:32

score 146 · Answer 3 · 2021-08-17 09:53:48

146

Views

If $ |f(z^{2})|\leq|f(z)| $ and $ f $ is entire, then is $ f $ constant?

Published on 17 Aug 2021 - 9:53

score 132 · Answer 4 · 2021-08-17 18:18:05

132

Views

Is a function with an essential singularity always some composition of an entire transcendental (not a polynomial) function?

Published on 17 Aug 2021 - 18:18

score 80 · Answer 5 · 2021-08-20 11:59:34

80

Views

What can we say about these two entire functions?

Published on 20 Aug 2021 - 11:59

score 396 · Answer 6 · 2021-08-23 14:29:06

396

Views

If $f$ is an entire function, $\lim_{z\to\infty} z^{-1}\operatorname{ Re} f(z)=0$, show $f$ is constant.

Published on 23 Aug 2021 - 14:29

score 86 · Answer 7 · 2026-02-26 18:16:40

86

Views

How to deduce that there doesn't exist any sequence of polynomials $P_n$ such that $P_n(z)\to \frac 1z$

Published on 26 Feb 2026 - 18:16

score 51 · Answer 8 · 2021-08-25 15:09:53

51

Views

What can we say about $f$ if $\int_{|z|=1}\frac{f(z)}{((k+1)z-1)^n}dz=0$?

Published on 25 Aug 2021 - 15:09

score 165 · Answer 9 · 2021-08-31 17:28:51

165

Views

Proving that the entire function which satisfies a given property is unique

Published on 31 Aug 2021 - 17:28

score 143 · Answer 10 · 2021-09-03 14:15:43

143

Views

Maximum value of an entire function in a closed set.

Published on 03 Sep 2021 - 14:15

score 45 · Answer 11 · 2026-03-25 12:50:48

45

Views

Suppose $f_1,f_2$ entire functions. Produce entire $h,g_1,g_2$ such that $f_1=hg_1$ and $f_2=hg_2$ with $g_1,g_2$ no common zeros.

Published on 25 Mar 2026 - 12:50

score 60 · Answer 12 · 2021-09-12 22:13:57

60

Views

More thoughts on if a entire function has at least one coefficient of the Taylor series being real

Published on 12 Sep 2021 - 22:13

score 77 · Answer 13 · 2021-09-15 15:18:04

77

Views

Representation exponential type functions

Published on 15 Sep 2021 - 15:18

score 953 · Answer 14 · 2021-09-16 15:10:35

953

Views

If $f$ is an entire function such that $|f(z)| \le 2 \ln (|z|+1)$ for all $z \in \Bbb{C}$, then $f(z) = 0$ for all $z \in \Bbb{C}$.

Published on 16 Sep 2021 - 15:10

score 204 · Answer 15 · 2021-09-26 08:22:44

204

Views

upper bound on $L_2$-norm of a power series, in terms of coefficients?

Published on 26 Sep 2021 - 8:22

List Question

$f,g$ entire such that $f(0)=g(0)\neq 0$ and $|f(z)|\leq |g(z)|$ for all $z\in\mathbb{C}$, then $f=g$.

Does there exist $z_0 \notin \{a_n\}_{n \geq 1}$ such that $f(z_0) = 0\ $?

If $ |f(z^{2})|\leq|f(z)| $ and $ f $ is entire, then is $ f $ constant?

Is a function with an essential singularity always some composition of an entire transcendental (not a polynomial) function?

What can we say about these two entire functions?

If $f$ is an entire function, $\lim_{z\to\infty} z^{-1}\operatorname{ Re} f(z)=0$, show $f$ is constant.

How to deduce that there doesn't exist any sequence of polynomials $P_n$ such that $P_n(z)\to \frac 1z$

What can we say about $f$ if $\int_{|z|=1}\frac{f(z)}{((k+1)z-1)^n}dz=0$?

Proving that the entire function which satisfies a given property is unique

Maximum value of an entire function in a closed set.

Suppose $f_1,f_2$ entire functions. Produce entire $h,g_1,g_2$ such that $f_1=hg_1$ and $f_2=hg_2$ with $g_1,g_2$ no common zeros.

More thoughts on if a entire function has at least one coefficient of the Taylor series being real

Representation exponential type functions

If $f$ is an entire function such that $|f(z)| \le 2 \ln (|z|+1)$ for all $z \in \Bbb{C}$, then $f(z) = 0$ for all $z \in \Bbb{C}$.

upper bound on $L_2$-norm of a power series, in terms of coefficients?

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