$f$ entire, $g(x, y) := |f(x+iy)|$ integrable on $\mathbb{R}^{2}$ $\implies$ $f \equiv 0$.
91304
An entire function whose integral is bounded is identically zero
147299
Prove that $g(x,y) = \frac{x^3y}{x^2+y^2}$ with $g(0,0)=0$ is continuous
58647
Show that $f$ is integrable and $\int_{I} f=0$
68811
If $f$ is Lebesgue integrable on $[0,2]$ and $\int_E fdx=0$ for all measurable set E such that $m(E)=\pi/2$. Prove or disprove that $f=0$ a.e.
10253
Proving that all entire & injective functions take the form $f = ax + b$?
98363
Expanding two-variable function $f(x,y)$ over the complete sets $\{ g_{i}(x) \}$ and $\{ h_{j}(y) \}$
152347
Proof that a certain entire function is a polynomial
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$
30219
Sufficient condition(s) for $f'$ to be Riemann integrable
82745
Showing Entire Function is Bounded
26207
If $f$ is a quadratic and $f(x)>0\;\forall x$, and $g= f + f' + f''$, prove $g(x)>0\; \forall x$
3818
Proving two entire functions are constant.
43733
$|f'(x)| \le g(x)$ implies $|f(b) - f(a)| \le \int_a^b g(x) dx$, without assuming $f'$ to be integrable.
77106
If $f$ is continuous and $g$ is integrable on $[a,b]$, with $g(x) \ge 0$ for all $x \in [a,b]$ ...
65319
For all $x,y∈\Bbb{R}$ define that $ x\equiv y$ if$ x^2=y^2$
69804
Prove that if $f$ is entire and $\vert f(z^2) \vert \leq 2 \vert f(z) \vert$, then $f$ is constant
34542
If $f$ is integrable then $|f|$ is also integrable