Prove that $g \in o(f)$ implies $f + g \in \Theta(f)$.  I'm pretty stuck on this one. I know that there is a constant $c_2$ such that $f \le c_2  (f+g)$, but I'm not sure how to prove that there exists a constant $c_1$ such that $c_1  (f+g) \le f$. If I can prove that then I can say that $f + g \in \Theta(f)$. Anyone have some suggestions?
123174
I know that big O is used to describe the upper bound of a function, so doesn't this mean that if the upper bounds f and g are the same, they are the same function? If this is the case, then we can say that  f ∈ Θ(g) because if that were not the case, then the two upper bounds would not be the same and thus one would out grow the other. Is this correct?
47373
Let $f\in C[0,1]$ be such  that $\forall \phi\in C[0,1]$ with $\int_0^1\phi \, dx=0$,we have $\int_0^1f\phi \, dx=0$.  Then $f$ is constant.  May be someone can help me proof this problem. Thanks!
54076
Let $f$ and $g$ be entire functions such that $f^n+g^n=1$, where $n\geq 3$ is an integer. Prove that $f$ and $g$ are constant.  I suppose I should somehow prove that either $f$ or $g$ is bounded so that I can apply Liouville's Theorem, but I don't see how. I tried setting the derivative of the left hand side equal to zero and work with that but that did not seem to work.
43306
I've encountered the following problem in complex analysis:  Prove that if $f$ is a real continuous function such that $|f|\le 1$ then $|\int_{|z|=1} f(z)dz| \le 4$.  I've decomposed $f$ to it's real and imaginary parts and got that it's enough to prove that:  $(\int_0^{2\pi}cos(\theta)g(\theta)d\theta)^2 +(\int_0^{2\pi}sin(\theta)g(\theta)d\theta)^2 \le 16$   Where $g(\theta)=f(e^{i\theta})$.   But now I'm stuck. Any ideas?  Thanks
10735
I am stuck with the following problem that says :     let $f \colon [a,b] \to \mathbb{R}$ be continuous on $[a,b]$    and $f''$ exist for all $x \in (a,b)$. Let $a \lt c \lt b$.    Prove that there exists a point $\xi \in (a,b)$    such that $$f(c)=\frac{b-c}{b-a}f(a)+\left( \frac{c-a}{b - a} \right)f(b)+\frac12(c-a)(c-b)f''(\xi).$$   Can someone explain in details? Thanks in advance for your time.
76511
Let $f_j \in C_c^\infty$ and assume there exists $M&gt;0$ such that $\text{supp}f_0 \subset \{ |x| \le M \}$ and $\text{supp} f \subset \{ 1/M \le |x| \le M \}$. Define $f_j$ by  $$ f_j (y) = f(y/2^j) $$ for $j =1, 2, \cdots$. Suppose that $$ \sum_{j=0}^\infty f_j (x) = 1.$$  Then how can I show that there exist $C, C'&gt;0$ such that$$ C' \sum_{j=0}^\infty \| f_j (g)\|_{L^2}^2 \le\| g \|_{L^2}^2 \le C \sum_{j=0}^\infty \| f_j (g)\|_{L^2}^2?$$ $C_c^\infty$ means $C^\infty$ functions with compact support and $\text{supp}$ means the support.
115495
I am studying for a qualifier exam in complex analysis and right now I'm solving questions from old exams. I am trying to prove the following:     Prove that if $f$ and $g$ are entire functions such that $f(z)^2 + g(z)^2 = 1$ for all $z \in \mathbb{C}$, then there exists an entire function $h$ such that $f(z) = \cos(h(z))$ and $g(z) = \sin(h(z))$.     My Attempt  The approach that occurred to me is the following. Since $f(z)^2 + g(z)^2 = 1$ then we have $(f(z) + ig(z))(f(z) - ig(z)) =  1$. Then each factor is nonvanishing everywhere in $\mathbb{C}$ and thus by the "holomorphic logarithm theorem" we know that since $\mathbb{C}$ is simply connected, there exists a holomorphic function $H:\mathbb{C} \to \mathbb{C}$ such that  $$e^{H(z)} = f(z) + ig(z)$$  and then we can write $\exp(H(z)) = \exp\left(i\dfrac{H(z)}{i} \right) = \exp(ih(z))$,    where  $h(z) := \dfrac{H(z)}{i}$.  Thus so far we have an entire function $h(z)$ that satisfies  $$e^{ih(z)} = f(z) + ig(z)$$  On the other hand, we also know that $e^{iz} = \cos{z} + i \sin{z}$ for any $z \in \mathbb{C}$, thus we see that   $$e^{ih(z)} = \cos{(h(z))} + i \sin{(h(z))} = f(z) + ig(z)$$  Thus at this point I would like to conclude somehow that we must have $f(z) = \cos(h(z))$ and $g(z) = \sin(h(z))$, but I can't see how and if this is possible.    My questions   Is the approach I have outlined a correct way to proceed, and if so how can I finish my argument? If my argument does not work, how can this be proved?   Thanks for any help.
51351
So the problem states that if $f(z)$ is entire, and satisfies the relation $f(z+i) = f(z)$ and $f(z+1) = f(z)$, show that $f(z)$ is constant. So I was thinking that since any point in $\mathbb{C}$ can be written as $\alpha * 1 + \beta * i $ we can say that $f(z + z_0) = f(z) $ in which case it is constant, but I'm having trouble breaking down the steps and using the fact that $f$ is entire, which makes me feel like I'm missing something. What should I review to figure this out?
72847
Let $G$ be a convex area and $f:G \rightarrow \mathbb C$ be a holomorphic function (with continious derivative). Prove: If $|f'(z)-1|\lt 1,$ then f is injective on $G$  Could anyone give me a hint ? I dont know how to tackle this problem
38819
Let $\Omega$ be a region in the complex plane and let $f_1$ and $f_2$ be holomorphic functions on $\Omega$ having no common zero. Show that there exist holomorphic functions $g_1$ and $g_2$ on $\Omega$ such that, $f_1g_1+f_2g_2$ is identically equal to one on $\Omega$.
9848
Suppose $D$ is a nonempty bounded subset of reals. Let $f:D \to \mathbb R$ and $g:D \to \mathbb R$. Define $(f+g)(x)=f(x)+g(x)$. Prove $\sup(f+g)(D) \le \sup f(D) + \sup g(D)$ (also prove that $\sup (f+g)$ exists).   I understand why this is the case, just not how to prove it. Left side is pretty much $\sup  (f(x)+g(x))$ and right side is $\sup (f(x) + g(y))$ for $x,\,y \in D$. Basically $f+g$ has to use the same variable and $f(D)+g(D)$ use different ones. But I don't know how to go about proving this.     The second part of the question is to find a specific example where strict inequality holds.    Let $D=\{a,b\}$ and $f: a \to 1,\, b\to 0, \,g: a \to 0,\, b\to 1$. $\sup f(D) = 1,\, \sup g(D) = 1,\, \sup f(D) + \sup g(D) = 2.$ $\sup (f+g)(D) = 1$ (if we choose a, $f+g = 1+0,\, b,\, f+g=0+1$).
87908
Prove that $f(z) = e^z$ maps $\mathbb{C} \setminus B $ onto $\mathbb{C} \setminus \{0\}$ where $B$ is a bounded subset of $\mathbb{C}$.  I have no idea how to even attempt this question so I would greatly appreciate some hints.
59868
Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $.  This is supposed to be question about continuity, but I’m not sure exactly what they mean, perhaps I misunderstood some concept along the way.
52170
How do you show that if $f$ is entire and if $\text{Im} f\ge0$, then $\text{Im} f$ is constant? Is this even true in the first place?
107853
Assuming $x,y\in\mathbb C^n$,consider:  $$f(x,y)=\sup_{θ,φ}\{||e^{i θ }x+ e^{i φ }y||^2: θ,φ\in\mathbb R\}$$  Which of the following is/are correct? 1.$ f(x,y)$≤$||x||^2+||y||^2+2|(x,y)|$ 2. $f(x,y)$= $||x||^2+||y||^2+2Re(x,y)$ 3. $f(x,y)$= $||x||^2+||y||^2+2|(x,y)|$ 4. $f(x,y)$> $||x||^2+||y||^2+2|(x,y)|$    How can I solve this problem ,I am completely stuck on it . can anyone help me?thanks
13726
Let $D\subseteq \mathbb{C}$ be open and connected and    $f,~g:D \rightarrow \mathbb{C}$ holomorphic functions such that    $|f|+|g|$ is constant on $D.$ Prove that $f,~g$ are constant on $D.$   Attempt. I noticed first that this problem is already set before, and is considered  to be duplicate () and we are sent directly to problem . However, the last reference deals with the sum $|f|^2+|g|^2$, which I do not see how is connected to the sum $|f|+|g|$, as stated in our title (for example, the equality $|f|^2+|g|^2=(|f|+|g|)^2-2|f||g|$ is not helpful here).  Thanks in advance!
10225
Suppose $f$ and $g$ are continuously differentiable functions such that $f(x) = g'(x)$ and $g(x) = f'(x)$ and that any product of $f, f', g$ and $g'$ is commutative for all $x ∈ R$. Show that $ f^2 − g^2 = C $ for some real constant C  I have actually no clue how to solve this, and would be really greatfull for all the help i can get