Rigidity results from Cauchy-Riemann constraints

Let $f$ be an entire function such that, for all $z$, $\left|f(z)\right| = \left|\sin z\right|$. Prove that there is a constant $C$ of modulus $1$ such that $f(z) = C\sin z$.

Let $f = u+iv$ be analytic in a connected open set $D$, where $u$ and $v$ are real valued. Suppose there are real constants $a$, $b$ and $c$ such that $a^{2}+b^{2} \neq 0$ and

in $D$. Show that $f$ is constant in $D$.

Let $s(y)$ and $t(y)$ be real differentiable functions of $y$, $-\infty <y<\infty$, such that the complex function

is complex analytic with $s(0) = 1$ and $t(0) = 0$. Determine $s(y)$ and $t(y)$.

Let the function $f$ be analytic in the entire complex plane, real valued on the real axis, and of positive imaginary part in the upper half-plane. Prove $f'(x)>0$ for $x$ real.

Let $f = u+iv$ and $g = p+iq$ be analytic functions defined in a neighborhood of the origin in the complex plane. Assume $\left|g'(0)\right|<\left|f'(0)\right|$. Prove that there is a neighborhood of the origin in which the function $h = f+\overline{g}$ is one-to-one.

Let the functions $f(z)$ and $g(z)$ be regular in some domain $G$. Prove that the sum $f(z)+\overline{g(z)}$ is real throughout the domain $G$ if and only if $f(z) = g(z)+C$, where $C$ is a real constant.

Let the functions $f(z)$ and $g(z)$ be regular in some domain $G$, and $g(z) \not\equiv 0$. Prove that the product $f(z) \overline{g(z)}$ is real (nonnegative) throughout the domain $G$ if and only if $f(z) = C g(z)$, where $C$ is a real (nonnegative) constant.