Power-series coefficients controlled by the real part

We assume that the function

Express the coefficients $a_{1},a_{2},\ldots ,a_{n},\ldots$ in terms of the real and the imaginary part resp.of $f(z)$ on the circle $\left|z\right|=r$, $0<r<R$.

The function

is assumed to be regular and to have positive real part in the open disk $\left|z\right|<1$. Then $\left|a_{n}\right|\leq 1$, $n=1,2,\ldots$ In none of these inequalities can we replace $1$ by a smaller number.

Suppose that the function

on the disk $\left|z\right|<R$. Then the inequality

holds for $0<r<R$. Example: $f(z)=\dfrac {z+1}{z-1}$, $R=1$, $A=0$.

We assume that the function $f(z) = a_{0}+a_{1}z+\cdots +a_{n}z^{n}+\cdots$ is regular in the disk $\left|z\right|<R$ and we denote by $\Delta (f)$ the largest oscillation of the real part of $f(z)$ in $\left|z\right|<R$, i.e.$\Delta (f)$ is the least upper bound of $\left|\re f(z_{1})-\re f(z_{2})\right|$ for $\left|z_{1}\right|<R$, $\left|z_{2}\right|<R$. Then we have

The factor $\dfrac {2}{\pi }$ cannot be replaced by a smaller one.

Interpret this proposition geometrically.

Let the function $f(z) = a_{0}+a_{1}z+\cdots +a_{n}z^{n}+\cdots$ be regular in the disk $\left|z\right|<R$ and denote by $D(f)$ the largest oscillation of $f(z)$ for $\left|z\right|<R$, i.e.$D(f)$ is the least upper bound of $\left|f(z_{1})-f(z_{2})\right|$ for $\left|z_{1}\right|<R$, $\left|z_{2}\right|<R$. Then we have

The factor $\frac{1}{2}$ can not be replaced by a smaller one. Interpret this proposition geometrically.