Further Results on $\mu(r)$ and $\nu(r)$

In a rectangular coordinate system mark the points

(omitting meaningless ones with $a_{n} = 0$) and from each point draw a ray directed vertically upwards. The smallest convex region $\mathfrak {R}$ which includes all these rays extends to infinity. Draw the supporting line with direction coefficient $\log r$ (the straight line on which lies at least one boundary point and no interior point of $\mathfrak {R}$ and which forms an angle with the positive real axis whose tangent is $\log r$). Interpret in the resulting figure $\log \mu (r)$, $\nu (r)$, $\log \rho_{n}$.

Let $\alpha$ be fixed, $0 < \alpha < 1$. The quotient $\mu (\alpha r)/\mu (r)$ is strictly decreasing with increasing $r$, apart from certain polynomials.

Let the power series

be such that each term in turn becomes the maximum term, i.e. to every index $n = 0, 1, 2, \ldots$ there belongs at least one real number $r$, $r > 0$, such that $\left|a_{n}\right| r^{n}$ is not exceeded by any term. For this it is necessary and sufficient that

Let $0 < \alpha < 1$. The central index $\nu (r)$ of the power series

which is convergent for $\left|z\right| < 1$ assumes in turn all the values $0, 1, 2, 3, \ldots$. Its maximum term satisfies

as $r$ tends to $1$.

(Continued from problem 44.) As $r$ tends to $1$,

Let $\alpha > 0$. Calculate $\rho_{1}, \rho_{2}, \rho_{3}, \ldots$ for the power series

which is convergent for all finite $z$, and show that its maximum term satisfies

as $r$ tends to $+\infty$.

(Continued from problem 46.) For fixed $\alpha$, $\alpha > 0$, as $r$ tends to $+\infty$