Let $f$ be a complete perform and outline
$$M(r)=maxlimits_le r|f(z)|=maxlimits_z|f(z)|.$$
Equally, additionally outline
$$m_0(r)=minlimits_le r |f(z)|,hspace{5mm} m(r)=minlimits_z|f(z)|.$$

I used to be serious about what we will say concerning the capabilities $M(r), m_0(r)$ and $m(r)$ as $rtoinfty.$ For instance, by the utmost modulus precept, we all know that $M(r)$ is definitely rising. Subsequently the restrict $limlimits_{rtoinfty}M(r)$ exists. It is usually clear that if $M(r)to c<infty$ then (Liouville’s theorem) $f$ have to be a relentless.It follows that $M(r)to infty.$ In actual fact, for polynomials it’s every to test that $M(r)approx r^n$ as $rto infty.$ Furthermore, if $frac{M(r)}{r^n}to c<infty$ then utilizing Cauchy’s estimate we will present that $f$ have to be a polynomial. If $f$ is non-polynomial whole perform then it’s clear that $M(r)/r^nto infty$ for each $nge 0.$

My query is can we strengthen it furthre? For instance can we are saying that for a non-polynomial whole perform $f,$ we will need to have $M(r)approx e^{r}$ or $M(r)geq Ce^{r^{1-epsilon}}$ for each $epsilon>0?$

Now coming to $m_0(r)$ and $m(r).$ We observe that $m_0(r)$ is decresing and therefore the restrict exists, however it’s not very attention-grabbing. If $f$ has any zero within the aircraft then $m_0(r)=0$ for all $r$ sufficiently massive and due to this fact the restrict might be zero. Alternatively, if $f$ doesn’t have any zero and $f$ is non-constant then $f$ should go arbitrary near $0$ by picard’s theorem. It follows that $m_0(r)to 0.$ In different phrases, $m_0(r)to c<infty,$ and $cneq 0$ if and provided that $f$ is a continuing.

Probably the most attention-grabbing one is $m(r).$ Allow us to begin with a easy case. If $f$ is a polynomial (of diploma $nge 1$) then $m(r)approx r^n$ for suffiently massive $r.$ Subsequently, the restrict $m(r)to infty.$ Furthermore, $frac{m(r)}{r^n}to cneq 0.$ If $f$ shouldn’t be a polynomial and $f$ doesn’t have a zero then it is extremely much like $m_0(r)$ and $m(r)to 0.$ On the whole, for a non-polynomial whole perform $f,$ we all know that the infinity is a necessary singularity. Particularly, there exists a sequence $z_nto infty$ such that $|f(z_n)|to 0.$ This tells us that $m(|z_n|)to 0.$ Particularly, if $lim m(r)$ exists, then it have to be $0.$ However, I’m not in a position to set up the existence of restrict of $m(r).$

Does the restrict $limlimits_{rto infty}m(r)$ at all times exist?

Can we make a extra refined assertion concerning the conduct of $m(r)?$ For instance, if $f$ has $n$ zeroes within the complicated aircraft can we are saying that $m(r)to 0$ like $r^{-n}?$ (I’m not hoping this assertion to be true, it’s only for the illustration of the type of assertion I wish to make about $m(r).$)