\documentstyle{article}
\def\grad{{\mathrm{grad}}}
\def\U{{\mathbf{U}}}
\title{An estimate for spherically $p$-valent functions}
\author{A. Eremenko}
\date{October 1998}
\begin{document}
\maketitle

Let us call a meromorphic function in a region $D$
spherically $p$-valent if the
spherical area of the image of this function, counting multiplicity,
is at most $\pi p$. In other words,
$$\int_D(f^\#)^2(z)dm_z\leq \pi p,\quad\mbox{where}\quad
f^\#=\frac{|f'|}{(1+|f|^2)}.$$
This class is related to mean $p$-valent holomorphic
functions as defined in \cite{Hayman}.
\medskip

\noindent
{\bf Theorem 1.} {\em The family of all spherically $p$-valent functions
in a region $D$ is quasinormal of order $p$ in $D$.}
\vspace{.1in}

{\em Proof.} We may assume without loss of generality that
$D$ is the unit disc. Let $f_n$ be a sequence of
spherically $p$-valent functions. Write $f_n=g_n/h_n$ so that 
$g_n$ and $h_n$ have no common zeros.
Let 
$$u_n=\log\sqrt{|g_n|^2+|h_n|^2}.$$
Then $u_n$ are positive subharmonic functions, and
$$\Delta u_n=4(f_n^{\#})^2$$
by direct computation. (This identity expresses the fact that
the spherical metric has curvature $4$).
Fix arbitrary $r<1$. Let 
$$u_n=v_n+w_n$$
be the Riesz decomposition, where $v_n$ are the least harmonic majorants
of $u_n$ in the disc $D(r)$, and $w_n$ are Green potentials.
As the Riesz measures of $u_n$ and $w_n$ in $D(r)$ are uniformly bounded,
one can select a subsequence such that
the corresponding potentials $w_n$ will converge to some
subharmonic function $w^*\not\equiv-\infty$.
Let $k_n$ be analytic functions without zeros
in $D(r)$ such that $v_n=-\log|k_n|$.
Then we have $f_n=g_nk_n/h_nk_n$, and
$$\log\sqrt{|g_nk_n|^2+|h_nk_n|^2}=w_n\to w^*.$$
So the sequences $g_nk_n$ and $h_nk_n$ are uniformly bounded
on compact subsets of $D(r)$ and by choosing another subsequence
we obtain that $g_nk_n\to g^*$ and $h_nk_n\to h^*$, where $g^*$ and $h^*$
are holomorphic in $D(r)$, one of them is not identical to zero.
Then $f_n\to g^*/h^*$ uniformly on compact subsets of $$D(r)\backslash
\{\mbox{common zeros of}\; g^*, h^*\}.$$
As $r<1$ is arbitrary, we obtain the statement about quasinormality.
Now, the number of common zeros of $g^*$ and $h^*$ cannot exceed $p$,
(each common zero contributes $\pi p$ to the 
Laplacian of $w^*$),
so we have quasinormality of order $p$.
\vspace{.1in}

Remark. Simple examples show that a family can be quasinormal
in the unit disc with an irregular point at zero,
but the functions of the family are not uniformly
spherically $p$-valent in any disc centered at $0$.
\vspace{.1in}

\noindent
{\bf Theorem 2}. {\em Let $f$ be a holomorphic
function without zeros
in the unit disc, which is spherically $p$-valent
in the ring $A=\{ 1/2<|z|<1\}$.
$$(1-|z|^2)f^{\#}(z)\leq 9p\log 2+3\log 2+2/\sqrt{e}\approx 6.24\,p+3.31,\quad
|z|<1, $$  
where $f^\#$ stands for the spherical derivative.}
\medskip

{\em Remarks}.
1. The example $f(z)=az$ which is univalent but may have large
derivative at $0$ shows that our additional assumption
that $f$ has no zeros is essential.
\newline
2. Our estimate is of interest only when $p\geq 1$ because it does not tend
to zero when $p\to 0$. When $p<1$ in fact no assumption about omitted zeros
is necessary and the best possible estimate for this case, due to
Duffresnoy, is given in 
\cite[Theorem 6.1]{Hayman2}\newline
3. 
A theorem of Montel
(\cite[p.66]{Montel}) shows that a quasinormal family of holomorphic
functions omitting one value is in fact normal. It follows that
under the assumptions of the Theorem there exists a bound for the derivative.
Our purpose was to obtain an explicit bound.\newline
4. A similar result with the same proof holds for
holomorphic curves in $n$-dimensional projective space, omitting
$n+1$ hyperplanes, the estimate depends on the omitted hyperplanes and
dimension. This also applies to Duffresnoy's theorem mentioned in 2.
\newline

{\em Proof}. A standard argument reduces the problem to
the case when $f$ is continuous in the closed disc $\bar\U$. In view of
conformal invariance it is enough to prove the estimate for $z=0$.
We also assume without loss of generality that 
\begin{equation}\label{leq}
|f(0)|\leq 1.
\end{equation}
Let $h$ be the least harmonic majorant of $\log^+|f|$ in $\U$. Then 
$h=-\log|f_0|$ for some holomorphic function without zeros in $\U$,
and we set $f_1=ff_0$, so that
$$f=f_1/f_0.$$
Now we define harmonic functions $u_j=\log|f_j|,\; j=0,1$ and notice that
these functions are negative and their least common harmonic majorant is $0$.
We also put $v=u_0\vee u_1$ (pointwise maximum) and
$u=\log\sqrt{|f_0|^2+|f_1|^2}$. Both functions $u$ and $v$ are subharmonic
and satisfy
\begin{equation}\label{estim}
v\leq u\leq v+(\log 2)/2,\quad v\leq 0\quad\mbox{and}\quad v(z)=0,|z|=1.
\end{equation}
In addition we have
$$\Delta u=f^\#.$$
Thus applying Jensen's Formula to $u$ we obtain
$$\frac{1}{2\pi}\int_{-\pi}^\pi u(re^{i\theta})\, d\theta+\frac{1}{\pi}\int_r^1A(t)\frac{dt}{t}
\geq
\frac{1}{2\pi}\int_{-\pi}^\pi u(e^{i\theta})\, d\theta,$$
where $A(t)$ is the spherical area of the $f$-image of the disc $|z|\leq t$,
and $r\in (0,1)$ is a number to be specified later.
Thus we can find a point $z_0$ with $|z_0|=r$, such that
$$u(z_0)\geq \int_{-\pi}^\pi u(e^{i\theta})\frac{d\theta}{2\pi}-p\log(1/r),
$$ 
where we used the assumption that $A(t)\leq p\pi.$ 
Using the unequalities (\ref{estim}) we conclude that
$$v(z_0)\geq -p\log (1/r)-(\log 2)/2.$$
Thus for some $j\in\{0,1\}$ we have
$$u_j(z_0)\geq -p\log(1/r)-(\log 2)/2,$$
and using the assumption (\ref{leq}),
which is equivalent to $u_0(0)\geq u_1(0)$, and Harnack's inequality
for negative harmonic function $u_j$, we obtain
$$u_0(0)\geq u_j(0)\geq\frac{1+r}{1-r}u_j(z_0)\geq-\frac{1+r}{1-r}
(p\log(1/r)+(\log 2)/2).$$
One can optimize with respect to $r$, but let us just put $r=1/2$:
$$u_0(0)\geq-3p\log 2-3(\log 2)/2=:-C(p).$$
\medskip

\noindent
{\bf Lemma}. {\em For negative harmonic functions $w$ in the unit disc}
$$|\grad\, w(0)|\leq 2|w(0)|.$$
\medskip

This follows from the Poisson formula.\hfill$\Box$
\medskip

Now we consider two cases.
\medskip

\noindent
{\em Case 1}. $|f_1(0)|\geq |f_0(0)|^2$.
This is equivalent to $u_1(0)\geq 2u_0(0)$. Thus by the Lemma
$|\grad\, u_0(0)|\leq 2C(p)$ and $|\grad\, u_1(0)|\leq 4C(p)$.
We have
$$f^\#(0)=\frac{|f_0f_1|}{|f_0|^2+|f_1|^2}\left|\frac{f_0^\prime}{f_0}+
         \frac{f_1^\prime}{f_1}\right|\leq
\frac{1}{2}(|\grad\, u_0|+|\grad\, u_1|)$$
and $f^\#(0)\leq 3C(p)$ follows.
\medskip

\noindent
{\em Case 2}. $|f_1(0)|\leq |f_0(0)|^2$. In this case we write
$$f^\#(0)\leq
\frac{|f_0f_1|}{|f_0|^2}\left(|\grad\, u_0|+|\grad\, u_1|\right)
\leq \left|\frac{f_1}{f_0}\right||\grad\, u_0|+
\frac{\sqrt{|f_1|}}{|f_0|}\sqrt{|f_1|}|\grad\, u_1|,$$
which gives in view of the Lemma the estimate
$$f^\#(0)\leq 2C(p)+\max_{0\leq x\leq 1}(2\sqrt{x}\log(1/x))=2C(p)+2/\sqrt{e}.$$

\begin{thebibliography}{1}

\bibitem{Hayman} W. Hayman, Multivalent Functions, Cambridge, 1994.
\bibitem{Hayman2} W. Hayman, Meromorphic Functions, Oxford, 1964.
\bibitem{Montel} P. Montel, Le\c{c}ons sur les familles normales de fonctions
analytiques at leurs applications, Paris, Gauthier-Villars, 1927.

\end{thebibliography}
\noindent
Purdue University,\newline
West Lafayette IN 47907\newline
{\em eremenko@math.purdue.edu}
\end{document}

