Circumferentially mean \(p\)-valent functions (Q1129750)

Let \(f(x)\) be a regular function in \(\Delta= \{z:| z|<1\}\), and let \(p\) be a positive number. The function \(f(z)\) is said to be circumferentially mean \(p\)-valent if \((1/2\pi) \int_0^{2\pi} n(re^{i\theta}, f)d\theta\leq p\) for all \(R>0\), where \(n(a,f)\) is the number of roots in \(\Delta\) of the equation \(f(z)=a\). ``Multivalent functions'' (1994; Zbl 0904.30001) by \textit{W. K. Hayman}, is a standard reference for this class of functions. It contains the principal tools used in the subject, including the Ahlfors length and area principle, the Hardy-Stein-Spencer identity, and several useful inequalities. Through proficient use of these methods, the author obtains sharp upper bounds on certain integrals. Suppose that \(f(z)\) is a circumferentially mean \(p\)-valent function and that either (1) \(f(z)\neq 0\) in \(\Delta\) or (2) \(p\) is a natural number and \(f(z)= z^p+ a_{p+1} z^{p+1}+\cdots\). Let \(L(r,\Gamma(\theta))\) denote the total length in \(\Delta\) of the curve \(\{z:| z|<r\), \(\arg f(z)=\theta\}\). Then \(\int_0^{2\pi} L(r,\Gamma(\theta)) d\theta\leq K(p)\) for \(0<r<1\). Similarly, let \(L(r,\Gamma(R))\) denote the total length in \(\Delta\) of \(\{z:| z|<r\), \(| f(z)|= R\}\), and, for \(\lambda\) in \((-\infty,\infty)\), and \(0< r<1\), let \(\ell(r,\lambda):= \int_0^\infty L(r,\Gamma(R)) R^\lambda dR\). When \(f(z)\neq 0\) in \(\Delta\), Theorem 1 gives separate sharp upper bounds on \(\ell(r,\lambda)\) for \(\lambda\) in \((-1,+1/2p,\infty)\), in \((-1-1/2p, -1+1/2p)\), in \((-\infty,-1-1/2p)\) and for \(\lambda=-1\pm 1/2p\). Theorem 2 assumes (2), and gives estimates when \(\lambda\) lies in \((-1+1/2p,\infty)\), \((-1-1/p, -1+1/2p)\), and \(\lambda=-1+1/2p\), and shows that if \(\lambda\leq -1-1/p\), then \(\ell(r,\lambda)=\infty\) for \(r\geq r_0= r_0(f)\).