Functions that are mean p-valent with respect to \(\mu\)-area (Q1063717)

Let f(z) be regular or meromorphic in a domain D and let n(w), \(w=re^{it}\), be the number of roots of the equation \(f(z)=w\), which lie in D. Let us put \[ p(r)=(1/2\pi)\int^{2\pi}_{0}n(re^{it})dt,\quad W_{\mu}(R)=\int^{R}_{0}p(r)d(r^{2\mu}). \] If for some positive numbers p,\(\mu\) we have \(W_{\mu}(R)\leq pR^{2\mu}\) for all \(R>0\), then the function f(z) is called mean p-valent with respect to the \(\mu\)-area in D. We denote by \(\Sigma_{p,\mu}\) the class of all regular functions F(\(\zeta)\) which are mean p-valent with respect to the \(\mu\)-area in \(\{\) \(\zeta\) : \(| \zeta | >1\}\) and have the representation \[ F(\zeta)=\zeta^ p(1+\alpha_ 1\zeta^{-1}+\alpha_ 2\zeta^{- 2}+...) \] and by \(S_{p,\mu}\) the class of all regular functions f(z) which are mean p-valent with respect to the \(\mu\)-area in \(\{\) \(z: | z| <1\}\) and have the representation \[ f(z)=z^ p(1+a_ 1z+a_ 2z^ 2+...). \] We also put \[ \Sigma '\!_{p,\mu}=\{F(\zeta)\in \Sigma_{p,\mu};\quad F(\zeta)\neq 0\quad for\quad | \zeta | >1\}. \] The following results are obtained: Theorem I. If \(F(\zeta)\in \Sigma'_{p,\mu}\) and \(c_ n(\lambda)\) are defined by the equality \[ (F(\zeta)/\zeta^ p)^{\lambda}=\sum^{\infty}_{n=0}c_ n(\lambda)\zeta^{-n},\quad c_ 0(\lambda)=1,\quad | \zeta | >1, \] then for \(\lambda\in (0,\mu]\) and not for all \(\lambda >\mu\) we have \[ \sum^{\infty}_{n=1}(n-p\lambda)| c_ n(\lambda)|^ 2\leq p\lambda. \] In addition \(| \alpha_ 1| \leq 2p\) for 2p\(\mu\geq 1\) and \(| \alpha_ 2| \leq p(2p-1)\) for \(p\mu\geq 1\). Theorem II. If \(f(z)\in S_{p,\mu}\), \(\lambda >0\) and \(c_ n(\lambda)\) are defined by the equality \[ (f(z)/z^ p)^{- \lambda}=\sum^{\infty}_{n=0}c_ n(\lambda)z^ n,\quad c_ 0(\lambda)=1,\quad | z| <1, \] then \[ \sum^{\infty}_{n=1}(n- p\lambda)| c_ n(\lambda)|^ 2\leq p\lambda. \] In addition \(| a_ 1| \leq 2p\).