Lemniscate zone and distortion theorems for multivalent functions. II
The paper continues earlier author's works on the relationship between the connectivity of lemniscates of a holomorphic function and its distortion. Let $f$ be a nonconstant meromorphic function on the unit disk $U=\{\zeta:|\zeta|<1\}$, and let $n(w,f)$ denote the number of roots of the equation $f(\zeta)=w$ in $U$. The function $f$ is circumferentially mean $p$-valent if \[ \frac{1}{2\pi}\int_0^{2\pi}n(\rho e^{i\varphi},f)d\varphi\leq p \] for any $\rho\in(0,1)$. The set $L_f(t)=\{\zeta\in U:|f(\zeta)|=t\}$, $0\leq t\leq\infty$, is a lemniscate of $f$. The union $L_f(t_1,t_2)=\bigcup_{t\in(t_1,t_2)}L_f(t)$, $0\leq t_1<t_2$, is a lemniscate zone of the first kind if each $L_f(t)$, $t\in(t_1,t_2)$, is connected and its image under $f$ is a $p$-fold cover of $\gamma(t):=\{w:|w|=t\}$. The set $L_f(t_1,t_2)$ is a lemniscate zone of the second kind if each $L_f(t)$, $t\in(t_1,t_2)$, contains no connected components compactly embedded in $U$. Finally, $L_f(t_1,t_2)$ is a lemniscate zone of the third kind if each $L_f(t)$ either contains no connected components compactly embedded in $U$ or is connected and its $f$-image forms a $p$-fold cover of $\gamma(t)$. For a positive integer $p$ and $\varkappa\in(0,1)$, let $Z_p(z)\equiv Z_p(z;\varkappa)$ be defined as \[ Z_p(\text{sn}(u;k);\varkappa):=\text{sn}(u\mathbf{K}(\varkappa)/\mathbf{K}(k);\varkappa),\;\;u\in\mathbb C,\;\;0<k<1, \] where the module $k$ is determined by $\mathbf{K}'(k)\mathbf{K}(\varkappa)=p\mathbf{K}'(\varkappa)\mathbf{K}(k)$, and $\mathbf{K}(\cdot)$ and $\mathbf{K}'(\cdot)$ are complete elliptic integrals of the first kind. Denote $F(z)\equiv F(z;p,\tau)=\Phi(Z_p(z;\varkappa))$, $p>1$, $\tau>1$, \[ \Phi(v)=\sqrt{\tau}\frac{v\sqrt{\varkappa}+1}{1-v\sqrt{\varkappa}},\;\;\;\sqrt{\varkappa}=\frac{\sqrt{\tau}-1}{\sqrt{\tau}+1}. \] All zeros of $F$ are simple and negative. Let $\alpha$ and $\beta$ denote the least and the greatest zero of $F$, respectively. The zeros of $F$ are numbered in the order of their occurrence: $\beta =z_1^*>z_2^*>\dots>z_p^*=\alpha$. By $|a,b|$ denote the closed interval on the real axis which joins $a$ and $b$ and does not pass through the origin. The author proves the following main theorem. \par Theorem 1. Let $p>1$, and let $f$ be a meromorphic circumferentially mean $p$-valent function on $U$ having a lemniscate zone $L_f(1,\tau)$ of the third kind. Suppose given some (not necessarily all) simple zeros $\zeta_1,\dots,\zeta_n$, $1\leq n\leq p$, of $f$ and different points $w_1,w_2$ in the $w$-plane. Suppose also that $f$ takes both $w_1$ and $w_2$ at most at $p-1$ points of $U$. Then, for any positive $\nu_1,\dots,\nu_n$, \[ \frac{\prod_{k=1}^n[(1-|\zeta_k|^2)|f'(\zeta_k)]^{\nu_k^2}}{\prod_{k=1}^n\prod_{l=1,l\neq k}^n\left|\frac {\zeta_k-\zeta_l}{1-\overline{\zeta}_k\zeta_l}\right|^{\nu_k\nu_l}}\leq\frac{\prod_{k=1}^n[(1-|\zeta_k^*|^2)|f_*'(\zeta_k^*)]^{(\nu_k^*)^2}} {\prod_{k=1}^n\prod_{l=1,l\neq k}^n\left|\frac {\zeta_k^*-\zeta_l^*}{1-\overline{\zeta_k^*}\zeta_l^*}\right|^{\nu_k^*\nu_l^*}}, \] where $\nu_k^*$ are such that $\nu_1^*>\dots>\nu_n^*$, $\{\nu_k^*\}_{k=1}^n=\{\nu_k\}_{k=1}^n$, $f_*=F\circ\varphi$, $\varphi(\zeta_k^*)=z_k^*$ for $k=1,\dots,n$, and $\varphi$ is any function mapping $U$ univalently and conformally onto the complement of the interval $|F^{-1}((-1)^p|w_1|),F^{-1}((-1)^p|w_2|)|^2$. Here $F^{-1}$ is the branch of the inverse to $F$ which maps $[-\infty,0]$ onto $|z|\geq-\alpha$ if $p$ is odd and $[0,\infty]$ onto $|z|\geq-\alpha$ if $p$ is even.
scientific article; zbMATH DE number 7037783