Computational Methods and Function Theory 4 (2004), No. 2, 447--460
Copyright Heldermann Verlag 2004
Igor Chyzhykov
tftj@franko.lviv.ua , Faculty of Mechanics and Mathematics, Ivan Franko National University, 1 Universytets'ka st., Lviv, 79000, Ukraine.
Oleh Skaskiv
tftj@franko.lviv.ua , Faculty of Mechanics and Mathematics, Ivan Franko National University, 1 Universytets'ka st., Lviv, 79000, Ukraine.
Let $w$ be a $\delta$-subharmonic function $w$ of zero genus, let $\mu_+$ and~$\mu_-$ be the positive and negative variations respectively of the Riesz charge associated with $w$ and let $n(t):=\mu_+(\{ |z|\le t\})+\mu_- (\{ |z|\le t\})$. In order that $$ \sup_{|z|=r} w(z)-\inf_{|z|=r} w(z)=\Landauo(1) \qquad\mbox{as }r=r_j\to+\infty, $$ it is sufficient that the condition $$ \liminf_{r\to+\infty} \left( \frac{1}{r} \int_1^r t\,dn(t)+ r\int_r^{+\infty}\frac{dn(t)}{t} \right)=0 $$ holds. In the case when the supports of $\mu_1$ and $\mu_2$ are concentrated on the opposite rays emanating from $z=0$ this condition is also necessary.
Keywords: δ-subharmonic function, meromorphic function, minimum modulus.
MSC 2000: 31A05, 30D15.
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