ON THE ABSENCE OF AN EXCEPTIONAL SET IN THE RELATION FROM WIMAN’S THEOREM

Abstract

Let $S(a,b),$\ $-\infty\le a<b\le +\infty,$\ be a class of functions analytic in $\Pi(a,b)=\{z: a<\text{\rm Re}\, z<b\}$ such that $$(\forall x\in(a,b))\colon\ M(x,F):=\sup\{|F(t+iy)|: a<t\le x, y\in{\mathbb R}\}<+\infty,$$ and $L(x,F)=(\ln M(x,F))'_+$ is the right derivative. By $S_{\infty}(a,b)$ we denote a subclass of the class $S(a,b)$, which consists of those functions $F\in S(a,b),$ such that $ L(x,F)\to +\infty \quad (x\to b-0)$, and by $S_0$ we denote the class of functions $F\in S_{\infty}(0, +\infty)$ for which there exists a function $\delta(r)\colon\mathbb{R}_{+}\to \mathbb{R}_{+}$ such that $\delta(r)\nearrow +\infty$ $(0\leq r\uparrow +\infty)$ and the inequality $$\big|L\big(r\pm \delta(r)/L(r,F),F\big)-L(r,F)\big|\leq L(r,F)/\delta(r) \quad (r\geq r_0) $$ is satisfied. Let  $$B_{F}(r)=\sup\{\hbox{Re~}F(z)\colon \hbox{Re~}z<r\},\quad A_{F}(r)=\inf\{\hbox{Re~}F(z)\colon \hbox{Re~}z<r\}. $$ The following theorem is proved: Let $F\in {S}_0$, then the asymptotic relations $$ M(r,F)=(1+o(1))B_{F}(r)=-(1+o(1))A_{F}(r), $$ hold as $r\to +\infty.$