ON THE ABSENCE OF AN EXCEPTIONAL SET IN THE RELATION FROM WIMAN’S THEOREM - Dubei, S.I.; Skaskiv, O.B.; Trusevych, O.M. (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2025)
Let S(a,b), -∞a<b+∞, be a class of functions analytic in Π(a,b)={z: a<Re z<b} such that ( x(a,b))M(x,F):={|F(t+iy)|: a<tx, yR}<+∞, and L(x,F)=(M(x,F))’_+ is the right derivative. By S_∞(a,b) we denote a subclass of the class S(a,b), which consists of those functions FS(a,b), such that L(x,F)→ +∞ (x→ b-0), and by S_0 we denote the class of functions FS_∞(0, +∞) for which there exists a function δ(r)R_+→ R_+ such that δ(r) earrow +∞ (0≤ r+∞) and the inequality |L(r± δ(r)/L(r,F),F)-L(r,F)|≤ L(r,F)/δ(r) (r≥ r_0) is satisfied. Let B_F(r)={Re F(z)Re z<r}, A_F(r)={Re F(z)Re z<r}. The following theorem is proved: Let FS_0, then the asymptotic relations M(r,F)=(1+o(1))B_F(r)=-(1+o(1))A_F(r), hold as r→ +∞.