On Julia directions of entire functions of small order (Q1609932)

A ray \(\arg z =\theta_0\) is called a Julia direction for an entire function \(f\) if for every \(\varepsilon>0\) and every complex \(a\) up to one exceptional value the equation \(f(z)-a=0\) has infinitely many solutions in the angle \(\arg z \in (\theta_0-\varepsilon,\theta_0+\varepsilon)\). In 1982 H.Yoshida conjectured that if a ray \(\arg z =\theta_0\) is not a Julia direction for an entire function \(f\) of order less then \(\frac{1}{2}\) then for some \(\varepsilon>0\) \(\lim|f(z)|=\infty\), \(z\rightarrow\infty\),\(\arg z \in (\theta_0-\varepsilon,\theta_0+\varepsilon)\).The author proves that Yoshida's conjecture is affirmative under an additional condition: the lower order of a function coincides with the order. The Nevanlinna theory and the estimation of Ahlfors for the harmonic measure are used to prove this.