On uniqueness theorems for entire functions tending to zero along disjoint arcs (Q1824060)

It is known that if an entire function f(z) is too small on a curve towards infinity, then \(f(z)\equiv 0\). Here a sequence of curves \(\{\gamma_ n\}\) is considered, each \(\gamma_ n\) connecting a point on \(| z| =L_ n\) with a point on \(| z| =\alpha L_ n\), \(0<\alpha <1\). Suppose f(z) has finite order and type \((\rho,\tau)\) and that the condition \(\ell n| f(z)| \leq -c| z|^{\rho}\) holds on \(\{\gamma_ n\}\). It c is large enough, then \(f(z)\equiv 0\). How large? The estimation of the critical \(c=c(\alpha)\) should be made closer than in the present paper. For instance, it is evident that \(c(\alpha)\) increases with \(\alpha\), but the expression given \(\to +\infty\) as \(\alpha\to 0!\)