Result: asymptotic behavior of entire functions with exceptional values in the borel relation: Asymptotic behavior of entire functions with exceptional values in the Borel relation

Summary: Let \(M_f(r)\) and \(\mu_f(r)\) be the maximum modulus and the maximal term of an entire function \(f\) and let \(l(r)\) be a continuously differentiable function convex with respect to \(\ln r\). We establish that in order for the relation \(\ln M_f(r)\sim \ln\mu_f(r)\), \(r\to\infty\) to hold for every entire function \(f\) such that \(\ln\mu_f(r)\sim l(r)\), \(r\to\infty,\) it is necessary and sufficient that \(\ln(rl'(r))=o(l(r))\), \(r\to\infty\).