On the ratio of the maximum term and the maximum modulus of the sum of two entire functions (Q1174914)

Let \(f(z)\) be an entire function, \(M(r,f)\) and \(\mu(r,f)\) be the maximum modulus of \(f(z)\) and the maximum term of \(f(z)\). Let \(M_ k\) be the class of all entire functions \(f\) for which \[ \limsup_{r\to\infty} {{\mu(r,f)} \over {M(r,f)}}=k. \] Though the set of all entire functions is closed under addition, the author here proves that the sum of any two transcendental entire functions belonging to \(M_ k\) need not belong to the same class \(M_ k\). For any \(f_ 1\in M_ k\) (\(0\leq k<1\)) he constructs an \(f_ 2(z)\in M_ k\) such that \(f_ 1(z)+f_ 2(z)\) is a transcendental entire function in \(M_ 1\). Given a transcendental entire function \(f_ 1\in M_ 1\) he finds another entire function \(f_ 2\in M\), such that \(f_ 1+f_ 2\in M_ k\), \(k\neq 1\).