On the growth of entire functions (Q1327519)

Let \(f\) be an entire function, denote by \(T(r,f)\) the Nevanlinna characteristic and by \(M(r,f)\) the maximum modulus of \(f\), and define \[ m^ +_ p(r,f)= \left({1\over 2\pi} \int^{2\pi}_ 0 \bigl(\log^ +| f(re^{i\theta})|\bigr)^ p d\theta\right)^{1/p} \] for \(1< p<\infty\). There is a number of papers (many of which are mentioned in this paper), where the growth of \(T(r,f)\) and \(\log M(r,f)\) is compared. Most of these results, however, hold only outside certain exceptional sets of \(r\)-values. This paper is concerned with results that hold for all \(r\). Suppose that \(\alpha> 1\) and \(| f(0)|\geq 1\). It is shown that there exists a constant \(d_ \alpha\) depending only on \(\alpha\) such that \[ \log M(r,f)\leq d_ \alpha T(r,f)^{1/2} T(\alpha r,f)^{1/2} \] and \[ m^ +_ p(r,f)\leq (d_ \alpha)^{(p-1)/p} T(r,f)^{(p+1)/2p} T(\alpha r,f)^{(p-1)/2p}. \] The first inequality improves the well-known inequality \[ \log M(r,f)\leq {R+ r\over R-r} T(R,f),\quad 0< r< R. \] A corollary is that if \(f\) has finite order \(\lambda\) and \(\varepsilon> 0\), then \(\log M(r,f)\leq r^{\lambda/2+\varepsilon} T(r,f)\) for all large \(r\). Examples are given to show that the results obtained are quite precise.