On the means of an entire function and their derivatives with index pair (p,q) (Q1109171)

For an entire function f(z) of index pair (p,q), the author defines a `unified mean' \(P^ m_{\delta,K}(r,f)\) as \[ \frac{K+1}{2\pi (\log^{[m]}r)^{K+1}}\int^{r}_{c}\int^{2\pi}_{0}\frac{| f(xe^{i\theta})|^{\delta}(\log^{[m]}x)^ K}{V_{[m-1]}(x)}dx d\theta \] where \(-1<K<\infty\), \(0<\delta <\infty\), \(\log^{[n]}x=\log (\log^{[n-1]}x)\), \(V_{[m]}(x)=\prod^{m}_{i=0}\log^{[i]}x\) and c is a constant depending on m. Relations have been obtained that indicate the relative growth of the unified means of the entire function f and its derivatives.