On \(p\)-th order of a function analytic in the unit disc (Q2276624)

Let \(f(z)=\sum^\infty_{n=0}c_nz^n\) define a function in the unit disc. Let \(M(r)\) be the maximum of \(|f(z)|\) on \(|z|=r\). If for some \(p\leq 2\), \(\limsup_{r\to 1}((\log^{[p]}M(r))/(-\log(1-r)))=\rho\), where \(\log^{[m]} x= \log(\log^{[m-1]}z)\), then \(\frac{\rho}{1+\rho}= \limsup_{n\to\infty} ((\log^{+ [p]}|c_n|)/(\log n))\). A characterization for lower order is also shown.