A decomposition theorem for an analytic function

AbstractLet f(z), an analytic function with radius of convergence R (0 < R < ∞) be represented by the gap series ∑k = 0∞ ckzλk. Set M(r) = max¦z¦ = r ¦f(z)¦, m(r) = maxk ⩾ 0{¦ ck ¦ rλk}, v(r) = max{λk ¦ ¦ ck ¦ rλk = m(r)} and define the growth constants ϱ, λ, T, t by ϱλ=lim supr→R inf{log[Rr /(R−r)]−1log+log+M(r)}, and if 0 < ϱ < ∞, Tt=lim supr→R inf{[Rr /(R−r)]−ϱlog+M(r)}. Then, assuming 0 < t < T < ∞, we obtain a decomposition theorem for f(z)