A generalisation of Turán's main theorems to binomials and logarithms

The Main Theorems of P. Turán's book Eine neue Methode in der Analysis und deren Anwendungen concern only sums of powers but are easily generalised to exponential sums with polynomial coefficients. It does not appear to have been observed however, that similar such theorems with analogous implication as to value distribution and arithmetical behaviour can be formulated for a wider class of functions. We prove a result for functions of the form [formula omitted] subsuming identities which Mahler has shown to contain transcendence results on the exponential and logarithmic functions and diophantine results of the Thue-Siegel-Roth type.