Generalisations of Turán's main theorems on lower bounds for sums of powers

In his book, Eine neue Methode in der Analysis und deren Andwendungen, P. Turán proved a number of new theorems given lower bounds for sums of powers. Since it was only his intention to demonstrate a new type of result, his bounds are by no means best possible nor are his proofs easily susceptible of improvement. We generalise Turán's so-called Main Theorems to exponential sums with polynomial coefficients by a simple method involving only the evaluation and estimation of certain determinants. This approach gives in each case a result known to be asymptotically correct in the various exponents, and when specialised to the case of constant coefficients it provides in each case best-known results. Our method moreover applies in more general circumstances and provided only that the determinants which arise can be conveniently estimated serves to provide lower bounds for other than exponential sums.