Consider a planar Hamiltonian system of the form \(\dot x=-g(y)\), \(\dot y=f(x)\) where both \(h=f\) and \(h=g\) satisfy the following assumptions: (i) \(h(0)=0\), \(h'(0)>0\), (ii) \((5(h'')^ 2-3h'h\prime'')(x)>0\) if \(h'(x)>0\), (iii) \(h(x)h''(x)<0\) if \(h'(x)=0\). Then (0,0) is a centerpoint and for the connected component of periodic orbits with (0,0) we can show that the period p(a) is strictly increasing with the initial condition a, \(x(0)=a\), \(y(0)=0\) \((p'(a)>0)\). Conditions (ii), (iii) can be shown to hold for any polynomial h of degree \(>1\) with all zeroes of h being real and simple, for all such polynomials in \(e^ x\) and for many more functions. The theorem can be applied to the Lotka-Volterra system and other examples.