Consider a production function of the form g(h) = ah^b, where h represents hours of input (h > 0), a is a positive parameter (a > 0), and the exponent b is between 0 and 1 (0 < b < 1). Construct a formal mathematical proof to demonstrate that the average product for this function is always diminishing. Your proof must explicitly show the relationship between the marginal product and the average product that leads to this conclusion.