Production functions are analyzed in two time frames:
Short-run: A period where at least one factor of production is fixed (e.g., capital, land) and others are variable (e.g., labor). The relationship between inputs and output in the short-run is explained by the Law of Variable Proportions (or Law of Diminishing Returns).
Long-run: A period where all factors of production are variable. The relationship is explained by the Law of Returns to Scale.
Law of demand and elasticity are related to consumption, not production.