Let X be a real Banach space with dual X ∗ and moduli of convexity and smoothness δ X ( ε) and ϱ X ( τ), respectively. For 1 < p <∞, J p denotes the duality mapping from X into 2 X ∗ with gauge function t p − 1 and j p denotes an arbitrary selection for J p . Let A= { φ: R + → + : φ (0) = 0, φ( t) is strictly increasing and there exists c > 0 such that φ(t) ⩾ cδ X( t 2 )} and F= { ϑ: R + → + : ϑ (0) = 0, ϑ( t) is convex, nondecreasing and there exists K > 0 such that ϑ( τ) ⩽ Kϱ X ( τ)}. It is proved that X is uniformly convex if and only if there is a φ ϵ A such that ∥ x + y∥ p ⩾ ∥ x∥ p + p〈 j p x, y〉 + σ φ ( x, y) ∀ x, y ϵ X and X is uniformly smooth if and only if there is a ϑ ∈ F such that ∥ x + y∥ p ⩽ ∥ x∥ p + p〈 j p x, y〉 + σ ϑ ( x, y) ∀ x, y ϵ X, where, for given function f, σ f ( x, y) is defined by σ f (x,y) = p∫ 0 1 ∥ x+ ty ∥ ∨ ∥ x ∥) p t f t ∥ y ∥ ∥ x+ty ∥ ∨ ∥ x ∥ dt These inequalities which have various applications can be regarded as general Banach space versions of the well-known polarization identity occurring in Hilbert spaces.