Inequalities

a₁ a₂ analogue argument b₁ b₂ best possible bilinear forms coefficients consider constant continuous and strictly convergent convex function curve decreases deduce Theorem defined definition effectively proportional equality equation example follows from Theorem Fourier series ƒ and g generalisations Hardy and Littlewood Hence Hilbert's Hölder's inequality homogeneous increasing function infinite interval J. E. Littlewood Journ Lebesgue Lebesgue integrals Math means Minkowski's inequality multilinear forms necessary and sufficient non-negative null null set obtain P₁ Pólya proof of Theorem prove Theorem proves the theorem r+s+t=0 replace result Riesz satisfied similarly Stieltjes integral strictly positive sufficient condition summation Suppose symmetrical Theorem 13 Theorem 326 Theorem 9 theory true unless values variables W. H. Young write x₁ y₁ zero α₁ α₂ Σ Σ Σα Σαν ΣΣ