Inequalities

a₁ a₂ ambn argument best possible bilinear forms coefficients consider constant continuous and strictly convergent convex function corresponding to Theorem curve decreasing deduce Theorem defined definition denote effectively proportional equality equation example f and g F₁ finite number follows from Theorem Fourier series fpdx generalisations gives Hardy and Littlewood Hence Hölder's inequality increasing function interval Lebesgue Lebesgue integrals M₁ means Minkowski's inequality monotonic functional multilinear forms necessary and sufficient non-negative null set obtain Pólya positive and finite positive number proof of Theorem prove Theorem proves the theorem r+s+t=0 rearrangement replace result Riesz satisfied Schur Stieltjes integral strictly positive sufficient condition summation Suppose symmetrical Theorem 13 Theorem 9 theory true unless f upper bound values vanishes variables W. H. Young write x₁ zero ακ Σα ΣΣ