Norm inequalities relating (i) a function and two of itsderivatives and (ii) a sequence and two of its differencesare studied. Detailed elementary proofs of basicinequalities are given. These are accessible to anyone witha background of advanced calculus and a rudimentaryknowledge of the Lp and lp spaces.The classical inequalities associated with the names ofLandau, Hadamard, Hardy and Littlewood, Kolmogorov, Schoenberg and Caravetta, etc., are discussed, as well astheir discrete analogues and weighted versions. Bestconstants and the existence and nature of extremals arestudied and many open...