Abstract:

The comparison of quantities is an essential tool in mathematics. The art of inequalities is found in the clever, often subtle methods used to generate and verify them. The science of inequalities lies in their careful interpretation and in the know...
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The comparison of quantities is an essential tool in mathematics. The art of inequalities is found in the clever, often subtle methods used to generate and verify them. The science of inequalities lies in their careful interpretation and in the knowledge of their scope and limitations. In this work we study some fundamental inequalities in mathematics. In the first chapter we prove the arithmetic  geometric mean inequality. A plenty of proofs of this inequality are known, induction, backward induction, use of Jensen inequality, swapping
terms, use of Lagrange multiplier, convexity. We include the Cauchy original proof and another one due to Polya. We deal with convexity in second chapter. The Jensen inequality give us a new proof of the arithmetic  geometric mean inequality with arbitrary weights. Also, from the inequality of
Young we derive the Holder inequality with positive weights. In the third chapter we study the behavior of weighted means of order p, including the extreme cases p = ±∞. We prove the equivalence between some studied inequalities: the arithmetic  geometric mean inequality, Holder inequality and weighted power mean inequality. The fourth chapter is dedicated to the Carleman’s inequality. In 1923, Carleman gave necessary and sufficient conditions for functions not to be quasianalytic. As a lemma (stated as a
theorem) for one of the implications, Carleman proved that
X∞k=1 √k a1a2 · · · ak < e X∞k=1 ak if (ak) is a sequence of real positive numbers and the sum on the righthand side is convergent. The constant e is sharp. We begin with Carleson original proof. In 1926 George Polya gave an elegant proof that depended on little more than the arithmetic  geometric mean inequality. We also include proofs by Knopp and Redheffer. We start the fifth chapter with a functional equation of the Γ function that is necessary to prove Hilbert inequality X∞ m=1 X∞ n=1 ambn m + n < 2π X∞ m=1 a2m
1/2 X∞ n=1 b2n 1/2. Several years after Hilbert’s discovery, Issai Schur provided a new proof which showed Hilbert’s inequality actually holds with constant π. Hilbert evaluated certain trigonometric integrals to prove his inequality. Nevertheless, we prove this inequality through an appropriate application of Cauchy  Schwarz inequality. We also give the integral version of this inequality and we derive from it a finer version of Hilbert’s inequality. We conclude the chapter demonstrating the generalization of Schur.
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