## InequalitiesThis classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and lucidly both the statement and proof of all the standard inequalities of analysis. The authors were well-known for their powers of exposition and made this subject accessible to a wide audience of mathematicians. |

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### Contents

INTRODUCTION | 1 |

ELEMENTARY MEAN VALUES | 12 |

equalities 3740 3739 | 37 |

MEAN VALUES WITH AN ARBITRARY | 65 |

OF THE CALCULUS | 102 |

INFINITE SERIES | 114 |

INTEGRALS | 126 |

SOME APPLICATIONS OF | 172 |

SOME THEOREMS CONCERNING | 196 |

HILBERTS INEQUALITY AND | 226 |

Theorems Paxje | 260 |

On strictly positive forms 406407 | 300 |

On Hilberts inequality | 308 |

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### Common terms and phrases

analogue of Theorem applications arbitrary argument asserts assume best possible bilinear form Calculus of Variations Cauchy's inequality chapter chord coefficients complete consider constant continuous and strictly convergent convex function corresponding to Theorem curve decreasing deduce Theorem defined definition denote effectively proportional equality equation equivalent example finite number follows from Theorem Fourier series generalisations gives Hardy and Littlewood Hence Holder's inequality homogeneous increasing function interval Lebesgue integrals Lemma limit linear maximum Minkowski's inequality monotonic functions multilinear forms necessary and sufficient negative non-negative null set obtain positive and finite positive number proof of Theorem prove Theorem proves the theorem quadratic form rearrangement reduces remarks replace result Riesz satisfied Schur similarly ordered Stieltjes integral strictly monotonic strictly positive sufficient condition summation Suppose symmetrical Theorem 13 Theorem 9 theory true unless upper bound vanishes variables W. H. Young weights write zero