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I am looking for a book that deals with inequalities like

$$\frac{\sin x}{x} < 1 - \frac{x}{ \pi} \qquad x \in (0,\pi)$$

perhaps some geometric exposition. I have heard that Hardy's book is good, any other suggestions?

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The book you might like is: Intro to geometric inequalities by Mitrinovic Kluwer Academic Press.

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Hardy, Little, and Polya is a classic.

Introduction to Inequalities: Edwin F. Beckenbach, R. Bellman.

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