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In this book I present new inequalities based on convexity. The first chapter is devoted to the Popovicius inequality for functions of several variables. The notion of 2Dconvex function is introduced, and several results are presented like the 2D analogue of Jensen's inequality, the version of HermiteHadamard type inequality, the analogue of midpoint convexity, convolution of 2Dconvex functions, and the nice proof of the classical Popovicius inequality, based on the mathematical induction. In the second chapter I studied in seven sections, the Hilberttype inequalities, HardyHilbert integral inequality etc. In the third chapter I present in six sections: the HermiteHadamard integral inequality, the Simpsontype, Newtontype and Gausstype inequalities. In the chapter four I discuss the Schur convexity of generalized Heronian means, involving two parameters. Chapter five has nine sections, and studies the BrunnMinkowski inequality.
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