Inequalities in calculus: methods of prooving results and problem solving

Abstract

This preprint is a text for students and teachers on inequalities. Some standard topics are covered on application of calculus to inequality proving. Many examples are considered, stated, solved or partially solved. Some problems are standard, but some are rare, new and original. The next topics are considered with many examples: monotonicity of functions, Lagrange theorem and inequalities proving, estimating of finite sums, inequalities of Schlömilch-LeMonnier type, proof of inequalities by method of mathematical induction, inequalities for the number $e$, exponentials, logarithmic and similar functions, some means and their inequalities, Cauchy-Bunyakovskii, Minkovskii, Young, Hölder (Rogers-Hölder-Riesz !) inequalities and some of their improvements and generalisations. Some new results include inequalities on exponentials, logarithmic and similar functions, generalisations of Cauchy--Bunyakovskii and Young inequalities, some mean inequalities including mean inequalities on the complex domain and more.