Inequalities in calculus: methods of prooving results and problem solving

TL;DR

This paper provides a comprehensive overview of inequalities in calculus, including standard and novel results, with numerous examples and methods for proving inequalities involving functions, sums, and means.

Contribution

It introduces new inequalities and generalizations, especially for exponentials, logarithms, and means, along with diverse proof techniques and applications.

Findings

New inequalities for exponentials and logarithms

Generalizations of classical inequalities like Cauchy-Bunyakovskii and Young

Mean inequalities extended to complex domains

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 Schl\"omilch-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, H\"older (Rogers-H\"older-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.