Titlebook: Inequalities; Theorems, Techniques Zdravko Cvetkovski Book 2012 Springer-Verlag Berlin Heidelberg 2012 Cauchy-Schwarz inequality.H?lder’s i

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,Generalizations of the Cauchy–Schwarz Inequality, Chebishev’s Inequality and the Mean Inequalities,In Chap.?. we presented the ., . and the .. In this section we will give their generalizations. The proof of first theorem is left to the reader, since it is similar to the proof of ..

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,Newton’s Inequality, Maclaurin’s Inequality,Let ..,..,…,.. be arbitrary real numbers..Consider the polynomial . Then the coefficients ..,..,…,.. can be expressed as functions of ..,..,…,.., i.e. we have . For each .=1,2,…,. we define ..

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GIBE

Two Theorems from Differential Calculus, and Their Applications for Proving Inequalities,In this section we’ll give two theorems (without proof), whose origins are part of differential calculus, and which are widely used in proving certain inequalities. We assume that the reader has basic knowledge of differential calculus.

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Sum of Squares (SOS Method),One of the basic procedures for proving inequalities is to rewrite them as a sum of squares (.) and then, according to the most elementary property that the square of a real number is non-negative, to prove a certain inequality. This property is the basis of the SOS method.

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,Bernoulli’s Inequality, the Cauchy–Schwarz Inequality, Chebishev’s Inequality, Surányi’s Inequalityl inequalities containing more variables, and inequalities which are difficult to prove with already adopted elementary inequalities. These inequalities are often used for proving different inequalities for mathematical competitions.

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