Mathematical Inequalities using Taylor Series
نویسنده
چکیده
1 Overview We begin by recalling the Rolle’s Theorem.1 Using this result, we shall derive the Lagrange Form of the Taylor’s Remainder Theorem. Subsequently, we shall derive several mathematical inequalities as a corollary of this result. For example, 1. We shall use the Taylor’s Remainder Theorem to upper and lower bound exponential functions using polynomials. 2. We shall use the Taylor’ Remainder Theorem to obtain the Jensen’s Inequality. • From the Jensen’s Inequality, we shall derive standard mathematical inequalities like the AM-GM-HM inequality, Cauchy-Schwarz inequality, Young’s inequality, and Hölder’s inequality, 3. In the future, we shall use the Taylor’s Remainder Theorem to prove the Bonami-GrossBeckner Hypercontractivity Theorem in Fourier Analysis.
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