Convex Functions of One Variable

Abstract

Section 1.1 just recalls well-known elementary facts which are essential for all the following chapters. Section 1.2 is devoted to proofs of basic inequalities in analysis by convexity arguments. The Legendre transform (conjugate convex functions) is discussed in Section 1.3 in a spirit which prepares for the case of several variables in Chapter II. Section 1.4 is an interlude presenting an interesting characterization of the Γ function by the functional equation and logarithmic convexity, due to Bohr and Mollerup. We introduce representation of convex functions by means of Green’s function in Section 1.5, as a preparation for the representation formulas for subharmonic functions. In Section 1.6 we discuss some weaker notions of convexity which occur in microlocal analysis. Section 1.4 and most of Section 1.6 can be bypassed with no loss of continuity. The last section, Section 1.7, studies when the minimum of a family of (convex) functions is convex. The extension to (pluri-)subharmonic functions in Chapters III and IV will be essential in Chapter VII.