Convexity Revisited: Methods, Results, and Applications Dorin Andrica, Sorin R ˘ adulescu, and Marius R ˘ adulescu Abstract We present some new aspects involving strong convexity, the pointwise and uniform convergence on compact sets of sequences of convex functions, circular symmetric inequalities and bistochastic matrices with examples and applications, the convexity properties of the multivariate monomial, and Schur convexity. 2010 AMS Subject Classification 41A36; 26D20 1 Introduction A function f : D → R defined on a nonempty subset D of a real linear space E is called convex, if the domain D of the function is convex and for every x,y ∈ D and every t ∈[0, 1] one has f(tx + (1 − t)y) ≤ tf (x) + (1 − t)f(y). If the above inequality is strict whenever x = y and 0 <t< 1, f is called strictly convex. A function f such that −f is convex is called concave. The simplest example of a convex function is an affine function f(x) = a T x + b. This function clearly is convex on the entire space R n , and the convexity inequality for it is equality. The affine function is also concave. One can easily prove that the function which is both convex and concave on the entire space is an affine function. Other examples of convex functions are given by the norms on the space R n , i.e. the real-valued functions which are nonnegative everywhere, positive outside of D. Andrica () Department of Mathematics, “Babe¸ s-Bolyai” University, Cluj-Napoca, Romania e-mail: dandrica@math.ubbcluj.ro S. R˘ adulescu · M. R˘ adulescu Institute of Mathematical Statistics and Applied Mathematics, Bucharest, Romania © Springer Nature Switzerland AG 2019 D. Andrica, T. M. Rassias (eds.), Differential and Integral Inequalities, Springer Optimization and Its Applications 151, https://doi.org/10.1007/978-3-030-27407-8_3 49