New Versions of Uniformly Convex Functions Via Quadratic Complete Homogeneous Symmetric Polynomials

We introduce new versions of uniformly convex functions, namely h_d strongly (weaker) convex functions. Based on the positivity of complete homogeneous symmetric polynomials with even degree, recently studied in Rovenţa and Temereanc (Mediterr J Math 16:1–16, 2019), Rovenţa et al. (A note on weighted Ingham’s inequality for families of exponentials with no gap, In: 24th ICSTCC, pp 43–48, 2020; Weighted Ingham’s type inequalities via the positivity of quadratic polynomials, submitted), and Tao ( https://terrytao.wordpress.com/2017/08/06/schur-convexity-and-positive-definiteness-of-the-even-degree-co-mplete-homogeneous-symmetric-polynomials/ ), we introduce stronger and weaker versions of uniformly convexity. In this context, we recover well-known type inequalities such as: Jensen’s, Hardy–Littlewood–Polya’s and Popoviciu’s inequalities. Some final remarks related to Sherman’s and Ingham’s type inequalities are also discussed.