LAUSR

Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In particular, we define what we called g -composite convexity as a generalization of log convexity. Then we prove that g -composite convex functions have better estimates in certain known inequalities like the Hermite-Hadamard inequality, super additivity of convex functions, the Majorization inequality and some means inequalities. Strongly related to this, we define the index of convexity as a measure of “how much the function is convex”. Applications including Hilbert space operators, matrices and entropies will be presented in the end. Mathematics subject classification (2020): Primary 26A51; Secondary 47A30, 39B62, 26D07, 47B15, 15A60.