Abstract
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 $g-$convexity as a generalization of
$\log-$convexity. Then we prove that $g-$convex
functions have better estimates in certain known inequalities like the
Hermite-Hadard 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.