This paper establishes several novel Hermite–Hadamard type inequalities for convex functions on closed intervals. In contrast to classical results, the proposed inequalities incorporate logarithmic weights together with variable subinterval boundaries \(a<a_{1}\le b_{1}<b\) providing a more flexible method for integral estimation. Furthermore, we develop a probabilistic interpretation by viewing a convex function as a probability density function, which enables the derivation of new bounds for event probabilities. In particular, we obtain estimates of the form \( \mathbb {P}\left( \left| X-\frac{3a+b}{4}\right| \le \frac{b-a}{8}\right) +\mathbb {P}\left( \left| X-\frac{a+3b}{4}\right| \le \frac{b-a}{8}\right) \le \frac{b-a}{16}\left( f(a)+f(b)\right) +\frac{3}{8}. \) A numerical example is included to demonstrate the improvement over the classical Hermite–Hadamard inequality. Finally, we propose several conjectures related to probability concentration for convex distributions, highlighting potential directions for future research.
