Improvements of Logarithmic and Identric Mean Inequalities for Scalars and Operators

In this article, we provide refined inequalities for a convex Riemann's integrable function using refinements of the classical Hermite-Hadamard inequality. The obtained results are applied on special functions to establish new improvements of inequalities on the weighted logarithmic mean and weighted identric mean. Moreover, corresponding operator inequalities are introduced based on the scalar inequalities and the monotonicity property for operators.