  A considerable amount of literature in the theory of inequality is devoted to
the study of Jensen's and Young's inequality. This article presents a number of
new inequalities involving the log-convex functions and the geometrically
convex functions. As their consequences, we derive the refinements for Young's
inequality and Jensen's inequality. In addition, the operator Jensen's type
inequality is also developed for conditioned two functions. Utilizing these new
inequalities, we investigate the operator inequalities related to the relative
operator entropy.
