Motivated by recent developments in the study of Jensen-Mercer type operator inequalities, this paper presents several new and more precise series of inequalities for self-adjoint operators on unital $C^{\ast }$-algebras. Our approach is based on the concept of strong convexity, which allows the introduction of additional correction terms and the derivation of refined upper and lower bounds. The obtained results generalize and strengthen the classical Jensen-Mercer operator inequalities, both in the presence and absence of operator convexity. As an application, we establish new estimates for operator quasi-arithmetic means of the Mercer type and derive corresponding refinements for operator power means, including the arithmetic, geometric, and harmonic means.