In the present paper, the authors derive sufficient conditions for functions to be in a certain general class of Carathéodory functions in the open unit disk by using the Miller--Mocanu lemma. As an application of our main result, we deduce sufficient conditions for functions belonging to the class $\mathcal{ST}\widehat{\mathcal S}^\mu_\beta$ which is introduced here. The various results presented here would generalize and extend many known results.