  We prove a sharp lower bound for the Tanaka-Webster holomorphic sectional
curvature of strictly pseudoconvex real hypersurfaces that are
"semi-isometrically" immersed in a K\"ahler manifold of nonnegative holomorphic
sectional curvature under an appropriate convexity condition. This gives a
partial answer to a question posed by Chanillo, Chiu, and Yang regarding the
positivity of the Tanaka-Webster scalar curvature of the boundary of a strictly
convex domain in $\mathbb{C}^2$ from 2012. In fact, the main result proves a
stronger positivity property, namely the $\frac12$-positivity in the sense of
Cao, Chang, and Chen, for compact "convex" real hypersurfaces in a K\"ahler
manifold of nonnegative holomorphic sectional curvature. Our approach is rather
simple and uses a version of the Gauss equation for semi-isometric CR
immersions of pseudohermitian manifolds into K\"ahler manifolds.
