  This paper presents necessary, sufficient, and equivalent conditions for the
spherical convexity of non-homogeneous quadratic functions. In addition to
motivating this study and identifying useful criteria for determining whether
such functions are spherically convex, we discovered surprising properties that
distinguish spherically convex quadratic functions from their geodesically
convex counterparts in both hyperbolic and Euclidean spaces. Since spherically
convex functions over the entire sphere are constant, we restricted our focus
to proper spherically convex subsets of the sphere. Although most of our
results pertain to non-homogeneous quadratic functions on the spherically
convex set of unit vectors with positive coordinates, we also present findings
for more general spherically convex sets. Beyond the general non-homogeneous
quadratic functions, we consider explicit special cases where the matrix in the
function's definition is of a specific type, such as positive, diagonal, and
Z-matrix.
