Let A and B be n×nσf:[0,∞)→[0,∞)f(0)=0f′(0)(AσB)≤f(m)m(AσB)≤f(A)σf(B)≤f(M)M(AσB)≤f′(M)(AσB),|||f(A)+f(B)|||≤f(M)M|||A+B|||≤|||f(A+B)|||M≤A+B positive definite complex matrices, let n×nσf:[0,∞)→[0,∞)f(0)=0f′(0)(AσB)≤f(m)m(AσB)≤f(A)σf(B)≤f(M)M(AσB)≤f′(M)(AσB),|||f(A)+f(B)|||≤f(M)M|||A+B|||≤|||f(A+B)|||M≤A+B be a matrix mean, and let n×nσf:[0,∞)→[0,∞)f(0)=0f′(0)(AσB)≤f(m)m(AσB)≤f(A)σf(B)≤f(M)M(AσB)≤f′(M)(AσB),|||f(A)+f(B)|||≤f(M)M|||A+B|||≤|||f(A+B)|||M≤A+B be a differentiable convex function with n×nσf:[0,∞)→[0,∞)f(0)=0f′(0)(AσB)≤f(m)m(AσB)≤f(A)σf(B)≤f(M)M(AσB)≤f′(M)(AσB),|||f(A)+f(B)|||≤f(M)M|||A+B|||≤|||f(A+B)|||M≤A+B. We prove that n×nσf:[0,∞)→[0,∞)f(0)=0f′(0)(AσB)≤f(m)m(AσB)≤f(A)σf(B)≤f(M)M(AσB)≤f′(M)(AσB),|||f(A)+f(B)|||≤f(M)M|||A+B|||≤|||f(A+B)|||M≤A+Bwhere m represents the smallest eigenvalues of A and B and M represents the largest eigenvalues of A and B. If f is differentiable and concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under certain mild conditions. In particular, if f(x)/x is increasing, then n×nσf:[0,∞)→[0,∞)f(0)=0f′(0)(AσB)≤f(m)m(AσB)≤f(A)σf(B)≤f(M)M(AσB)≤f′(M)(AσB),|||f(A)+f(B)|||≤f(M)M|||A+B|||≤|||f(A+B)|||M≤A+Bholds for all A and B with n×nσf:[0,∞)→[0,∞)f(0)=0f′(0)(AσB)≤f(m)m(AσB)≤f(A)σf(B)≤f(M)M(AσB)≤f′(M)(AσB),|||f(A)+f(B)|||≤f(M)M|||A+B|||≤|||f(A+B)|||M≤A+B. Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski’s determinant inequality.