In this paper, we prove that each of the following functions is convex on R : f (t) = w N (A t XA 1−t ± A 1−t XA t), (t) = w N (A t XA 1−t), and h(t) = w N (A t XA t) where A > 0, X ∈ M n and N(.) is a unitarily invariant norm on M n. Consequently, we answer positively the question concerning the convexity of the function t → w(A t XA t) proposed by in (2018). We provide some generalizations and extensions of w N (.) by using Kwong functions. More precisely, we prove the following w N (f (A)X(A) + (A)X f (A)) ≤ w N (AX + XA) ≤ 2w N (X)N(A), which is a kind of generalization of Heinz inequality for the generalized numerical radius norm. Finally, some inequalities for the Schatten p-generalized numerical radius for partitioned 2 × 2 block matrices are established, which generalize the Hilbert-Schmidt numerical radius inequalities given by Aldalabih and Kittaneh in (2019).