Artinianness of Certain Graded Local Cohomology Modules

We show that if $R\,=\,{{\oplus }_{n\in \mathbb{N}0}}\,{{R}_{n}}$ is a Noetherian homogeneous ring with local base ring $({{R}_{0}},\,{{m}_{0}})$ , irrelevant ideal ${{R}_{+}}$ , and $M$ a finitely generated graded $R$ -module, then $H_{{{m}_{0}}R}^{j}\,(H_{R+}^{t}\,(M))$ is Artinian for $j\,=\,0,\,1$ where $t\,=\,\inf $ { $i\in {{\mathbb{N}}_{0}}:H_{R+}^{i}(M)$ is not finitely generated}. Also, we prove that if $\text{cd(}{{R}_{+}},M)\,=\,2$ , then for each $i\,\in \,{{\mathbb{N}}_{0}},\,H_{{{m}_{0}}R}^{i}\,(H_{R+}^{2}\,(M))$ is Artinian if and only if $H_{{{m}_{0}}R}^{i+2}(H_{R+}^{1}(M))$ is Artinian, where $ \text{cd(}{{R}_{+}},\,M)$ is the cohomological dimension of $M$ with respect to ${{R}_{+}}$ . This improves some results of R. Sazeedeh.