Case 3 p ∈ [3,+∞). Then Lemma 2.7(5) shows that g′(x) < 0 for all x ∈ (0,mp∗). This together with equations (3.18)-(3.22) leads to the conclusion that f (x) < 0 for all x ∈ (0,mp∗), namely, inequality (3.16) holds for all x ∈ (0,mp∗) and p ∈ [3,+∞). □ Acknowledgements. The research was supported by the Natural Science Foundation of China (Grant Nos. 11701176, 61673169, 11301127, 11626101, 11601485), the Natural Science Foundation of Zhejiang Province (Grant No. LQY18G030001) and the research project for college students of Huzhou University (Grant No. 2019-111).
