In this paper, we prove some bounds for the p$$ p $$‐numerical radius as natural extensions of certain known bounds for the numerical radius. This complements many recent studies in this direction.

Among many results, we show that if A$$ A $$ is an n×n$$ n\times n $$ complex matrix, with Aluthge transform Ã$$ \overset{\widetilde }{A} $$, then for p≥1$$ p\ge 1 $$, ω​p(A)≤21p−1Ap+ωpÃ,$$ {\omega}_p(A)\le {2}^{\frac{1}{p}-1}\left({VerbarAVerbar}_p+{\omega}_p\left(\overset{\widetilde }{A}\right)\right), $$where ωp(·)$$ {\omega}_p\left(\cdotp \right) $$ and ‖·‖p$$ {\left\Vert \cdotp \right\Vert}_p $$ denote the p$$ p $$‐numerical radius and the Schatten p$$ p $$‐norm, respectively. This extends a celebrated result by Yamazaki, who showed the same result when p=∞$$ p=\infty $$.​

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