Let $A$ be a square Matrix and $||\cdot ||_p$ the induced Matrixnorm for $1 \leq p \leq \infty$. Is it true that $$||A||_p\leq \max(||A||_1,||A||_{\infty})?$$ For $p=2$ the answer is yes because $||A||_2^2\leq ||A||_1||A||_{\infty}$.

The motivation is that I have a family of matrices where I can bound the infinity and 1-Norm but I dont know how to bound the $p$-Norm.

Thx for any help in advance