An operator A∈B(H)\documentclass[12pt]{minimal}
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\begin{document}$A\in B(\mathcal{H})$\end{document}, the algebra of bounded linear transformations on a complex infinite dimensional Hilbert space H\documentclass[12pt]{minimal}
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\begin{document}$\mathcal{H}$\end{document}, belongs to class A(n)\documentclass[12pt]{minimal}
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\begin{document}$\mathcal{A}(n)$\end{document} (resp., A(∗−n)\documentclass[12pt]{minimal}
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\begin{document}$\mathcal{A}(*-n)$\end{document}) if |A|2≤|An+1|2n+1\documentclass[12pt]{minimal}
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\begin{document}$\vert A\vert^{2}\leq\vert A^{n+1}\vert^{\frac{2}{n+1}}$\end{document} (resp., |A∗|2≤|An+1|2n+1\documentclass[12pt]{minimal}
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\begin{document}$\vert A^{*}\vert^{2}\leq \vert A^{n+1}\vert^{\frac{2}{n+1}}$\end{document}) for some integer n≥1\documentclass[12pt]{minimal}
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\begin{document}$n\geq1$\end{document}, and an operator A∈B(H)\documentclass[12pt]{minimal}
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\begin{document}$A\in B(\mathcal{H})$\end{document} is called n-paranormal, denoted A∈P(n)\documentclass[12pt]{minimal}
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\begin{document}$A\in \mathcal{P}(n)$\end{document} (resp., ∗−n\documentclass[12pt]{minimal}
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\begin{document}$*-n$\end{document}-paranormal, denoted A∈P(∗−n)\documentclass[12pt]{minimal}
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\begin{document}$A\in \mathcal{P}(*-n)$\end{document}) if ∥Ax∥n+1≤∥An+1x∥∥x∥n\documentclass[12pt]{minimal}
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\begin{document}$\Vert Ax\Vert ^{n+1}\leq \Vert A^{n+1}x\Vert \Vert x\Vert ^{n}$\end{document} (resp., ∥A∗x∥n+1≤∥An+1x∥∥x∥n\documentclass[12pt]{minimal}
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\begin{document}$\Vert A^{*}x\Vert ^{n+1}\leq \Vert A^{n+1}x\Vert \Vert x\Vert ^{n}$\end{document}) for some integer n≥1\documentclass[12pt]{minimal}
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\begin{document}$n\geq 1$\end{document} and all x∈H\documentclass[12pt]{minimal}
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\begin{document}$x \in\mathcal{H}$\end{document}. In this paper, we prove that if A∈{A(n)∪P(n)}\documentclass[12pt]{minimal}
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\begin{document}$A\in\{\mathcal{A}(n)\cup \mathcal{P}(n)\}$\end{document} (resp., A∈{A(∗−n)∪P(∗−n)}\documentclass[12pt]{minimal}
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\begin{document}$A\in\{\mathcal{A}(*-n)\cup \mathcal{P}(*-n)\}$\end{document}) is a contraction without a non-trivial invariant subspace, then A, |An+1|2n+1−|A|2\documentclass[12pt]{minimal}
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\begin{document}$\vert A^{n+1}\vert^{\frac{2}{n+1}}-\vert A\vert^{2}$\end{document} and |An+1|2−n+1n|A|2+1\documentclass[12pt]{minimal}
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\begin{document}$\vert A^{n+1}\vert^{2}- {\frac{n+1}{n}}\vert A\vert^{2}+ 1$\end{document} (resp., A, |An+1|2n+1−|A∗|2\documentclass[12pt]{minimal}
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\begin{document}$\vert A^{n+1}\vert^{\frac{2}{n+1}}-\vert A^{*}\vert^{2}$\end{document} and |An+2|2−n+1n|A|2+1≥0\documentclass[12pt]{minimal}
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\begin{document}$\vert A^{n+2}\vert^{2}- {\frac{n+1}{n}}\vert A\vert^{2}+ 1\geq0$\end{document}) are proper contractions.
