Abstract In this paper, we mainly give some quadratic refinements of Young type inequalities. Namely: (va+(1−v)b)2−v∑j=1N2j(b−ab2j−12j)2≤(avb1−v)2+v2(a−b)2 $$\begin{array}{} \displaystyle (va+(1-v)b)^{2}-v{{\sum\limits_{j=1}^N}}2^{j}\Big(b- \sqrt[2^{j}]{ab^{2^{j}-1} }\, \Big)^{2}\leq(a^{v}b^{1-v})^{2}+v^{2}(a-b)^{2} \end{array}$$ for v ∉ [0, 12N+1 $\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array}$], N ∈ ℕ, a, b > 0; and (va+(1−v)b)2−(1−v)∑j=1N2j(a−a2j−1b2j)2≤(avb1−v)2+(1−v)2(a−b)2 $$\begin{array}{} \displaystyle (va+(1-v)b)^{2}-(1-v){{\sum\limits_{j=1}^N}}2^{j}\Big(a- \sqrt[2^{j}]{a^{2^{j}-1}b}\, \Big)^{2}\leq(a^{v}b^{1-v})^{2}+(1-v)^{2}(a-b)^{2} \end{array}$$ for v ∉ [1 − 12N+1 $\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array}$, 1], N ∈ ℕ, a, b > 0. As an application of these scalars results, we obtain some matrix inequalities for operators and Hilbert-Schmidt norms.

中文翻译：

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Young 型不等式的二次细化

摘要 本文主要给出Young型不等式的一些二次细化。即： (va+(1−v)b)2−v∑j=1N2j(b−ab2j−12j)2≤(avb1−v)2+v2(a−b)2 $$\begin{array}{} \displaystyle (va+(1-v)b)^{2}-v{{\sum\limits_{j=1}^N}}2^{j}\Big(b- \sqrt[2^{j} ]{ab^{2^{j}-1}}\, \Big)^{2}\leq(a^{v}b^{1-v})^{2}+v^{2}( ab)^{2} \end{array}$$ for v ∉ [0, 12N+1 $\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array }$], N ∈ ℕ, a, b > 0; 和 (va+(1−v)b)2−(1−v)∑j=1N2j(a−a2j−1b2j)2≤(avb1−v)2+(1−v)2(a−b)2 $ $\begin{array}{} \displaystyle (va+(1-v)b)^{2}-(1-v){{\sum\limits_{j=1}^N}}2^{j}\ Big(a- \sqrt[2^{j}]{a^{2^{j}-1}b}\, \Big)^{2}\leq(a^{v}b^{1-v })^{2}+(1-v)^{2}(ab)^{2} \end{array}$$ for v ∉ [1 − 12N+1 $\begin{array}{} \displaystyle \ frac{1}{2^{N+1}} \end{array}$, 1], N ∈ ℕ, a, b > 0. 作为这些标量的应用结果，