Quadratic refinements of Young type inequalities

Abstract

Abstract In this paper, we mainly give some quadratic refinements of Young type inequalities. Namely: ( v a + ( 1 − v ) b ) 2 − v ∑ j = 1 N 2 j ( b − a b 2 j − 1 2 j ) 2 ≤ ( a v b 1 − v ) 2 + v 2 ( 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, 1 2 N + 1 $\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array}$ ], N ∈ ℕ, a, b > 0; and ( v a + ( 1 − v ) b ) 2 − ( 1 − v ) ∑ j = 1 N 2 j ( a − a 2 j − 1 b 2 j ) 2 ≤ ( a v b 1 − 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 − 1 2 N + 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.