Quadratic refinements of Young type inequalities

Abstract

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