Some results of Young-type inequalities

Abstract

In this paper, one of our main targets is to present some improvements of Young-type inequalities due to Alzer et al. (Linear Multilinear Algebra 63(3):622–635, 2015) under some conditions. That is to say: when $0< \nu , \tau <1,\ a,b>0$, we have

$$\begin{aligned} \frac{a\nabla _{\nu }b-a\sharp _{\nu }b}{a\nabla _{\tau }b-a\sharp _{\tau }b}\le \frac{\nu (1-\nu )}{\tau (1-\tau )} \ \ { \mathrm {and}} \ \ \frac{(a\nabla _{\nu }b)^{2}-(a\sharp _{\nu } b)^{2}}{(a\nabla _{\tau }b)^{2}-(a\sharp _{\tau }b)^{2}}\le \frac{\nu (1-\nu )}{\tau (1-\tau )} \end{aligned}$$

for $(b-a)(\tau -\nu )\ge 0;$ and the inequalities are reversed if $(b-a)(\tau -\nu )\le 0.$ In addition, we show a new Young-type inequality

$$\begin{aligned} (1-v^{N+1}+v^{N+2})a+(1-v^{2})b\le v^{vN-(N+1)}a^{v}b^{1-v}+(\sqrt{a}-\sqrt{b} \ )^{2} \end{aligned}$$

for $0\le v\le 1, N\in {\mathbb {N}}$ and $a,b>0.$ Then we can get some related results about operators, Hilbert–Schmidt norms, determinants by these scalars results.