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 $$0< \nu , \tau <1,\ a,b>0$$ 0 < ν , τ < 1 , a , b > 0 , we have $$\begin{aligned} \frac{a\nabla _{\nu }b-a\sharp _{\nu }b}{a\nabla _{\tau }b-a\sharp _{\tau }b}\le \frac{\nu (1-\nu )}{\tau (1-\tau )} \ \ { \mathrm {and}} \ \ \frac{(a\nabla _{\nu }b)^{2}-(a\sharp _{\nu } b)^{2}}{(a\nabla _{\tau }b)^{2}-(a\sharp _{\tau }b)^{2}}\le \frac{\nu (1-\nu )}{\tau (1-\tau )} \end{aligned}$$ a ∇ ν b - a ♯ ν b a ∇ τ b - a ♯ τ b ≤ ν ( 1 - ν ) τ ( 1 - τ ) and ( a ∇ ν b ) 2 - ( a ♯ ν b ) 2 ( a ∇ τ b ) 2 - ( a ♯ τ b ) 2 ≤ ν ( 1 - ν ) τ ( 1 - τ ) for $$(b-a)(\tau -\nu )\ge 0;$$ ( b - a ) ( τ - ν ) ≥ 0 ; and the inequalities are reversed if $$(b-a)(\tau -\nu )\le 0.$$ ( b - a ) ( τ - ν ) ≤ 0 . In addition, we show a new Young-type inequality $$\begin{aligned} (1-v^{N+1}+v^{N+2})a+(1-v^{2})b\le v^{vN-(N+1)}a^{v}b^{1-v}+(\sqrt{a}-\sqrt{b} \ )^{2} \end{aligned}$$ ( 1 - v N + 1 + v N + 2 ) a + ( 1 - v 2 ) b ≤ v v N - ( N + 1 ) a v b 1 - v + ( a - b ) 2 for $$0\le v\le 1, N\in {\mathbb {N}}$$ 0 ≤ v ≤ 1 , N ∈ N and $$a,b>0.$$ a , b > 0 . Then we can get some related results about operators, Hilbert–Schmidt norms, determinants by these scalars results.

中文翻译：

---

年轻型不等式的一些结果

在本文中，我们的主要目标之一是提出 Alzer 等人对 Young-type 不等式的一些改进。(Linear Multilinear Algebra 63(3):622–635, 2015) 在某些条件下。也就是说：当 $$0< \nu , \tau <1,\ a,b>0$$ 0 < ν , τ < 1 , a , b > 0 ，我们有 $$\begin{aligned} \frac {a\nabla _{\nu }ba\sharp _{\nu }b}{a\nabla _{\tau }ba\sharp _{\tau }b}\le \frac{\nu (1-\nu )}{\tau (1-\tau )} \ \ { \mathrm {and}} \ \ \frac{(a\nabla _{\nu }b)^{2}-(a\sharp _{\nu } b)^{2}}{(a\nabla _{\tau }b)^{2}-(a\sharp _{\tau }b)^{2}}\le \frac{\nu (1 -\nu )}{\tau (1-\tau )} \end{aligned}$$ a ∇ ν b - a ♯ ν ba ∇ τ b - a ♯ τ b ≤ ν ( 1 - ν ) τ ( 1 - τ ) 和 ( a ∇ ν b ) 2 - ( a ♯ ν b ) 2 ( a ∇ τ b ) 2 - ( a ♯ τ b ) 2 ≤ ν ( 1 - ν ) τ ( 1 - τ ) 对于 $$( ba)(\tau -\nu )\ge 0;$$ ( b - a ) ( τ - ν ) ≥ 0 ; 如果 $$(ba)(\tau -\nu )\le 0.$$ ( b - a ) ( τ - ν ) ≤ 0 ，则不等式反转。此外，我们展示了一个新的 Young-type 不等式 $$\begin{aligned} (1-v^{N+1}+v^{N+2})a+(1-v^{2})b\le v^{vN-(N+1)}a^{v}b^{1-v}+(\sqrt{a}-\sqrt{b} \ )^{2} \end{aligned}$$ ( 1 - v N + 1 + v N + 2 ) a + ( 1 - v 2 ) b ≤ vv N - ( N + 1 ) avb 1 - v + ( a - b ) 2 $0\le v\le 1 , N\in {\mathbb {N}}$$ 0 ≤ v ≤ 1 , N ∈ N 和 $$a,b>0.$$ a , b > 0 。然后我们可以通过这些标量结果得到一些关于算子、Hilbert-Schmidt 范数、行列式的相关结果。我们展示了一个新的 Young-type 不等式 $$\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}$$ ( 1 - v N + 1 + v N + 2 ) a + ( 1 - v 2 ) b ≤ vv N - ( N + 1 ) avb 1 - v + ( a - b ) 2 对于 $$0\le v\le 1, N\在 {\mathbb {N}}$$ 0 ≤ v ≤ 1 ，N ∈ N 和 $$a,b>0.$$ a , b > 0 。然后我们可以通过这些标量结果得到一些关于算子、Hilbert-Schmidt 范数、行列式的相关结果。我们展示了一个新的 Young-type 不等式 $$\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}$$ ( 1 - v N + 1 + v N + 2 ) a + ( 1 - v 2 ) b ≤ vv N - ( N + 1 ) avb 1 - v + ( a - b ) 2 对于 $$0\le v\le 1, N\在 {\mathbb {N}}$$ 0 ≤ v ≤ 1 ，N ∈ N 和 $$a,b>0.$$ a , b > 0 。然后我们可以通过这些标量结果得到一些关于算子、Hilbert-Schmidt 范数、行列式的相关结果。