For \(a,b>0\) with \(a\ne b\), the Toader mean of a and b is defined by

$$\begin{aligned} T\left( a,b\right) =\frac{2}{\pi }\int _{0}^{\pi /2}\sqrt{a^{2}\cos ^{2}t+b^{2}\sin ^{2}t}dt. \end{aligned}$$

In this paper, we prove that the double inequality

$$\begin{aligned} \left( \frac{3}{2}\left( \frac{a+b}{2}\right) ^{q}-\frac{1}{2}\left( \sqrt{ab }\right) ^{q}\right) ^{1/q}<T\left( a,b\right) <\left( \frac{3}{2}\left( \frac{a+b}{2}\right) ^{p}-\frac{1}{2}\left( \sqrt{ab}\right) ^{p}\right) ^{1/p} \end{aligned}$$

holds if and only if \(0<p\le 3/2\) and \(q\ge \ln \left( 3/2\right) /\ln \left( 4/\pi \right) \). This gives new sharp lower and upper bounds for the Toader mean, and improves several known results.

中文翻译：

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Toader均值在算术和几何均值方面的锐利边界

对于\（a，b> 0 \）与\（a \ ne b \），a和b的Toader均值定义为

$$ \ begin {aligned} T \ left（a，b \ right）= \ frac {2} {\ pi} \ int _ {0} ^ {\ pi / 2} \ sqrt {a ^ {2} \ cos ^ {2} t + b ^ {2} \ sin ^ {2} t} dt。\ end {aligned} $$

在本文中，我们证明了双重不等式

$$ \ begin {aligned} \ left（\ frac {3} {2} \ left（\ frac {a + b} {2} \ right）^ {q}-\ frac {1} {2} \ left（ \ sqrt {ab} \ right）^ {q} \ right）^ {1 / q} <T \ left（a，b \ right）<\ left（\ frac {3} {2} \ left（\ frac { a + b} {2} \ right）^ {p}-\ frac {1} {2} \ left（\ sqrt {ab} \ right）^ {p} \ right）^ {1 / p} \ end {已对齐} $$

仅当\（0 <p \ le 3/2 \）和\（q \ ge \ ln \ left（3/2 \ right）/ \ ln \ left（4 / \ pi \ right）\）时才成立。这为Toader平均值提供了新的尖锐上下限，并改善了一些已知结果。