Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas ( IF 1.8 ) Pub Date : 2021-03-22 , DOI: 10.1007/s13398-021-01031-5 Wei-Mao Qian , Miao-Kun Wang , Hui-Zuo Xu , Yu-Ming Chu
In this paper, we present the best possible parameters \(\alpha _{1}\), \(\alpha _{2}\), \(\alpha _{3}\), \(\alpha _{4}\), \(\beta _{1}\), \(\beta _{2}\), \(\beta _{3}\), \(\beta _{4} \in {\mathbb {R}}\) such that
$$\begin{aligned} \frac{\alpha _{1}}{H(x, y)}+\frac{1-\alpha _{1}}{L(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{1}}{H(x, y)}+\frac{1-\beta _{1}}{L(x, y)},\\ \frac{\alpha _{2}}{H(x, y)}+\frac{1-\alpha _{2}}{P(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{2}}{H(x, y)}+\frac{1-\beta _{2}}{P(x, y)},\\ \frac{\alpha _{3}}{H(x, y)}+\frac{1-\alpha _{3}}{N S(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{3}}{H(x, y)}+\frac{1-\beta _{3}}{N S(x, y)},\\ \frac{\alpha _{4}}{H(x, y)}+\frac{1-\alpha _{4}}{T(x, y)}&<\frac{1}{V(x, y)}<\frac{\beta _{4}}{H(x, y)}+\frac{1-\beta _{4}}{T(x, y)} \end{aligned}$$hold for all \(x, y>0\) with \(x \ne y\), where H(x, y), G(x, y), L(x, y), A(x, y), NS(x, y), P(x, y) and T(x, y) are respectively the harmonic, geometric, logarithmic, arithmetic, Neuman-Sándor, and first and second Seiffert means of two distinct positive numbers x and y, and
$$\begin{aligned} V(x,y)=\pi G^{2}(x, y) /\left[ 2\int _{0}^{\pi /2}\sqrt{A^{2}(x,y) \cos ^{2}\varphi +G^{2}(x,y)\sin ^{2}\varphi }d\varphi \right] \end{aligned}$$is a new Seiffert-like mean. As applications, some new inequalities for the complete elliptic integral of the second kind are given.
中文翻译：
第二个$$ \ hbox {Kind} $$ Kind的完整椭圆积分的近似值
在本文中，我们提出了可能的最佳参数\（\ alpha _ {1} \），\（\ alpha _ {2} \），\（\ alpha _ {3} \），\（\ alpha _ {4 } \），\（\ beta _ {1} \），\（\ beta _ {2} \），\（\ beta _ {3} \），\（\ beta _ {4} \ in {\ mathbb {R}} \）这样
$$ \ begin {aligned} \ frac {\ alpha _ {1}} {H（x，y）} + \ frac {1- \ alpha _ {1}} {L（x，y）}＆<\ frac {1} {V（x，y）} <\ frac {\ beta _ {1}} {H（x，y）} + \ frac {1- \ beta _ {1}} {L（x，y） }，\\ \ frac {\ alpha _ {2}} {H（x，y）} + \ frac {1- \ alpha _ {2}} {P（x，y）}＆<\ frac {1} {V（x，y）} <\ frac {\ beta _ {2}} {H（x，y）} + \ frac {1- \ beta _ {2}} {P（x，y）}，\ \ \ frac {\ alpha _ {3}} {H（x，y）} + \ frac {1- \ alpha _ {3}} {NS（x，y）}＆<\ frac {1} {V（ x，y）} <\ frac {\ beta _ {3}} {H（x，y）} + \ frac {1- \ beta _ {3}} {NS（x，y）}，\\ \ frac {\ alpha _ {4}} {H（x，y）} + \ frac {1- \ alpha _ {4}} {T（x，y）}＆<\ frac {1} {V（x，y ）} <\ frac {\ beta _ {4}} {H（x，y）} + \ frac {1- \ beta _ {4}} {T（x，y）} \ end {aligned} $$保持所有\（x，y> 0 \）与\（x \ ne y \），其中H（x， y），G（x， y），L（x， y），A（x， y） ，NS（x， y），P（x， y）和T（x， y）分别是两个不同的正数x和y的调和，几何，对数，算术，诺曼-桑多以及第一个和第二个Seiffert均值，以及
$$ \ begin {aligned} V（x，y）= \ pi G ^ {2}（x，y）/ \ left [2 \ int _ {0} ^ {\ pi / 2} \ sqrt {A ^ { 2}（x，y）\ cos ^ {2} \ varphi + G ^ {2}（x，y）\ sin ^ {2} \ varphi} d \ varphi \ right] \ end {aligned} $$是一种类似于塞弗特的新均值。作为应用，给出了第二种完全椭圆积分的一些新的不等式。
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