Affiliation:
1. College of Civil Engineering , Hunan University , 410082 , Changsha , China
2. Department of Mathematics , Hangzhou Normal University , 311121 , Hangzhou , China
3. Department of Mathematics , Huzhou University , 313000 , Huzhou , China
4. School of Mathematics and Statistics , Changsha University of Science & Technology , 410114 , Changsha , China
Abstract
Abstract
In the article, we present the best possible parameters α
1, β
1, α
2, β
2 ∈ ℝ and α
3, β
3 ∈ [1/2, 1] such that the double inequalities
α
1
C
(
a
,
b
)
+
(
1
−
α
1
)
A
(
a
,
b
)
<
T
3
(
a
,
b
)
<
β
1
C
(
a
,
b
)
+
(
1
−
β
1
)
A
(
a
,
b
)
,
α
2
C
(
a
,
b
)
+
(
1
−
α
2
)
Q
(
a
,
b
)
<
T
3
(
a
,
b
)
<
β
2
C
(
a
,
b
)
+
(
1
−
β
2
)
Q
(
a
,
b
)
,
C
(
α
3
;
a
,
b
)
<
T
3
(
a
,
b
)
<
C
(
β
3
;
a
,
b
)
$$\begin{array}{}
\begin{split}
\displaystyle
\alpha_{1}C(a, b)+(1-\alpha_{1})A(a, b) & \lt T_{3}(a, b) \lt \beta_{1}C(a, b)+(1-\beta_{1})A(a, b),
\\
\alpha_{2}C(a, b)+(1-\alpha_{2})Q(a, b) & \lt T_{3}(a, b) \lt \beta_{2}C(a, b)+(1-\beta_{2})Q(a, b),
\\
C(\alpha_{3}; a, b) & \lt T_{3}(a, b) \lt C(\beta_{3}; a, b)
\end{split}
\end{array}$$
hold for a, b > 0 with a ≠ b, and provide new bounds for the complete elliptic integral of the second kind, where A(a, b) = (a + b)/2 is the arithmetic mean,
Q
(
a
,
b
)
=
a
2
+
b
2
/
2
$\begin{array}{}
\displaystyle
Q(a, b)=\sqrt{\left(a^{2}+b^{2}\right)/2}
\end{array}$
is the quadratic mean, C(a, b) = (a
2 + b
2)/(a + b) is the contra-harmonic mean, C(p; a, b) = C[pa + (1 – p)b, pb + (1 – p)a] is the one-parameter contra-harmonic mean and
T
3
(
a
,
b
)
=
(
2
π
∫
0
π
/
2
a
3
cos
2
θ
+
b
3
sin
2
θ
d
θ
)
2
/
3
$\begin{array}{}
T_{3}(a,b)=\Big(\frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt{a^{3}\cos^{2}\theta+b^{3}\sin^{2}\theta}\text{d}\theta\Big)^{2/3}
\end{array}$
is the Toader mean of order 3.