ANSWERS TO THREE CONJECTURES ON CONVEXITY OF THREE FUNCTIONS INVOLVING COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND
In the article, we prove that the function x → ( 1 − x ) p K ( x ) is logarithmically concave on (0, 1) if and only if p ≥ 7/32, the function x → K ( x ) / log ( 1 + 4 / 1 − x ) is convex on (0, 1) and the function x → d 2 d x 2 [ K ( x ) − log ( 1 + 4 1 − x ) ] is absolutely monotonic on (0, 1), wh...
Saved in:
| Published in | Applicable analysis and discrete mathematics Vol. 14; no. 1; pp. 255 - 271 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
University of Belgrade, Serbia
01.04.2020
|
| Online Access | Get full text |
| ISSN | 1452-8630 2406-100X 2406-100X |
| DOI | 10.2298/AADM190924020W |
Cover
| Summary: | In the article, we prove that the function
x
→
(
1
−
x
)
p
K
(
x
)
is logarithmically concave on (0, 1) if and only if p ≥ 7/32, the function
x
→
K
(
x
)
/
log
(
1
+
4
/
1
−
x
)
is convex on (0, 1) and the function
x
→
d
2
d
x
2
[
K
(
x
)
−
log
(
1
+
4
1
−
x
)
]
is absolutely monotonic on (0, 1), where
K
(
x
)
=
∫
0
π
/
2
(
1
−
x
2
sin
2
t
)
−
1
/
2
d
t
is the complete elliptic integral of the first kind. |
|---|---|
| ISSN: | 1452-8630 2406-100X 2406-100X |
| DOI: | 10.2298/AADM190924020W |