Inequalities and monotonicity of sequences involving n√(n + k)!/k!
This is the Published version of the following publication
Qi, Feng (1999) Inequalities and monotonicity of sequences involving n√(n + k)!/k! RGMIA research report collection, 2 (5).
The publisher’s official version can be found at
Note that access to this version may require subscription.
Downloaded from VU Research Repository https://vuir.vu.edu.au/17233/
LQHTXDOLWLHV DQG PRQRWRQLFLW\ RI VHTXHQFHV LQYROYLQJ sq
+q.n,$@n$
6,B "W
Ot|iBW| Nt?} 5|h*?}<t uLh4*@c uLh @** ?L??i}@|i ?|i}iht & @?_ ?@|h@*
?4Miht ? @?_ 6c i ThLi |@|
# ?n&
\
'&n
$*?/#?n6n&
\
'&n
$*E?n6
$
v ? n &
? n 6 n &
6hL4 |tc tL4i 4L?L|L?U|)hit*|t Lu ti^i?Uit ?L*?} sq
E? n &-*&- @hi _i_Ui_c @?_ |i hi*@|i_ ?i^@*|it @hi hi?i_
41 YA|iNaW|NA
Lq ^7`/ K1 Plqf dqg O1 Vdwkuh suryhg wkdw/ liu lv d srvlwlyh lqwhjhu dqg!+u, @ +u$,4@u/ wkhq
4? !+u.4,@!+u,?+u.4,@u=
+4,
Lq ^4/ 6`/ K1 Do}hu dqg M1 V1 Pduwlqv uhqhg wkh uljkw lqhtxdolw| lq +4, dqg vkrzhg wkdw/ liqlv d srvlwlyh lqwhjhu/ wkhq zh kdyh iru doo srvlwlyh uhdo qxpehuv u/
q q. 4?
#4 q
[q l@4
lu / 4
q. 4
q.4[
l@4
lu
$4@u
? q
sq$
q.4s
+q.4,$= +5,
Erwk erxqgv lq +5, duh wkh ehvw srvvleoh1
Ohwqdqgpeh qdwxudo qxpehuv/nd qrqqhjdwlyh lqwhjhu1 Wkh dxwkru jhqhudol}hg lq ^;` wkh ohiw lqhtxdolw| lq +5, dqg rewdlqhg
q.n q.p.n ?
#4 q
q.n[
lu / 4
q.p
q.p.n[
lu
$4@u
>
+6,
zkhuhulv d jlyhq srvlwlyh uhdo qxpehu1 Wkh orzhu erxqg lq +6, lv wkh ehvw srvvleoh1 E| pdwkhpdwlfdo lqgxfwlrq/ Q1 Hoh}rylf dqg M1 Shfdulf ^5` dqg wkh dxwkru ^9/ :`
ixuwkhu jhqhudol}hg wkh ohiw vlgh ri lqhtxdolwlhv lq +5, dqg wkh lqhtxdolw| +6, wr d odujh fodvv ri srvlwlyh/ lqfuhdvlqj dqg frqyh{ vhtxhqfhv dqg ri srvlwlyh/ lqfuhdvlqj dqg orjdulwkplfdoo| frqfdyh vhtxhqfhv lq glhuhqw gluhfwlrqv/ uhvshfwlyho|1
Uhfhqwo|/ wkh lqhtxdolwlhv lq +4, zhuh jhqhudol}hg e| wkh dxwkru dqg T10P1 Oxr lq ^43` dqg wkh iroorzlqj lqhtxdolwlhv zhuh rewdlqhg=
q.n
q.p.n ? q
s+q.n,$@n$
q.ps
+q.p.n,$@n$ ?4>
+7,
bb @|i4@|Ut 5MiU| *@ttU@|L? h4@h) 2S#D( 5iUL?_@h) 2SeH ki) Lh_t @?_ Th@tit W?i^@*|)c 4L?L|L?U|)c 5|h*?}<tuLh4*@c ti^i?Ui
Ai @|Lh @ttTTLh|i_ ? T@h| M) 56 Lu Oi?@? hL?Uic Ai iLT*i<t+iTM*U Lu ?@
AtT@Tih @t|)Titi| t?} }4wA,j
zkhuhqdqg pduh qdwxudo qxpehuv/nlv d qrqqhjdwlyh lqwhjhu1 Phdqzkloh/ vrph prqrwrqlflw| uhvxowv ri wkh vhtxhqfhv lqyroylqj sq
+q.n,$@n$ zhuh suhvhqwhg1 Lq wklv duwlfoh/ xvlqj lqhtxdolwlhv ghgxfhg iurp Vwluolqj*v irupxod/ zh jlyh d uh0 qhphqw ri wkh ohiw lqhtxdolw| lq +4, dqg wkh uljkw lqhtxdolw| lq +7,/ wkdw lv Wkhruhp 41 Ohwneh d qrqqhjdwlyh lqwhjhu/ qdqgp eh qdwxudo qxpehuv/ wkhq
# q.n
\
l
$4@q/#q.p.n
\
l
$4@+q.p,
u q.n q.p.n= +8,
Zkhqq@p@ 4/ wkh htxdolw| lq +8,lv ydolg1 Wkhruhp 51 Wkh vhtxhqfhv
# q.n
\
l
$4@q/ sq.n +9,
duh vwulfwo| lqfuhdvlqj zlwk qdqgn/ uhvshfwlyho|1 Wkh vhtxhqfhv
# q.n
\
l
$4@q/#q.p.n
\
l
$4@+q.p,
+:,
duh vwulfwo| lqfuhdvlqj lq niru doo jlyhq qdwxudo qxpehuvqdqgp1 51 wj66Bt
Lq rughu wr yhuli| rxu wkhruhpv/ zh qhhg vrph ohppdv1 Ohppd 41 Iru doo qdwxudo qxpehuvq/ zh kdyh
qoqqqoqq$
q.4
5
oqqq. 4>
+;,
s5qq h
q
h{s 4
45q. 4 ? q$?s 5qq
h q
h{s 4 45q= +<,
Iru ghwdlov derxw lqhtxdolwlhv +;, dqg +<,/ sohdvh uhihu wr ^8/ s1 4;7 dqg s1 4<7`1 Ohppd 51 Iru doo qdwxudo qxpehuvn dqgq A4/ zh kdyh
q+q4,
5 oq
4 . 4
q.n
.5n. 4 5 oq
4 .q
n
q. 45q4 45+q.n,+45n.4,= +43,
Surri1 Ohw
)E%c + ' %E% 3 2 *?
n
% n +
n2+ n 2 *?
n% +
3 2% 3
2E% n +E2+ n 3%c % D 2c + D
Glhuhqwldwlqj dqg vlpsoli|lqj surgxfhv Y)E%c +
Y% ' 2%3 2 *?
n %n+
n S+2n D+3 n ED n S+32+2%3f+%23H%
2E%n+2E%n+n Xvlqj wkh lqhtxdolw| oq+4 .w,5+4.w,w+5.w, iruw3 lq ^<`/ zh kdyh
C*+{> |,
C{ 5|4 . 5{9|{59{6 45+{.|,5+{.|.4, 3>
wkhuhiruh*+{> |, lv ghfuhdvlqj zlwk{51
Lw lv hdv| wr vhh wkdw
*+5> |, @ oq
4 . 4
|. 5
.5|. 4 5 oq
4 .5
|
56
45+|.5,+45|.4,5>
*+5>4, @ 5 oq 5 .4
5oq 6 56
79;5?3=
Ghqrwh
#+|, @*+5> |, . 56
45+|.5,+45|.4,>
+44, wkhq
#3+|, @ oq
4 .5
|
5|5. ;|. 6
|+|.5,+|.6,>
#33+|, @ 4;9|8|5
^|+|.5,+|.6,`5=
Wkxv #3+|, lv ghfuhdvlqj zlwk | 51 Vlqfh olp|$.4#3+|, @ 3/ wkhq#3+|, 3 iru
|5 dqg#+|, lqfuhdvhv iru |51
Dovr vlqfh olp|$.4#+|, @ 3/ zh jhw #+|, ? 3 iru | 5 dqg *+5> |, ? 3 iru
| 51 Khqfh wkh vhtxhqfh
*+5> n,.4
n@4 lv qhjdwlyh/ dqg vr lv wkh vhtxhqfh *+q> n,.4
q@5>n@41 Ohppd 5 iroorzv1
61 iNNut Nu CjNij6t Sursrvlwlrq 41 Ohwqeh d qdwxudo qxpehu/ wkhq
sq
q$
q.4s
+q.4,$
u q q. 4= +45,
Zkhqq@ 4/ wkh htxdolw| lq +45, krogv1 Surri1 Ohw
i+{, @{+{4, oq+4 .{,+{5{4, oq{5{. 5> {4=
E| vwdqgdug dujxphqwv/ zh jhw i3+{, @ +5{4, oq
4 .4 {
.{+{4,
4 .{ {5{4
{ 5
5{4
{ .{+{4,
4 .{ {5{4
{ 5
@ 4{ 4 .{ 3>
i+4, @ 3=
Wkhuhiruh/i+{, lv ghfuhdvlqj dqg i+{,3 iru{41 Dqg iruq4/ zh rewdlq q+q4, oq+4 .q, . 5+q5q4, oqq. 5q>
wklv lv htxlydohqw wr
q.4 5
oqqq. 4 q+q.4,
5 oqqq+q4,
5 oq+4 .q,=
Wkxv/ iurp wkh uljkw0kdqg vlgh ri lqhtxdolw| +;,/ zh rewdlq oqq$ q+q.4,
5 oqqq+q4,
5 oq+4 .q,>
wkdw lv/
q$ qq+q.4,@5 +q.4,q+q4,@5>
q$
+q.4,q qq+q.4,@5 +q.4,q+q.4,@5>
u q q. 4
+q$,q.4
^+q.4,$`q
4@q+q.4,
= Wkh surri lv frpsohwh1
Sursrvlwlrq 51 Ohwndqgq eh qdwxudo qxpehuv/ wkhq
# q.n
\
l
$4@q/#q.n.4
\
l
$4@+q.4,
u q.n q.n. 4= +46,
Li q@ 4/ wkh htxdolw| lq +46,lv ydolg iru doon41 Surri1 Iruq A4/ wkh lqhtxdolw| +43, pd| eh uhzulwwhq dv
q+q4,
5 oqq.n. 4
q.n .5n. 4
5 oqq.n
n q. 45q4
45+q.n,+45n.4,>
q+q4,
5 oq+q.n.4, .qq5. 5n. 4
5 oq+q.n,
n.4 5
oqn.q. 45q4 45+q.n,+45n.4,>
q.n.4 5
oq+q.n,
n.4 5
oqnq. 4
45+q.n, 4 45n. 4 q+q.4,
5 oq+q.n,q+q4,
5 oq+q.n.4,=
Wkhq/ vxevwlwxwlqj lqhtxdolwlhv lq +<, lqwr wkh qdo lqhtxdolw| deryh |lhogv oq+q.n,$oqn$q+q.4,
5 oq+q.n,q+q4,
5 oq+q.n.4,>
wklv lqhtxdolw| fdq eh uhduudqjhg dv +q.n,$
n$ +q.n,q+q.4,@5 +q.n.4,q+q4,@5>
q.n\
l /
+q.n.4,q
q.n q.n. 4
q+q.4,@5
>
u q.n q.n. 4
%#q.n
\
l
$q# q.n
\
l
$/#q.n.4
\
l
$q&4@q+q.4,
= Iurp wklv/ wkh lqhtxdolw| +46, iroorzv1
Surriv ri Wkhruhp 4 dqg Wkhruhp 51 Wkh lqhtxdolw| +8, fdq eh uhzulwwhq dv q.nT
[email protected]l 4@q sq.n
q.p.nT
4@+q.p, sq.p.n >
zklfk lv htxlydohqw wr
q.nT
[email protected]l 4@q sq.n
q.n.4T
4@+q.4, sq.n. 4 >
wkdw lv/
# q.n
\
l
$4@q/#q.n.4
\
l
$4@+q.4,
u q.n q.n. 4=
Wklv iroorzv iurp wkh frpelqdwlrq ri Sursrvlwlrq 4 dqg Sursrvlwlrq 51 Wkxv/ Wkhr0 uhp 4 dqg wkh prqrwrqlflw| ri wkh vhtxhqfhv
q.nT
4@q!s
q.nehlqj lqfuhdv0 lqj zlwkqduh suryhg1
Lw lv hdv| wr vhh wkdw q.n.4T
[email protected]l 4@q sq.n. 4 @
u q.n q.n. 4
q.n. 4 n. 4
4@q q.nT
[email protected]l 4@q sq.n = Ohw
j+{, @ 4
{4 5
oq+{.n.4,4
{oq+n.4, .4
5oq+{.n,> {4=
Xvlqj 5.w5w oq+4 .w, iruw A3 lq ^8/ ss1 5:6~5:7` dqg ^<` dqg glhuhqwldwlqj |lhogv j3+{, @ 4
{5
{+6{. 5n,
5+{.n,+{.n.4,.oq n. 4 {.n. 4
4
{5
{+6{. 5n,
5+{.n,+{.n.4, 5{
{.5+n.4,
@ 5{
5+{.n,+{.n.4,^{.5+n.4,`= Wkxv/j+{, lv ghfuhdvlqj zlwk{51 Vlqfh olp
{$.4j+{, @ 3 dqgj+4, @ 45oqn.5n.4 A3/
wkhq wkh vhtxhqfh
j+q,.4
q@4lv srvlwlyh1 Khqfh u q.n
q.n. 4
q.n. 4 n. 4
4@q A4>
q.n.4T
[email protected]l 4@q sq.n. 4 A
q.nT
[email protected]l 4@q
sq.n = Wkh vhtxhqfhv
q.nT
4@q!s
q.nehlqj lqfuhdvlqj zlwkn duh yhulhg1
Lw lv fohdu wkdw sq
+q.n.4,$@+n.4,$
q.ps
+q.p.n.4,$@+n.4,$
@
+q.n.4,q.p +n.4,p+q.p.n.4,q
4@q+q.p, sq
+q.n,$@n$
q.ps
+q.p.n,$@n$= Ohw
k+{, @ +{.p, oq+{.n.4,{oq+{.p.n.4,poq+n.4,> {3=
Gluhfw fdofxodwlrq ohdgv wr
k3+{, @ p+5{.p.n.4,
+{.n.4,+{.p.n.4,oq
4 . p
{.n. 4 p+5{.p.n.4,
+{.n.4,+{.p.n.4, p {.n. 4
@ p{
+{.n.4,+{.p.n.4, 3>
k+3, @ 3=
Vr/ wkh ixqfwlrqk+{,3 iru{3/ dqg zh kdyh +q.n.4,q.p
+n.4,p+q.p.n.4,q A4>
#q.n.4
\
$4@q/#q.p.n.4
\
$4@+q.p,
A
# q.n
\
l
$4@q/#q.p.n
\
l
$4@+q.p,
= Wkh surri ri Wkhruhp 5 lv frpsohwh1
-jujijAWjt
do O *3ihc ? @? ?i^@*|) Lu O ?U @?_ w 5@|hic a @| ?@* TT* .b Ebbc bSef2
d2o ,*i3LU @?_ a iU@hUc ? *3ih<t ?i^@*|)ca @| ?@* TT* 22 EbbHc SSSb do a 5 @h|?tc h|4i|U @?_ }iL4i|hU 4i@?tc @? @TT*U@|L?t |L wLhi?|3 ti^i?Ui tT@Uitc
@| @Uh b EbHHc 2H2HH
deo O ?U @?_ w 5@|hic 5L4i ?i^@*|it ?L*?} Eo-*oc hLU ,_?Mh} @| 5LU e EbSe%SDc eeS
dDo # 5 |h?LUc ?@*)|U W?i^@*|itc 5Th?}ihVih*@}c i vLh!%Oi_i*Mih}%ih*?c b.f dSo 6i?} "c ? @*}iMh@U ?i^@*|)c +BW +iti@hU +iTLh| L**iU|L? 2 Ebbbc ?L c ..2 d.o 6i?} "c 6h|ih }i?ih@*3@|L? Lu O*3ih<t ?i^@*|)cW?|ih? a @| : @| 5U 2D
E2fffc ? |i Thitt
dHo 6i?} "c Bi?ih@*3@|L? Lu O*3ih<t ?i^@*|)c a @| ?@* TT* Ebbbc ? |i Thitt dbo 6i?} " @?_ @L?} i?c L?L|L?U|it Lu |L ti^i?Uitc @|i4@|Ut @?_ W?uLh4@|Ut
"@h|ih*) Ebbbc ? |i Thitt
dfo 6i?} " @?_ "?} wLc Bi?ih@*3@|L? Lu O?U @?_ a5@|hi<t ?i^@*|)c A@4!@?}
aLh?@* Lu @|i4@|Ut E2fffc ?L 2c ? |i Thitt
do 6i?} "c Bi?ih@*3@|L?t Lu *3ih<t @?_ k@?}<t ?i^@*|)c tM4||i_ uLh TM*U@|L?
$jVBi|6jA| Nu B|j6B|Wte cBN5N YAt|||j Nu CjWAN,N}+e cBN5N |+e QjABA gFghhhe Cj jNV,j?t -jVO,W Nu AB
,4@* @__hittG Zqf={71zf[Q=