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On Alzer's Inequality
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
223, 366]369 Ž1998.
AY985985
NOTE On Alzer’s Inequality N. Elezovic* ´ Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia
and J. Pecaric ˇ ´† Faculty of Textile Technology, Uni¨ ersity of Zagreb, Pierottije¨ a 6, 10000 Zagreb, Croatia Submitted by Ra¨ i P. Agarwal Received September 15, 1997
A generalization of Alzer’s inequality is proved. It is shown that this inequality is Q 1998 Academic Press satisfied for a large class of increasing convex sequences. Key Words: Alzer’s inequality; convex sequence.
Alzer w1x proved the following inequality: n
n
ž
F Ž n q 1.
nq1
Ý
1rr
nq1
i
r
is1
n
Ý is1
i
r
/
,
Ž 1.
where r is a positive real number and n is a natural number. In this paper we shall prove Ž1. in a more general context, for a sequence Ž a n . which satisfies an additional condition. In w2, 3x, an elementary proof for Ž1. is given. The proof in w3x is based on the following lemma: LEMMA 1.
If r is a positi¨ e real number, then
1 F Ž1 q w.
r
w q Ž1 y w.
*E-mail: [email protected]. † E-mail: [email protected]. 366 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
rq1
for 0 F w F 1.
Ž 2.
367
NOTE
We note that the proof of Lemma 1, given in w3x, can also be simplified. Indeed, for each positive x 1 , x 2 , r, p1 , and p 2 Ž p1 q p 2 s 1.,
Ž p1 x 1rq1 q p2 x 2rq1 .
1r Ž rq1 .
G p1 x 1 q p 2 x 2
Ž 3.
Žinequality between weighted means of order r q 1 and 1.. It is now sufficient to take in Ž3. p1 s wrŽ w q 1., p 2 s 1rŽ w q 1., x 1 s 1, and x 2 s 1 y w. Instead of Ž1. we shall prove an a nq 1
F
ž
a nq1 Ý nis1 a ir 1 r a n Ý nq is1 a i
1rr
/
,
Ž 4.
where r is a positive real number and Ž a n ., n G 1, is a sequence of positive real numbers. THEOREM 1. fies 1F
If the sequence Ž a n ., n G 1, of positi¨ e real numbers satis-
a nq 2
r
a nq2
ž / a nq 1
a nq1
y1q
an
rq1
ž /
n G 0, a0 s 0,
,
a nq1
Ž 5.
then Ž4. holds. Proof. The proof is by induction. For n s 1, Ž4. becomes a1r a2r
F
a2 a1r a1 Ž a1r q a2r .
,
which is equivalent to 1F
a2
r
ž /ž a1
a2 a1
y1 ,
/
and this is exactly Ž5. for n s 0. Let us prove the next step of induction. The inequality Ž4. is equivalent to nq1
Ý ks1
a kr G
rq1 a2nq 1 rq1 rq1 a nq 1 y an
,
i.e., nq2
Ý ks1
a kr
G
rq1 r rq1 r rq1 a2nq 1 q a nq2 a nq1 y a nq2 a n rq1 rq1 a nq 1 y an
.
368
NOTE
Therefore, it suffices to prove rq1 r rq1 r a2nq1 q a nq2 a nq1 y a nq2 a nrq1 rq1 rq1 a nq 1 y an
G
rq1 a2nq2 rq1 rq1 a nq2 y a nq1
.
Ž 6.
But Ž6. is equivalent to Ž5.. Let the sequence Ž a n . of positi¨ e real numbers satisfy
COROLLARY 1.
a2 a1
G
r
a1
q 1,
ž / a2
Ž 7.
a n y 2 a nq1 q a nq2 G 0,
n G 1.
Ž 8.
Then Ž4. holds. Proof. Equation Ž7. is equivalent to Ž5. for n s 0. Let us denote w s a nq 2ra nq1 y 1. Then w ) 0, since the convex sequence Ž a n . which satisfies a2 ) a1 must be increasing. If a nq2 F 2 a nq1 then w F 1 holds as well. But if a nq 2 G 2 a nq1 then we have a nq 2
r
a nq2
ž / a nq 1
G
a nq1 a nq 2
y1q r
ž / a nq 1
a nq2 a nq1
an
rq1
ž / a nq1
a nq2
y1 G
r
ž / a nq1
G 1.
So, we can suppose 0 - w F 1. But, then, Ž8. implies an a nq 1
G1y
ž
a nq2 a nq1
y1
/
and, hence, by Lemma 1, a nq 2
ž / a nq 1
G
r
a nq2 a nq1
y1q
a nq 2
r
ž / a nq 1
s Ž1 q w.
a nq2 a nq1
r
an
rq1
ž / a nq1
½ ž
y1q 1y
w q Ž1 y w.
rq1
a nq2 a nq1
rq1
y1
/5
G 1.
Thus, Ž a n . satisfies Ž5., and the corollary follows. COROLLARY 2. For each strictly increasing con¨ ex sequence Ž a n . of positi¨ e real numbers there exists an r ) 0 such that Ž4. holds.
369
NOTE
Proof. The statement follows since Ž7. is satisfied for sufficiently large r. EXAMPLE 1. The sequence a n s n satisfies Ž7. and Ž8.. Hence, Theorem 1 generalizes Alzer’s inequality. EXAMPLE 2. The sequence a n s 2 n y 1 satisfies Ž7. and Ž8.. Therefore, we have 2n y 1
1rr
r
Ž 2 n q 1 . Ý nis1 Ž 2 i y 1 . F r 2n q 1 Ž 2 n y 1 . Ý nq1 is1 Ž 2 i y 1 .
ž
/
.
EXAMPLE 3. The sequence a n s k Ž n y 1. q 1, k ) 0, satisfies Ž8.. Further, Ž7. is equivalent to r
k Ž k q 1 . G 1.
Ž 9.
Therefore, Ž4. holds for this sequence whenever Ž9. is valid. EXAMPLE 4. The sequence a n s a n, a ) 1, satisfies Ž8.. Further, Ž7. is equivalent to aG
1 ar
q 1.
Ž 10 .
As in the previous example, for each r ) 0 there exists a ) 1 for which Ž10. is valid.
REFERENCES 1. H. Alzer, On an inequality of H. Minc and L. Sathre, J. Math. Anal. Appl. 179 Ž1993., 396]402. 2. J. Sandor, On an inequality of Alzer, J. Math. Anal. Appl. 192 Ž1995., 1034]1035. ´ 3. J. S. Ume, An elementary proof of H. Alzer’s inequality, Math. Japon. 44Ž3. Ž1996., 521]522.
223, 366]369 Ž1998.
AY985985
NOTE On Alzer’s Inequality N. Elezovic* ´ Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia
and J. Pecaric ˇ ´† Faculty of Textile Technology, Uni¨ ersity of Zagreb, Pierottije¨ a 6, 10000 Zagreb, Croatia Submitted by Ra¨ i P. Agarwal Received September 15, 1997
A generalization of Alzer’s inequality is proved. It is shown that this inequality is Q 1998 Academic Press satisfied for a large class of increasing convex sequences. Key Words: Alzer’s inequality; convex sequence.
Alzer w1x proved the following inequality: n
n
ž
F Ž n q 1.
nq1
Ý
1rr
nq1
i
r
is1
n
Ý is1
i
r
/
,
Ž 1.
where r is a positive real number and n is a natural number. In this paper we shall prove Ž1. in a more general context, for a sequence Ž a n . which satisfies an additional condition. In w2, 3x, an elementary proof for Ž1. is given. The proof in w3x is based on the following lemma: LEMMA 1.
If r is a positi¨ e real number, then
1 F Ž1 q w.
r
w q Ž1 y w.
*E-mail: [email protected]. † E-mail: [email protected]. 366 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
rq1
for 0 F w F 1.
Ž 2.
367
NOTE
We note that the proof of Lemma 1, given in w3x, can also be simplified. Indeed, for each positive x 1 , x 2 , r, p1 , and p 2 Ž p1 q p 2 s 1.,
Ž p1 x 1rq1 q p2 x 2rq1 .
1r Ž rq1 .
G p1 x 1 q p 2 x 2
Ž 3.
Žinequality between weighted means of order r q 1 and 1.. It is now sufficient to take in Ž3. p1 s wrŽ w q 1., p 2 s 1rŽ w q 1., x 1 s 1, and x 2 s 1 y w. Instead of Ž1. we shall prove an a nq 1
F
ž
a nq1 Ý nis1 a ir 1 r a n Ý nq is1 a i
1rr
/
,
Ž 4.
where r is a positive real number and Ž a n ., n G 1, is a sequence of positive real numbers. THEOREM 1. fies 1F
If the sequence Ž a n ., n G 1, of positi¨ e real numbers satis-
a nq 2
r
a nq2
ž / a nq 1
a nq1
y1q
an
rq1
ž /
n G 0, a0 s 0,
,
a nq1
Ž 5.
then Ž4. holds. Proof. The proof is by induction. For n s 1, Ž4. becomes a1r a2r
F
a2 a1r a1 Ž a1r q a2r .
,
which is equivalent to 1F
a2
r
ž /ž a1
a2 a1
y1 ,
/
and this is exactly Ž5. for n s 0. Let us prove the next step of induction. The inequality Ž4. is equivalent to nq1
Ý ks1
a kr G
rq1 a2nq 1 rq1 rq1 a nq 1 y an
,
i.e., nq2
Ý ks1
a kr
G
rq1 r rq1 r rq1 a2nq 1 q a nq2 a nq1 y a nq2 a n rq1 rq1 a nq 1 y an
.
368
NOTE
Therefore, it suffices to prove rq1 r rq1 r a2nq1 q a nq2 a nq1 y a nq2 a nrq1 rq1 rq1 a nq 1 y an
G
rq1 a2nq2 rq1 rq1 a nq2 y a nq1
.
Ž 6.
But Ž6. is equivalent to Ž5.. Let the sequence Ž a n . of positi¨ e real numbers satisfy
COROLLARY 1.
a2 a1
G
r
a1
q 1,
ž / a2
Ž 7.
a n y 2 a nq1 q a nq2 G 0,
n G 1.
Ž 8.
Then Ž4. holds. Proof. Equation Ž7. is equivalent to Ž5. for n s 0. Let us denote w s a nq 2ra nq1 y 1. Then w ) 0, since the convex sequence Ž a n . which satisfies a2 ) a1 must be increasing. If a nq2 F 2 a nq1 then w F 1 holds as well. But if a nq 2 G 2 a nq1 then we have a nq 2
r
a nq2
ž / a nq 1
G
a nq1 a nq 2
y1q r
ž / a nq 1
a nq2 a nq1
an
rq1
ž / a nq1
a nq2
y1 G
r
ž / a nq1
G 1.
So, we can suppose 0 - w F 1. But, then, Ž8. implies an a nq 1
G1y
ž
a nq2 a nq1
y1
/
and, hence, by Lemma 1, a nq 2
ž / a nq 1
G
r
a nq2 a nq1
y1q
a nq 2
r
ž / a nq 1
s Ž1 q w.
a nq2 a nq1
r
an
rq1
ž / a nq1
½ ž
y1q 1y
w q Ž1 y w.
rq1
a nq2 a nq1
rq1
y1
/5
G 1.
Thus, Ž a n . satisfies Ž5., and the corollary follows. COROLLARY 2. For each strictly increasing con¨ ex sequence Ž a n . of positi¨ e real numbers there exists an r ) 0 such that Ž4. holds.
369
NOTE
Proof. The statement follows since Ž7. is satisfied for sufficiently large r. EXAMPLE 1. The sequence a n s n satisfies Ž7. and Ž8.. Hence, Theorem 1 generalizes Alzer’s inequality. EXAMPLE 2. The sequence a n s 2 n y 1 satisfies Ž7. and Ž8.. Therefore, we have 2n y 1
1rr
r
Ž 2 n q 1 . Ý nis1 Ž 2 i y 1 . F r 2n q 1 Ž 2 n y 1 . Ý nq1 is1 Ž 2 i y 1 .
ž
/
.
EXAMPLE 3. The sequence a n s k Ž n y 1. q 1, k ) 0, satisfies Ž8.. Further, Ž7. is equivalent to r
k Ž k q 1 . G 1.
Ž 9.
Therefore, Ž4. holds for this sequence whenever Ž9. is valid. EXAMPLE 4. The sequence a n s a n, a ) 1, satisfies Ž8.. Further, Ž7. is equivalent to aG
1 ar
q 1.
Ž 10 .
As in the previous example, for each r ) 0 there exists a ) 1 for which Ž10. is valid.
REFERENCES 1. H. Alzer, On an inequality of H. Minc and L. Sathre, J. Math. Anal. Appl. 179 Ž1993., 396]402. 2. J. Sandor, On an inequality of Alzer, J. Math. Anal. Appl. 192 Ž1995., 1034]1035. ´ 3. J. S. Ume, An elementary proof of H. Alzer’s inequality, Math. Japon. 44Ž3. Ž1996., 521]522.